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Formal Logic

Pawel Garbacz

October 1, 2012

Contents

1 Want to know about logic 51.1 Do you want to argue with me? . . . . . . . . . . . . . . . . . . . 51.2 You shall not be inconsistent! . . . . . . . . . . . . . . . . . . . . 61.3 I want to be precise. . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Further reading/hearing/watching . . . . . . . . . . . . . . . . . 7

1.4.1 books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.2 Web pages . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Semiotics 92.1 Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 The good, the bad, and the ugly . . . . . . . . . . . . . . 102.1.4 Formal vs informal logic . . . . . . . . . . . . . . . . . . . 15

2.2 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 7000+ languages and the simple-minded logician . . . . . 172.2.2 The Liar and the logician . . . . . . . . . . . . . . . . . . 192.2.3 Sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.5 Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Propositional Logic 253.1 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Truth-tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Tableaux technique . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Axiomatic method . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Predicate Logic 474.1 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Tableaux technique . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.1 New rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.2 Some laws of predicate logic . . . . . . . . . . . . . . . . . 59

3

4 CONTENTS

4.4 Predicate logic at work . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Axiomatic method . . . . . . . . . . . . . . . . . . . . . . . . . . 654.6 “Truth-tables” for unary predicate logic . . . . . . . . . . . . . . 66

5 Definitions 775.1 Definition of definition and other funny stuff . . . . . . . . . . . . 775.2 Typology of definitions . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.1 Nominal and real definitions . . . . . . . . . . . . . . . . . 795.2.2 Definitions in normal form and other forms of definitions 80

5.3 Normal definitions and their soundness . . . . . . . . . . . . . . . 81

Chapter 1

Everything you alwayswanted to know about logicbut were afraid to ask

1.1 Do you want to argue with me?

We argue because:

• we have beliefs,

• our beliefs differ,

– sometimes my beliefs are inconsistent with yours

– sometimes I believe in something you do not believe,

– sometimes you believe in something I do not believe.

So argumentation is driven by the difference in beliefs. Of course this is onlyone of many drivers because we also argue to express our emotions, to gainthe political support, etc. Still, the logician usually restrict his attention to thedifference of beliefs.

One possibility that explains why this difference makes us argue assumes ourbenevolence:

• we believe that our beliefs are true,

• we believe that beliefs are important:

– true beliefs are beneficial for the believer,

– false beliefs are dangerous for the believer,

• we are benevolent so we want that everybody benefit and avoid danger,thus:

5

6 CHAPTER 1. WANT TO KNOW ABOUT LOGIC

– we argue in favour of those beliefs that we find true,

– we argue against those beliefs that we find false.

Another possibility assumes that we are malevolent.

• we are malevolent so we want that everybody suffer and encounter danger(as often as possible), thus:

– we argue in favour of those beliefs that we find false,

– we argue against those beliefs that we find true.

There exist also other strategies that explain why the difference in beliefs makesus argue.

The domain of argumentation is seen here sufficiently broad to contain thescientific proofs. In that case the aim of argumentation is not to persuadeother scientists that this and this is the case, but rather to show that a certainconclusion follows from the (previously accepted) premises or that the premisessupport the conclusion. We will focus mainly on proofs.

Logic is happy to help you with your argumentation because it aims

1. to describe the phenomenon of argumentation,

2. to identify its basic types, and

3. to define the most important properties of arguments, including those thatdecide whether the argument is to evaluated as good or bad,

4. to provide tools for finding good arguments.

The notion of argument does not cover however quarrels or heated debatessuch domestic rows.

1.2 You shall not be inconsistent!

If you believe that

1. God exists.

2. God does not exist.

then you have a problem with your beliefs irrespective of the fact whether Godexists or not. If he exists, then you second belief is false. If he does not exist,then your first belief is false. So, in either case you will turn out to be a loser.

Logic describes the above beliefs/sentences/propositions as inconsistent. Thisroughly means that they cannot be both true no matter what. Having incon-sistent beliefs is pointless, as we can be sure that the ”inconsistent” believer isalways wrong.

Sometimes the inconsistency of beliefs is not straightforward.

1.3. I WANT TO BE PRECISE. 7

1. Some men are not atheists.

2. All men are sinners.

3. Every sinner is an atheist.

Sometimes the beliefs that we find absurd are not inconsistent (from the pointof view of logic):

1. Smoking is bad for your health.

2. Smoking does not hurt my health.

Logic is happy to help you with inconsistency because it aims

1. to define what it means that certain beliefs are (in)consistent,

2. to provide tools for checking whether certain beliefs are (in)consistent.

1.3 I want to be precise.

Sometimes you want to be precise. This may happen when you want to be fullyand effectively understood, e.g. when you act as a surgeon during a complexoperation. On other occasions you want to be imprecise. This may happen whenyou want to hide some information, e.g. when you are being interrogated at theGuantanamo Bay Detention Camp.

In both cases it is good to know how to formulate your beliefs in the preciseway.

Logic is happy to help you with precision because it aims

1. to define what it means the certain forms of speech are precise and othersnot

2. to identify and precisify the imprecise expressions,

3. to assist you in using the precise language of logic, which is defined insemiotics.

1.4 Further reading/hearing/watching

1.4.1 books

1. M Zegarelli, Logic For Dummies, Wiley Publishing 2006

2. W. Hodges, Logic, Penguin Books, 2001

3. A. Tarski, Introduction to Logic and to the Methodology of Deductive Sci-ence, Dover Publications 1995

8 CHAPTER 1. WANT TO KNOW ABOUT LOGIC

books on-line

1. www.fecundity.com/logic/download.html

1.4.2 Web pages

1. www.logicmatters.net/

2. www.science.uva.nl/~seop/entries/logic-classical/

3. www.olemiss.edu/courses/logic/logic2.html

Podcasts

1. http://itunes.apple.com/podcast/logic-audio-w-.pdfs/id207699294

YouTube’s channels

1. http://youtube.com/user/PhilosophyFreak

2. http://youtube.com/user/drjasonjcampbell

Chapter 2

Men are from Mars, womenare from Venus, andlogicians are from Semiotics

2.1 Argumentation

2.1.1 Examples

1. Let us reflect in another way, and we shall see that there is great reasonto hope that death is a good, for one of two things: – either death is astate of nothingness and utter unconsciousness, or, as men say, there isa change and migration of the soul from this world to another. Now ifyou suppose that there is no consciousness, but a sleep like the sleep ofhim who is undisturbed even by the sight of dreams, death will be anunspeakable gain. For if a person were to select the night in which hissleep was undisturbed even by dreams, and were to compare with thisthe other days and nights of his life, and then were to tell us how manydays and nights he had passed in the course of his life better and morepleasantly than this one, I think that any man, I will not say a privateman, but even the great king, will not find many such days or nights, whencompared with the others. Now if death is like this, I say that to die isgain; for eternity is then only a single night. But if death is the journey toanother place, and there, as men say, all the dead are, what good, O myfriends and judges, can be greater than this? If indeed when the pilgrimarrives in the world below, he is delivered from the professors of justice inthis world, and finds the true judges who are said to give judgment there,Minos and Rhadamanthus and Aeacus and Triptolemus, and other sonsof God who were righteous in their own life, that pilgrimage will be worthmaking. What would not a man give if he might converse with Orpheusand Musaeus and Hesiod and Homer? Nay, if this be true, let me die again

9

10 CHAPTER 2. SEMIOTICS

and again. (Plato, Apology)

2. All puddings are nice and this dish is a pudding. However, no nice thingsare wholesome. So it seems that this dish is not wholesome.

3. Everything has its cause. Thus, there is a cause for everything.

2.1.2 Structure

The logicians are simple-minded creatures and they like simplify things. Whena logician sees an argument, then he tries to find its premises and its conclusion.(When he fails, he claims that he is not interested in such formless arguments).The premises of an argument are ideas/beliefs/sentences from which the argu-mentation start or which the argumentation takes for granted. The conclusionis the idea/belief/sentence at which the argumentation rests. It should be ob-vious the premises are somehow related to the conclusion. One can say thatthe premises support the conclusion or rather that the premises are intended tosupport the conclusion.

Consider the second argument above. The logician distinguishes its twopremises and the conclusion in the following way:

All puddings are nice and this dish is a pudding.

However, no nice things are wholesome.

————————————————————–

This dish is not wholesome.

(2.1)

For the sake of simplicity, the logician assumes that all arguments consist ofbeliefs that are expressed in sentences. So, he considers an argument as a pair< P,C >, where P is a set of premises (i.e. of sentences) and C is the conclusion(i.e. a sentence).

For example, the argument 2.1 is represented as < P1, C1 >, where

• P1 = {All puddings are nice and this dish is a pudding,However, no nice things are wholesome},

• C1 = This dish is not wholesome.

2.1.3 The good, the bad, and the ugly

Some arguments are good, others are bad, and still others are simply ugly.Argumentation is a complex phenomenon and may be described from a num-

ber of viewpoints. For example, one may evaluate whether the following argu-ment:

• is persuasive or not,

• is funny or boring,

2.1. ARGUMENTATION 11

• helps the arguer to express his emotions

• etc.

The logicians evaluate the argument from their own peculiar perspective.

Definition 1. An argument is valid when it is not possible that

1. all its premises are true,

2. its conclusion is false.

Consider argument 2.1. Is it a valid argument? Assume that it is not.According to definition 1, this means that it is possible that the premisses of2.1 are true and its conclusion is false. So it is possible that:

1. This dish is a pudding.

2. All puddings are nice.

3. No nice things are wholesome.

4. This dish is wholesome.

4 is the negation of 2.1’s conclusion.Now, it follows from 1 and 2 that this dish is nice. So by 3 this dish is not

wholesome, but this is inconsistent with 4. We arrived at the inconsistency -naughty, naughty, naughty! This means that we reasoned very badly, but whenwe review our reasoning, there seems to be nothing wrong with it except perhapsfor the starting point when we assumed that 2.1 is not a valid argument. Thislooks like a source of all our problems - we scorned 2.1 as invalid and this ledus into inconsistency. This means that our initial assumption was wrong - 2.1is a valid argument.

Consider another argument:

It is false that all puddings are nice.

————————————————————–

No pudding is nice.

(2.2)

Is this a valid argument? Assume that it is not. According to definition 1, thismeans that it is possible that the premise of 3.4 is true and its conclusion isfalse. So it is possible that:

1. It is false that all puddings are nice.

2. It is false that no pudding is nice.

Is this possible? Imagine that John always cook nice puddings and Ann alwayscooks awful puddings. Then because of Ann’s awful puddings 1 is true andbecause of John’s nice puddings 2 is true as well. No inconsistency occurs,which means that our imaginary situation is possible. Therefore, 3.4 is not avalid argument.


Definition 2. An argument is sound when it is valid and all of its premisesare true.

This second logical property of arguments is outside the reach of the mightyhands of our logician. He is in a position to say when an argument is valid, butin most cases it is beyond his competence to find whether a given premise is trueor not. It is sad to admit but the logician is not well-prepared to say whether allpuddings are nice. Alas logic is not about puddings. Nevertheless, the categoryof soundness is an important factor of good argumentation. In the logician’seyes when you provide unsound arguments, you do not prove anything.

Definition 3. An argument begs the question when one of its premises isimplicitly or explicitly included in its conclusion.

As opposed to the other two definitions above, definition 3 specifies theclass of “ugly” arguments. Moreover, again as opposed to the other definitions,actually it does not match up to the high logical standards because of theunclarity of the relation of inclusion.

Consider the following ”argument”.

God exists.

—————-

God exists.

(2.3)

Clearly, there is something wrong with this. It resembles one of Martin Scors-ese’s characters, Jimmy Two Times, who always repeated each phrase two times.What is good for Jimmy may not be good for the argumentation. 3.5 does notprove anything because the premise does not support its conclusion in any sense.When you do not believe the conclusion of 3.5, you will not believe its premiseso there is no point in arguing along these lines. You will not persuade theatheist with 3.5 and the theist will not be impressed although he acknowledgesboth the premise and the conclusion. It might seem funny, but it can be shownthat 3.5 is a valid argument.

Sometimes this fallacy is more complex. Consider the following conversation:

John: I believe that God exists.

Paul: Why?

John: Because Bible says so.

Paul: So what?

John: Bible always tells the truth.

Paul: How do you know that?

John: Because God said that it does.


Paul: But how can you know that God exists?

Paul: Because Bible say so.

In this conversation John tries to persuade Paul that God exists using a circularargumentation that aims to establish that God exists on the basis of the premisethat Bible say that God exists. The problem with this argumentation is thatthis premise is justified by this very premise. If Paul does not believe that Bibleis trustworthy, he will not be persuaded by this kind of circularity. And it israther improbable that Paul believes in Bible despite the fact that he is anatheist. In sum, logic distinguishes those sound arguments that do not beg thequestion.

In what follows we will focus only on validity.Definition 1 say that in a valid argument it is not possible that true premises

support a false conclusion. But what does it mean? For example, is the followingargument valid?

My car runs on biodiesel.

Your car runs on petrol.

——————————————————————–

My car produces less carbon dioxide than your car.

(2.4)

Surely, it is actually the case that the premises of 3.6 are true and its conclusionis true as well. But couldn’t it happen that the premises are true and theconclusion is false, i.e. are there any circumstances under which the premisesare true, but the conclusion is false? From what we currently know aboutnature we can infer that this is not possible in our world because burningbiodiesel produces much less carbon dioxide than burning petrol. So it seemsthat given the laws of nature, it is not possible that the premises of 3.6 aretrue and the conclusion is false. The logician is not however satisfied withthis answer. Namely, we can imagine that the chemical structures of biofuelsand/or fossil fuels were slightly different from what they are now. This differencewould imply, among other things, that burning biodiesel produces more carbondioxide than burning petrol. It might happen if the historic processes that ledto fossil fuels were different from the actual ones. So it is imaginable that thepremises of 3.6 are true and the conclusion is false. This seems to imply thatit is possible that the premises of 3.6 are true and the conclusion is false. Thisshows us that what we mean by “it is possible” heavily depends on the notion ofpossibility we use and that there is more than one reasonable construal of thisnotion. Furthermore this analysis also suggests that the notion of possibilityis related to a set of laws. Given the currently known set of chemical laws,3.6 is valid, but if we start with a different selection of laws, we might get adifferent understanding of what is possible and, consequently, a different set ofvalid arguments.

It is needless to say that definition 1 above employs the logical notion ofpossibility that is related to the set of logical laws. Sometimes, the logicians


say that they are interested only in the logical type of validity, which definedby the logical possibility, which is determined by the set of logical laws, whichare determined by logic itself.

Definition 4. An argument is logically valid when it is not logically possiblethat

1. all its premises are true,

2. its conclusion is false.

Definition 5. It is logically possible that φ and ψ if and only if (iff, for short)φ is consistent with ψ.

In order to check whether a certain argument is (logically) valid we will usethe equivalent definition:

Definition 6. An argument < {φ1, φ2, . . . , φn}, ψ > is logically valid when theimplication “if φ1 and φ2 and . . . and φn, then ψ” is a substitution of a logicallaw.

In the following chapters we will learn how to recognise logical laws (or lawsof logic). Now it should suffice to say that any sentential expression (includ-ing the so-called sentential forms) that is true and is composed only of logicalsymbols and variables is a logical law.

Consider the following example.

If Adam speaks English, then he is able to read the International Herald Tribune.

Adam is not able to read the International Herald Tribune.

————————————————————————————————-

Adam does not speak English.(2.5)

This argument is logically valid. To see this, do the following:

1. put ♣ in place of the sentence “Adam speaks English”,

2. put ♠ in place of the sentence “Adam is able to read the InternationalHerald Tribune.”.

Then from 3.7 you will get 2.6:

If ♣, then ♠.

Not ♠.

—————-

Not ♣.

(2.6)

2.6 is a propositional schema (or form) of 3.7 because you can get 2.6 from 3.7 bythe “reverse” substitution. The logician thinks that 2.6 shows the structure of


3.7. Note that 2.6 consists of three sentences. Now, out of these three sentenceswe create one (complex) sentence:

If (If ♣, then ♠) and Not ♠, then Not ♠. (2.7)

I put the brackets for fun and for clarity.According to definition 6, 3.7 is logically valid if 2.7 is a logical law. Althoughthe logicians are rather odd, they do not use such odd symbols as ♣ or ♠. InLogicspeak 2.7 will look as below:

(p→ q) ∧ ¬q → ¬p. (2.8)

As you can suspect,

1. “p” plays the role of “♣”,

2. “q” plays the role of “♠”,

3. “¬” plays the role of “not”,

4. “∧” plays the role of “and”,

5. “→” (sort of) plays the role of “if . . . , then . . . ”.

In a few months we will be able to prove that 2.8 is a logical law, so 3.7 islogically valid.

It is worth to note that the honour of logical validity belongs not only to3.7, but also to every argument that can be obtained from 2.6. This shows usthe importance of schemas. In fact, the logicians crave rather for the schemasof valid arguments then for the valid arguments themselves.

Since it is believed that schemas capture the form of an argument, the branchof logic that deals with validity is called “formal logic”. We might thus expectthat there also exists informal logic.

2.1.4 Formal vs informal logic

Informal logic aims to evaluate arguments with respect rather to their contentthan to their form. The most easy to read introductions to informal logic lista number of informally “bad” argument, which are usually called fallacies. Itneeds to be remembered that some of these fallacies may under certain condi-tions be legitimate ways of argumentation.

Informal fallacies - sample

Argumentum ad baculum This fallacy occurs when one appeals to thethreat of force to bring about the acceptance of a conclusion.

Example:


• On October 10, 1971, Secretary of State William P. Rogers cautionedforeign ministers that Congress might force the United States reduce itsfinancial contributions to the United Nations if Nationalist China is ex-pelled.

In fact, in most cases this kind of arguments is not an argument at all becausewhat it brings about is a change in someone’s behaviour and emotions and notin his or her beliefs.

Argumentum ad populum This fallacy occurs when one arouses the feelingand enthusiasms of the multitude in order to popular assent to a conclusion.

Example:

• ”But officer, I don’t deserve a ticket; everyone goes this speed.”

This kind of arguments is vicious because popular assent based on emotionsrarely yields true beliefs.

Argumentum ad ignorantiam This fallacy occurs when one proves his orher thesis on the basis that the thesis has not been disproved. This fallacy hastwo forms:

That p is not proved.

———————————–

Not p.

(2.9)

That not- p is not proved.

———————————–

p.

(2.10)

Examples:

• Ghosts exist because none has ever proved that they do not exist.

• Since the class has no questions concerning the topics discussed in class,the class is ready for a test.

This kind of arguments is vicious because

1. for a number of (scientifically) important truths, there were prolongedtime intervals when no one was able to prove them,

2. it has been proved that a number of truths are unprovable,

2.2. LANGUAGE 17

2.2 Language

2.2.1 7000+ languages and the simple-minded logician

The domain of languages flourished since the Tower of Babel was built. It isestimated that there are more than 7000 natural languages around the world.Besides their numerous dialects, there is a substantial number of artificial lan-guages, e.g. programming languages for computers. This phenomenon is com-plex also from the qualitative point of view. Not only do we have differentalphabets, but also the human race uses different systems of notation.

This implies that the adequate definition of language is far beyond our un-derstanding. But the logician is eager to evaluate arguments, so he needs tohave a rough and ready definition. Since he is fairly simple-minded, he insiston describing any language by means of a couple.

Definition 7. A language is (represented as) a pair < X,R >, where

1. X is a set of (atomic) symbols,

2. R is a set of rules on compositions of complex symbols out of other symbols.

This definition defines a language by expanding the initial set of atomicsymbols by means of (quasi-grammatical) rules.

An atomic symbol is a symbol such that no part of this symbol is itself asymbol. The rules are contained in a language define how one should createmore complex phrases from the less complex ones.

Consider the following example. Let an FX language be defined as below:

1. XFX is a set that contains four sets:

(a) ISO currency symbols, i.e. “USD”, “GBP”, “PLN”, etc.,

(b) all rational numbers between 0 and 1 (including 0 and 1) that canbe represented by up to 5 digits (and the dot), e.g., 0.1234, 0.0099,0.12, 1, 0.

(c) the singleton, i.e. the set that contain only one member, of “=”(read: is equivalent to),

(d) the singleton of “ ” (blank).

2. RFX contains five rules:

(a) all elements of XFX belong to the FX language,

(b) if α is a currency symbol and β is a 5-digit rational number, then“α β” (note the blank inside!) is a composite currency symbol thatbelongs to the FX language,

i. there are no other composite currency symbols besides those de-fined in 2b,


(c) if α and is a currency symbol and β is a composite currency symbol,then “1 α=β” (note the blank after “1”!) is exchange rate expressionthat belongs to the FX language,

(d) no other element belongs to the FX language.

Then the following objects belong to the FX language:

• USD,

• 0.23 GBP,

• 1 EUR=3.3625 PLN.

But all of the following objects do NOT belong to the FX language:

• AC,

• GBP USD,

• 2 EUR=6.7250 PLN.

The logician insists that all languages be divided into two categories: (object)languages and meta-languages. A meta-languages is a languages that speaksabout a language. An (object) language is a language that is not a meta-language.

Most natural languages are mixtures of expressions from (object) languagesand meta-languages. Consider for example English. It contains both the meta-linguistic terms, e.g. ”word”, ”sentences”, ”The word ”John” has four letters”,”The word ”letter” has six letters”, and the (object) linguistic terms, e.g. ”run”,”John runs”, ”John has two legs”.

The logician likes to use the quotation marks to create meta-expressions forfree. Take the word ”John”. This object belongs to an (object) language sinceit speaks about a being that is not itself part of any language. Then the logicianapplies his quotation mark and he gets ” ”John” ”. In the logician’s eyes, thelatter word speaks about the (word) ”John”, so it is a meta-linguistic term. Thequotation marks in the hands of the smart logician is a really magic tool. From”John” you can create not only a meta-linguistic term ” ”John” ”, but alsoa meta-meta-linguistic term ” ” ”John” ” ”. The latter phrase is meta-meta-linguistic since it speaks about the meta-linguistic term ” ”John” ”, which speaksabout the (object)-linguistic term ”John”. This shows us that the distinctionlanguage/meta-language is relative. Language L1 can be a meta-language of alanguage L0, but there can exist a language L2 that is a meta-language of L1.So L1 is both an (object) language to L2 and a meta-language to L0.

The distinction between (object) languages and meta-languages is importantbecause of the so-called semantic antinomies.

2.2. LANGUAGE 19

2.2.2 The Liar and the logician

Assume that you meet someone that tells you ”I am lying”. Is he telling youthe truth? Assume that he is. This means that it is true that he is lying, so heis not telling you the truth. Assume now that he is not telling you the truth.This means that it is false that he is lying, so he is telling the truth. Naughty,naughty, naughty! Again we got a contradiction.

This piece of reasoning is known from the ancient times as the Liar Paradox.There are many version thereof, some of which do not resort to the complexnotion of lying. 1 Consider the following sentence:

The sentence 2.11 is not true (2.11)

In a similar way it can be shown that this sentence is both true and is not true.Naughty, naughty, naughty!

What shall we do?The logician’s solution stipulates that 2.11 (and any expression similar to it)

is meaningless. It is meaningless to say that two plus two equals John. Similarly,the logician says that it is meaningless to say that this sentence is not true. Thereason is that 2.11 confuses an (object) language and its meta-language. Theterm ”is true” is a meta-linguistic term to any term it is applied. And 2.11applies it to itself. So the logician advises anyone (who is willing to listen) thatwe should pay the special attention when we use a mixed language in which its(object) linguistic part is interspersed among its meta-linguistic part. And thesafest rule is to avoid such languages altogether. The logician is proud that anylanguage he produces is free from such gobbledygook. 2.11 is an example of aclass of similar paradoxes called semantic antinomies.

Although the use of the quotation marks seems is a useful tool, after ex-plaining all the subtleties involved the logicians like to avoid its overuse. Wewill follow this tradition and will drop it unless this does not lead into ambiguity.

2.2.3 Sentences

Under normal circumstances normal people express their beliefs by means ofsentences. When I believe that Warsaw is a capital of Russia and when I wantto

• record this belief,

• share this belief with others,

• make all Russians laugh,

• etc.,

1In fact, when you lie, you do not tell what you believe as true. Then it might happenthat you tell the truth while lying. For example, assume that you believe that there is thegreatest prime number. Then assume that you decided to lie (to one of your enemies) thatthere is no greatest prime number. Then, by you ignorance, you are telling the truth!


I can either write down or utter the sentence ”Warsaw is a capital of Russia.”.

Definition 8. A sentence is any (linguistic) expression that is either true orfalse.

Examples of sentences:

• God exists,

• 2+2=4,

• If John is a politician, he always lies.

• Mercury is the biggest planet in the Solar system.

• Every philosopher knows some mathematician.

• . . .

We also assume the principle of bivalence.

Assumption 1 (Principle of bivalence). Every sentence is either true or falseand no sentence is both true and false.

According to definition 8 all and only all expressions bearing a truth valueare sentences. Thereby we exclude questions and requests from the scope of thenotion of sentence, so only the so called declarative sentences are sentences in thesense of definition 8. Moreover, we also presuppose that either the content of thesentence or the context of its utterance remove all traces of semantic ambiguity.Assume that John says: It is raining. This utterance of John qualifies as asentence only if its context makes it clear when and where it is raining. Thedefault assumption is, of course, that it is raining when John utters this sentenceand where John stands. Note that such expressions as 2.11 are not sentences inthe sense of definition 8. Since the logician is more interested in the forms ofarguments than in the arguments themselves, he wants to be able to speak aboutforms of sentences. To this end, he invented the notion of sentential frame:

Definition 9. A sentential frame is any expression that satisfies two conditions:

1. it contains free variables,

2. it becomes a sentence when all free variables are substituted with constants.

This definition contains three notions that might be unfamiliar to the averagetaxpayer:

1. variable

2. free variable

3. constant

2.2. LANGUAGE 21

4. substitution of a variable with a constant

Take the simple equation 1 + 1 = 2 ∗ 1. All its parts, i.e.: 1, +, =, 2, and *are constants, i.e. they are not variables. This equation instantiates a simplerelationship between addition and multiplication. There are of course otherinstantiation of this relationship:

• 2 + 2 = 2 ∗ 2,

• 3 + 3 = 2 ∗ 3,

• 4 + 4 = 2 ∗ 4,

• . . . .

One of the key features of human beings is their ability to generalise. An averageadult will soon get bored writing down such platitudes as the ones above. Thenhe might come to the following conclusion: If you take any number and add itto itself, the result will be equal to the multiplication of 2 and this number. Ifhe is sufficiently educated, he may even know how to express this generalisationin an elegant way:

x+ x = 2 ∗ x. (2.12)

It is the latter expression that contain variables and moreover all of those vari-ables are free! In fact it contain just one variable x. When we say that this isa variable, then mean that anyone can substitute something for it. So out of2.12 you can get all of the above equations. In a sense this expression containsthem in a compressed/zipped form. Of course by definition 11 equation 2.12 isa sentential frame.

What about 2.13?For all x, x+ x = 2 ∗ x (2.13)

First, note that this utterance is true, so it follows from definition 8 that itis a sentence.

Secondly, note that it contains the same variable as the previous expression.However, this time this variable is not free. The idea of variable freedom isbound to the notion of substitution: a variable is free if it makes sense tosubstitute something for it. For the mathematician it makes little sense to saythat for all 1, 1 + 1 = 2. Consider a different example of sentential frames: ”xis a terrorist”. Obviously, it makes sense to say that G. W. Bush is a terrorist,even if this claim is false. But it is meaningless to say that for all G. W. Bush, G.W. Bush is a terrorist. The logical explanation of the difference between thesetwo cases is that the variable x is free in a sentential frame ”x is a terrorist”and is not free (i.e. is bound) in the sentence ”for all x, x is a terrorist”. Insum:

• 2.12 is a sentential frame

– only x is a variable


– x is free in 2.12

• 2.13 is a sentence

– only x is a variable

– x is not free in 2.13

2.2.4 Names

All sentences (in the sense of definition 8) have parts. For example, the sentence“John is a liar” contains the following words:

• “John”,

• “is”,

• “a”

• “liar”.

Some of these expressions refer to the real world objects. Among those referringwords the logician selects names. There are various definitions of the notion ofname. Here we will use the following one:

Definition 10. A name is any (linguistic) expression can play the role of subjectin the sentence of the form “(A/an/the) A is (a/an/the) B”.

Examples of names:

• “John”,

• “liar”,

• “Clinton’s wife”,

• “national debt”.

As in the case of sentences, we will be also interested in nominal frames:

Definition 11. A nominal frame is any expression that satisfies two conditions:

1. it contains free variables,

2. it becomes a name when all free variables are substituted with constants.

Every names names. Sometimes, it may happen that the objects namedby a name does not exist. For example, “the tenth symphony of Ludwig vanBeethoven” does not name any object because Ludwig van Beethoven composedonly nine symphonies. If a name names an object that does not exist, we willsay that it is an empty name. Other names name exactly one object. Forexample, “first president of the USA” name exactly one object, namely GeorgeWashington. If a name names exactly one object, we will say that it is a

2.2. LANGUAGE 23

singular name. Consequently, all other names, i.e. those that are neither emptynor singular, name multiple objects. We will name these names general names.Note that “name” is a general name because there are many names. And “emptyname” is not an empty name because there are many empty names.

Sometimes the set of all objects named by a name is called the extension ordenotation of this name.

2.2.5 Connectives

Complex phrases are composed out of simpler phrases. However, without somekind of linguistic glue the latter would not stick together to compose the former.Consider the following sentence:

John is lazy. (2.14)

This sentence contains two names glued together by the verb “is”.

John︸︷︷︸name

is︸︷︷︸connective

lazy︸︷︷︸name

(2.15)

The verb is here a connective.There is a huge variety of connectives. Some glue together names. Other

glue together sentences. Still others glue names with sentences. There are evensuch connectives that join other connectives. In order to put some order into thisconceptual jungle, we describe connectives by means of indexes. For example,the index for “is” is s

n,n . Assume that a connective γ glues together expressionsφ1, φ2, . . . , φn and forms an expression ψ. Let the index of ψ be x and let theindices of φ1, φ2, . . . , φn be, respectively, y1, y2, . . . yn. Then the index of theconnective at stake is x

y1,y2,...yn.

Consider now the following sentence:

If John is lazy, then he is not rich. (2.16)

First, observe that 2.16 is built up from two sentences:

If John is lazy︸︷︷︸sentence

, then he is not rich︸︷︷︸sentence

. (2.17)

So what is left is a connective of the index ss,s . We analysed the first sentence

above, so let us focus on the second sentence. Again, note that this sentencemight be construed as containing another sentence:

Not: he is rich︸︷︷︸sentence

(2.18)

So what is left is a connective with the index ss .

We learned three types of connectives: sn,n , s

s,s , and ss . For the time being,

we will be interested in the so called propositional or sentential connectives, i.e,


the ones that compose sentences out of sentences. So for the time being we willforget about such phrases as “is”.

The class of all propositional connectives contains an interesting subclass.Consider the connective “not”, which is called negation. This connective hasthe following property:

• If you insert it into a true sentence, then the sentence will become false.

• If you insert it into a false sentence, then the sentence will become true.

The important thing is that these two claims hold for any sentence whatsoever.We will call such connectives truth-functional.

Definition 12. A propositional connective is truth-functional if the truth valueof any sentence this connective composes depends solely on the truth values ofits components.

The logician likes truth-functional connectives because of their decent, pre-dictable behaviour.

The notion of truth-functional connectives is thus of the utter-most impor-tance for this course. The reason is that the logical theory of truth-functionalconnectives is, first, presupposed by most logical theories, and is in fact thesimplest logical theory. So any student of logic should know truth-functionalconnectives and their nice behaviour.

Are there any non-truth-functional propositional connectives? Alas, there isquite a lot of them. Consider the following sentence:

It is wrong that John is lazy.

Let’s focus on the connective “it is wrong that” ( ss ). Note that sometimeswhen attached to a true sentence, it yields another true sentence and on otheroccasions it yields a false sentence. Assume that our John is lazy but intelligent.

1. It is wrong that John is lazy.

2. It is wrong that John is intelligent.

Since 1 is true and 2 is false, “it is wrong that” is an example of a non-truthfunctional connective.

Chapter 3

Propositional Logic

Now we are ready to get acquainted with the simplest logical system by thename of propositional logic.

Each logical theory will be introduced here in two steps:

1. First, we will specify its language, i.e. we say which expressions are well-formed and which are not. After this step, we know how to say what wewant to say, but this does not give us any means to decide whether whatwe say is true or not.

2. So at the second step, we will provide these means, i.e. we specify whichexpressions are true and which are false.

Propositional logic is really simple. One of the outstanding logicians oncesaid that it can tell you that either it rains or it doesn’t rain and any systemthat provides such trivial information cannot be complex. In fact, the logicalanalysis of our reasoning we find in this system is very shallow. The analysisterminates at the levels of sentences, thus its name - “propositional logic”.

3.1 Language

Following definition 7 above we define the language of the propositional logicin two steps. First, we will define the alphabet of propositional logic and thenwe will formulate the composition rules that, on the one hand, determine howcomplex expressions are formed out of simple ones, and, on the other hand, thelanguage of the propositional logic itself.

Definition 13. The alphabet of propositional logic is the set that contains thefollowing elements:

1. (an infinite number of) sentence letters: p, q, r, s, t, p1, q1, r1, s1, t1,p2, r2, . . . ,

2. (five) truth-functional connectives: ¬, ∧, ∨, →, and ≡,

25

26 CHAPTER 3. PROPOSITIONAL LOGIC

Language LogicGeorge W. Bush is not a communist. ¬ George W. Bush is a communist.

He didn’t tell any lie. ¬ He told a lie.I hardly believe in what he says. ¬ I believe in what he says.

Table 3.1: Negation in logic and natural language

3. parentheses: (, ).

Obviously, the alphabet of propositional logic contains all and only atomicsymbols in this logic. Each finite string of atomic symbols is called a propo-sitional inscription. For example, all of the strings below are propositionalinscriptions (although some of them have no sense at all!):

• p

• p→ q

• pr¬

• p1 →→ q3

Definition 14. The language of propositional logic is the smallest subset of theset of all propositional inscriptions that satisfies all of the following conditions:

1. All sentence letters belong to the language of propositional logic.

2. If a propostional inscription φ belongs to the language of propositionallogic, then inscription ¬φ belongs to this language as well.

3. If propostional inscriptions φ and ψ belongs to the language of proposi-tional logic, then all of the following inscriptions (φ∧ψ), (φ∨ψ), (φ→ ψ),(φ ≡ ψ) belong to this language as well.

Negation ¬ stands for negation. ¬φ is to be read as “it is not the case thatφ”, e.g. it is not the case that 2 + 2 = 5. In logic the use of negation is simple.Outside, however, its use is governed by various linguistic rules. Table 3.1 showssome simplifications we need to presuppose in order for negation to be appliedin out everyday reasonings.

Conjunction ∧ stands for conjunction. φ∧ψ is to be read as “φ and ψ”, i.e.John is rich and Ann is poor. In logic the use of conjunction is simple. Outside,however, its use is governed by various linguistic rules. Table 3.2 shows somesimplifications we need to presuppose in order for conjunction to be applied inout everyday reasonings.

3.1. LANGUAGE 27

Language

Logic

John

isri

chand

Ann

isp

oor.

John

isri

ch∧

Ann

isp

oor

This

pow

der

conta

ins

sulp

hurand

iron.

This

pow

der

conta

ins

sulp

hur∧

This

pow

der

conta

ins

iron.

Pro

posi

tional

logic

issi

mple

but

eff

ecti

ve.

Pro

posi

tional

logic

issi

mple∧

Pro

posi

tional

logic

iseff

ecti

ve.

Neither

Inor

you

speak

Chin

ese

.I

do

not

speak

Chin

ese∧

You

do

not

speak

Chin

ese

.H

eis

our

chie

fte

chnic

al

offi

cer,

who

isre

sponsi

ble

for

the

ITin

frast

ructu

re.

He

isour

chie

fte

chnic

al

offi

cer∧

He

isre

sponsi

ble

for

the

ITin

frast

ructu

re.

Tab

le3.2

:C

on

jun

ctio

nin

logic

an

dn

atu

ral

lan

gu

age

Language

Logic

He

isa

foolor

aliar.

He

isa

fool∨

He

isa

liar

We

will

fire

you

or

we

will

go

bankru

pt.

We

will

fire

you∨

we

will

go

bankru

pt.

Tab

le3.3

:D

isju

nct

ion

inlo

gic

an

dn

atu

ral

lan

gu

age

Language

Logic

IfJohn

iscle

verth

en

he

will

manage

toso

lve

this

pro

ble

m.

John

iscle

ver→

John

will

manage

toso

lve

this

pro

ble

mAssumin

gth

at

he

retu

rns

wit

hin

one

hour,

we

will

go

shoppin

gto

day.

He

retu

rns

wit

hin

one

hour→

We

will

go

shoppin

gto

day

You

will

get

adri

vin

glicense

pro

vided

that

you

will

pass

the

dri

vin

gte

st.

You

wil

lpass

the

dri

vin

gte

st→

You

will

get

adri

vin

glicense

Tab

le3.4

:Im

pli

cati

on

inlo

gic

an

dn

atu

ral

lan

gu

age


Well-formed Ill-formedp pp

(p→ p) p→→ p(p ∨ q) ∨q

((p ∧ q)→ ¬p) p ∧ q → ¬p

Table 3.5: Examples of well-formed and ill-formed propositional inscriptions

Disjunction ∨ stands for disjunction. φ ∨ ψ is to be read as “φ or ψ”, i.e.John is clever or he always pretends. In logic the use of disjunction is simple.Outside, however, its use is governed by various linguistic rules. Table 3.3 showssome simplifications we need to presuppose in order for disjunction to be appliedin out everyday reasonings.

Implication → stands for implication. φ→ ψ is to be read as “If φ then ψ”,i.e. If John is clever then he will manage to solve this problem. In logic theuse of implication is simple. Outside, however, its use is governed by variouslinguistic rules. Table 3.4 shows some simplifications we need to presuppose inorder for implication to be applied in out everyday reasonings.

Equivalence → stands for equivalence. φ → ψ is to be read as “φ if andonly if ψ”, i.e. Mary is a spinster if and only if he is an old unmarried lady. Inlogic the use of equivalence is simple. Outside, however, its use is governed byvarious linguistic rules. This time the task of finding suitable examples is leftto the readers.

Each element of the language of propositional logic is called a well-formedpropositional inscription (or propositional formula, for short). Table 3.5 men-tions some examples of well-formed and ill-formed propositional inscriptions.

Note that definition 14 requires a huge number of parentheses. This maymake longer formulas unreadable. In order to simplify the structure of proposi-tional formulas, we will use the following convention.

Definition 15. The following sequence defines the relative scope of the truth-functional connectives in propositional logic:

¬,∧,∨,→,≡ .

That is to say, “¬” has the narrowest scope, i.e., its range is the shortest oneamong the five connectives, and “≡” has the broadest scope.

This convention makes it possible to delete redundant parentheses. Forexample, instead of

((p ∧ q)→ ¬p)we could write

p ∧ q → ¬p.

3.2. TRUTH-TABLES 29

Full propositional formulas Simplified propositional formulas(φ ∧ ψ) ∨ χ φ ∧ ψ ∨ χχ ∨ (φ ∧ ψ) χ ∨ φ ∧ ψ(φ ∧ ψ)→ χ φ ∧ ψ → χχ→ (φ ∧ ψ) χ→ φ ∧ ψ(φ ∧ ψ) ≡ χ φ ∧ ψ ≡ χχ ≡ (φ ∧ ψ) χ ≡ φ ∧ ψ(φ ∨ ψ)→ χ φ ∨ ψ → χχ→ (φ ∨ ψ) χ→ φ ∨ ψ(φ ∨ ψ) ≡ χ φ ∨ ψ ≡ χχ ≡ (φ ∨ ψ) χ ≡ φ ∨ ψ(φ→ ψ) ≡ χ φ→ ψ ≡ χχ ≡ (φ→ ψ) χ ≡ φ→ ψ

Table 3.6: Convention 15 at work

In general we may use the following simplifications:

3.2 Truth-tables

What is the role of sentence letters in propositional logic? In short, they aresort of pigeonholes for sentences. Thus, for example, propositional formula

p→ q

pigeonholes, among many more others, all of the following sentences:

• If John is clever then he will manage to solve this problem.

• If he returns within one hour, then we will go shopping today.

• If x = 1, then x+ 1 = 2.

• . . .

Now the truth of such complex sentences depends on the truth-values of theircomponents. However, since all connectives in propositional logic are truth-functional, we do not need to ponder over each sentence separately in order tosay whether it is true or not. What is important is not what a given sentence isabout, but its truth-value. This fact is presupposed in the idea of truth-table.Informally, a truth-table is a prescription that defines how the truth-value of acomplex sentences depends on the truth-values of its components.

Consider the truth-table for negation (table 3.7), where:

1. 1 stands for truth (or a true sentence)

2. 0 stands for falsehood (or a false sentence)


φ ¬φ1 00 1

Table 3.7: Truth-table for negation

φ ψ φ ∧ ψ1 1 11 0 00 1 00 0 0

Table 3.8: Truth-table for conjunction

This table works as described below.

1. If a sentence, say φ, is true, then its negation, i.e., ¬φ is false.

2. If a sentence, say φ, is false, then its negation, i.e., ¬φ is true.

The truth tables for other connectives look slightly different because theyconcern connectives that require two arguments. Let us start with conjunction(table 3.8).

This table works as described below.

1. If one sentence, say φ, is true and another sentence, say ψ is true, thentheir conjunction, i.e., φ ∧ ψ is true.

2. In all other cases, i.e. for all other truth-values of φ and ψ, their conjunc-tion φ ∧ ψ is false.

Please find below the truth-tables for other connectives.Note that propositional formulas may change their truth-values in accor-

dance with the truth-values of their components. Consider again p→ q.

1. p→ q is false only if

φ ψ φ ∨ ψ1 1 11 0 10 1 10 0 0

Table 3.9: Truth-table for disjunction

3.2. TRUTH-TABLES 31

φ ψ φ→ ψ1 1 11 0 00 1 10 0 1

Table 3.10: Truth-table for implication

φ ψ φ ≡ ψ1 1 11 0 00 1 00 0 1

Table 3.11: Truth-table for equivalence

• sentence letter p stands for a true sentence, i.e., it has the value 1,and

• sentence letter q stands for a false sentence, i.e., it has the value 0.

2. in all other cases, p→ q is true.

This leads us to the notion of tautology. Namely, it may happen that apropositional formula is always true irrespective of the truth-values assigned toits components. We will call such formulas tautologies. In order to define themin a more rigorous manner, we need the notion of valuation.

Definition 16. A mapping val from the set of sentence letters to the set {1, 0}is called a propositional valuation.

In other words, a valuation is an assignment of truth-values to sentenceletters. Instead of val(p) = 1, I will also write p = 1. The same holds for p = 0.

Definition 17. If a formula φ is true, i.e. has value 1, under each propositionalvaluation (given the above truth-tables), then φ is a tautology.

Notice that one do not need consider all values of propositional variables inorder to find out the truth-value of a complex formula. If a formula φ contains,say, sentence letter p, q, and r, then one needs to take into account only thevalues assigned to those letters by a valuation in question. This reduces thenumber of valuations one need take into account in his search for tautologies.

1. if a formula consists of one variable, say, p1, then only two valuations arerelevant for the truth-value of the formula:

(a) one under which p1 = 1


(b) one under which p1 = 0

2. if a formula consists of two variables, say, p1 and p2, then only four valu-ations are relevant for the truth-value of the formula:

(a) one under which p1 = 1 and p2 = 1

(b) one under which p1 = 1 and p2 = 0

(c) one under which p1 = 0 and p2 = 1

(d) one under which p1 = 0 and p2 = 0

n. . . .

3. if a formula consists of n variables, say, p1, p2, . . . pn, then only 2n valua-tions are relevant for the truth-value of the formula:

(a) one under which p1 = 1, p2 = 1, . . . , pn = 1

(b) one under which p1 = 1, p2 = 1, . . . , pn = 1

(c) . . .

(2n) one under which p1 = 0, p2 = 0, . . . , pn = 0

We already know that p → q is not a tautology because if val(p) = 1 andval(q) = 0, then it has value 0. What about

p ∨ ¬p ?

Well, since

1. if p = 1, then p ∨ ¬p = 1.

2. if p = 0, then p ∨ ¬p = 1.

p ∨ ¬p is a tautology.It goes without saying that all tautologies are laws of logic.

3.3 Tableaux technique

The truth-table method is so simple that even computers can do it. However, itis time-consuming since for a formula with n propositional letters, you need todo 2n calculations. So, for 4 letters, you will need repeat almost the same step16 times. The semantic tableaux method allows us to find the laws of logic in aquicker (and less boring) way. The price we will have to pay for it is a certainartificiality of the method itself and the increased amount of intellectual effortone needs to spare. Fun costs!

The semantic tableaux method consists in constructing certain formal struc-tures, namely semantic tableaux, that look like plant roots or, when you turnthem upside down, like trees. Thus, sometimes this method is also known asa truth-tree method. The important thing about these table, roots or trees, is

3.3. TABLEAUX TECHNIQUE 33

that they are not composed out of fruits, cellulose, or any organic matter, butof well-formed propositional formulas. Here is an example of such tableaux.

¬((p→ q)→ ¬p)

p→ q

¬¬p

p

¬p q

]

The top of each semantic tableaux is called its root. (You need to turn thepage upside down to understand this!) Starting from the root, each semantictableaux is constructed along certain rules - see below. Let’s start with a simplerule (named DDN = detachment of double negation) that is described in thefollowing diagram:

DDN: ¬¬φ

φ

This rule is supposed to tell you that if your semantic tableaux containsa formula ¬¬φ, you may add to the tableaux φ. This rule has been applied,among others, when the above tree was constructed - see the formulas in thebold face below.

¬((p→ q)→ ¬p)

p→ q

¬¬p

p

¬p q

]

Unfortunately, there are much more rules in this method than just DDN.Please find below the full list.

DDN: ¬¬φ

φ

DC: φ ∧ ψ

φ

ψ


DS: φ ∨ ψ

φ ψ

DI: φ→ ψ

¬φ ψ

DE: φ ≡ ψ

φ

ψ

¬φ

¬ψ

DNC: ¬(φ ∧ ψ)

¬φ ¬ψ

DND: ¬(φ ∨ ψ)

¬φ

¬ψ

DNI: ¬(φ→ ψ)

φ

¬ψ

DNE: ¬(φ ≡ ψ)

φ

¬ψ

¬φ

ψ

It is easy to note that some rules are hammer-like, i.e., that they split se-mantic tableaux. For example, when one applies DC to the following tree:

¬((p→ q)→ ¬p)

p→ q

¬¬p

]

one will have to split it:


¬((p→ q)→ ¬p)

p→ q

¬¬p

p

¬p q

]

The results of such hammer work are called branches. However, one needsto bear in mind that each branch starts at the very beginning of the semantictableaux at stake. Thus, in our example we have the two branches.

1. “left-hand side”: ¬((p→ q)→ ¬p)

p→ q

¬¬p

p

¬p

]

2. “right-hand side”: ¬((p→ q)→ ¬p)

p→ q

¬¬p

p

q

]

You must remember that your semantic tableaux is not finished until thereis no rule you can apply. The tableaux to which no rule applies is called full.

Now once you constructed your semantic tableaux, the rest is easy. Youalready should remember that inconsistency means death in logic. In the contextof our method this allows us to classify branches and tableaux as either deador live. A branch is dead iff it contains (at least) two inconsistent formulae: φ,¬φ; otherwise it is called live. A semantic tableaux is dead iff all of its branchesare dead. If at least one branch is live, then the whole tableaux is live.

In the example above, the “left-hand side” branch is dead and the “right-hand side” is live. So the whole tableaux is live.

Finally, we are ready to specify all details of the semantic tableaux method.Assume that you were asked to find out whether a formula φ is a law of logic.

1. Start constructing the semantic tableaux for φ with negation of φ, i.e.,with ¬φ in the root!


2. Construct the full semantic tableaux according to the rules: DDN, DC,DD, DI, DE, DNC, DND, DNI, DNE.

3. Once the semantic tableaux is full, check by inspection whether it is dead.

(a) If it is dead, then φ is a law of logic.

(b) If it is live, then φ is not a law of logic.

Let’s return to our example. Suppose that we were asked to check whether(p → q) → ¬p is a law of logic. We take the negation of this formula, i.e., wetake ¬((p → q) → ¬p) and put it as the root of the semantic tableaux to bebuilt. Then we apply the following rules: DNC, DDN, and DC. As a result, weget the full semantic tableaux:

¬((p→ q)→ ¬p)

p→ q

¬¬p

p

¬p q

]

Since one of its branches is live, the whole tableaux is live as well. Conse-quently, (p→ q)→ ¬p is not a law of logic.

3.4 Natural deduction

We already know that logic is about inference. When an inference is reallyserious, we call it a proof. Most of the proofs are confined to science, in particularto mathematics. Consider, for example, the following arithmetical claim:

Every number that divides by 231 divides also by 77.How shall we prove it?

1. Assume that there is a natural number n that divides by 231.

2. This implies that n divides by 3, 7 and 11.

3. Consequently n divides by 7 ∗ 11 = 77, so we are done.

In logic such proofs are called direct proofs. Sometimes mathematicians usemore elaborated technique. Consider the following mathematical claim:√

2 is not a rational number.Usually, this claim is proved by means of the so-called reductio ad absurdum

or indirect proof.

1. Assume otherwise, i.e., assume that√

2 is a rational number.

3.4. NATURAL DEDUCTION 37

2. This implies that√

2 is equal to an irreducible fraction km , where m 6= 0

and k and m do not have any factors in common except for 1.

3. Thus, 2 = k2

m2 .

4. Then k2 = 2 ∗m2.

5. So k2 divides by 2.

6. So k divides by 2.

7. Consequently, there is a natural number i such that k = 2 ∗ i.

8. As a result, 4 ∗ i2 = 2 ∗m2.

9. So m2 = 2 ∗ i2, i.e., m divides by 2.

10. But this is a contradiction because k and m have a common factor, i.e. 2,contrary to what step 2 says.

In both cases a proof is a series of sequence of sentences. In propositionallogic a proof will be a series of propositional formulas accompanied by numbersand rule annotations. The numbers will organise all consequent stages in theproof and rule annotations are to explain where the annotated formula comesfrom.

Definition 18. A derivation line consists of:

1. line number n

2. propositional formula φ

3. annotation

Definition 19. A (direct) proof of formula

φ1 → (φ2 → . . . (φn−1 → φn) . . . ).

is a series of derviation lines such that

1. The proof begins with n − 1 premises where φk occurs in the derivationline with number k (1 ≤ k < n). Each such line is annotated by the phase“premise”.

2. Each other line in the proof either

(a) is a propositional formula already proven or

(b) it is derived from some earlier derivation lines according to one ofthe primitive rules of derivation.

3. The proof is finished when the last derivation line contains formula φn.The last line is not numbered.


There are seven primitive derivation rules for propositional logic. Let usstart with the rule that allows us to eliminate an implication from a proof if theproof actually has one. This rule will be denoted by “E→”. Please find belowits schema. .

E→ φ→ ψφψ

This schema is to be read as follows.

1. If one derivation line (of a proof) has implication φ→ ψ and

2. (another) derivation line has φ, then

3. one can add to a proof another line with formula ψ.

Let us see how this rule works outside the propositional logic proper. Sup-pose that you believe that:

1. If John murdered his wife, he should be persecuted.

2. John murdered his wife.

Then E → allows you to infer from these two premises that John should bepersecuted. Easy and simple, isn’t it?

Besides the rule for elimination of implication the proof method for propo-sitional logic uses six more derivation rules. The come in pairs: one rule in apair introduces (I) a connective and the second rule eliminates (E) it.

I∧ φψφ ∧ ψ

E∧ φ ∧ ψ φ ∧ ψφ ψ

Note that the above rule, which eliminates conjunctions, has two schemas:one schema allows you to detach the left-hand side conjunct and the other allowsyou to detach the right-hand side conjunct.

I∨ φ ψφ ∨ ψ φ ∨ ψ

E∨ φ ∨ ψ¬φψ

I≡ φ→ ψψ → φφ ≡ ψ


E≡ φ ≡ ψ φ ≡ ψφ→ ψ ψ → φ

.How does this complex machinery work? Let us have a look!Assume that you were asked to prove the following formula:

(p→ q) ∧ (q → r)→ (p→ r). (3.1)

So you start your prove with premises. You are lucky as there are only twoof them!

1. (p→ q) ∧ (q → r) premise2. p premise

Then you start thinking how to reach r from these premises. Here is how!3. p→ q E∧ : 14. q → r E∧ : 25. q E→ : 3, 2

r E→ : 4, 5Note how we annotate derivation lines. Namely, we mention the rule used

and the (previous) derivation lines to which the rule has been applied. If therule at stake has only one premise, we write down only one number. If the rulehas two premises, then we write down two numbers, and so on.

It turns out that direct proofs are not very good at proving. There are logicallaws that cannot be proved by means of them. Therefore, the propositional logicdefines also the second type of proof: indirect proof or reductio ad absurdum -see the second example above.

Definition 20. An indirect proof ( reductio ad absurdum) of formula

φ1 → (φ2 → . . . (φn−1 → φn) . . . ).

is a series of derviation lines such that

1. The proof begins with n−1 premises where φk occurs in the derivation linewith number k (1 ≤ k < n). Each such line is annotated by the phrase“premise”.

2. The next derivation line contains the formula ¬φn. This derivation lineis annotated by the phrase “reductio”.

3. Each other line in the proof either

(a) is a propositional formula already proven or

(b) it is derived from some earlier derivation lines according to one ofthe primitive rules of derivation.

4. The proof is finished when we arrive at a contradiction, i.e., two inconsis-tent derivation lines: ψ and ¬ψ.


And here is an example of reductio. Again, we were asked to prove thefollowing claim.

(¬p→ q) ∧ ¬q → p. (3.2)

We might do it as follows:1. (¬p→ q) ∧ ¬q premise2. ¬p reductio3. ¬p→ q E∧ : 14. ¬q E∧ : 15. q E→ : 3, 2

And we are done because line 4 is inconsistent with line 5.It is now the highest time to say a few words about metalogic. As you might

have already noticed, logic is a funny branch of science. One of the reasonsis that logic is a scientific reflection about science or a science about science.Mathematicians do prove, so in logic we define the notion of proof. Biologistdo define, so in logic we define definitions. Physicist do experiments, so in logicwe define experiments. And so on. In sum, logic is different from other sciencesin that it concerns the basic methods and procedures that are used in them.Therefore, we call it a metascience. All of the notions defined above, e.g., thenotion of argument, its validity, the idea of logical laws, etc. are metascientificnotions.

However, this is not the whole story about the peculiarity of logic. Sincelogic is a science about science and logic itself is a science, then logic is aboutitself. Indeed, there is an important part of logic by the name of metalogic thatdefines the very logical notions. In fact, the notion of logical law is metalogical.We have not defined it yet and even the following definition is not entirelysatisfactory; still, to provide with an adequate definition would involve a muchtoo heavy formalism for a student to digest.

Definition 21. A logical law (or a law of logic) is any true expression that isbuilt entirely out of

1. variables

2. logical constants

3. and possibly some auxiliary symbols such as parentheses.

Sometimes, laws of logic are also known as tautologies. We already knowthat such propositional formulas as p ∨ ¬p are logical laws. Oddly enough suchclaims like that 1 + 1 = 2 is not a logical law because it contains non-logicalconstants. Although the notion of logical constant is left undefined here, ourunderstanding thereof will be subsequently built step by step as we discoverthe realm of logic. For the time being, it should be sufficient to know that alltruth-functional connectives are logical constants.

If we want to say that propositional formula φ is a law of logic, we will write φ. Thus, p ∨ ¬p.


On the other hand, if we are able to indirectly prove φ, then we will write` φ. Thus, ` (¬p→ q) ∧ ¬q → p. Each proven formula will be called a thesis.1

A curious reader may ask what is the relation between tautologies and thesis.That is, he or she might wonder whether

1. each tautology is a thesis

2. each thesis is a tautology.

In other words, we might ask whether

1. we can prove all tautologies

2. everything we can prove is a tautology.

It is one of the most fundamental metalogical facts that although we canalways guarantee that each thesis is a tautology, in some cases some tautologiesare not provable. In a sense this shows the limits of our knowledge: there aretruths that we cannot prove. Moreover, we can prove that, i.e., we can provethat in the context of some formal theories there are laws that are not provable.

However, in propositional logic tautologies and theses coincide. That is, eachpropositional tautology is a thesis and each thesis is a tautology. In short, inpropositional logic

=` .

Can we prove this? Usually, the proofs of this kind consists of two parts:

1. each tautology is a thesis

2. each thesis is a tautology.

The former proof is complex and will be skipped here - the reader mayconsult the literature to convince him- or herself that it is complex. The latterproof is much easier and is presented below.

Theorem 1 (Soundness). Let φ be a propositional formula. If φ is a thesis,then it is also a tautology.

Proof. Let φ be a formula of the form

φ1 → (φ2 → . . . (φn−1 → φn) . . . ).

We prove our theorem in two steps.First, let us assume that we deal with such formulas so that we always can

pick up an indirect proof of φ that does not contain other theses, i.e., such thatwe have not used in this proof condition 3a of definition 20. Suppose that φ isnot a tautology. If you look at the truth table for implication (see table 3.10above), you will see that the latter assumption implies that all antecedents in φare true and that the last consequent is false. In other words,

1Note that theses are based on indirect proofs, so even if a formula does not have a directproof, this does not mean that it is not a thesis.


1 φ1 is true

2 φ2 is true

. . . . . .

n− 1 φn−1 is true

n φn is false.

So assuming that φ is a thesis but not a tautology, leads us to the proof (of φ)that starts with n true assumptions because

1 first premise is φ1 is true,

2 second premise φ2 is true

. . . . . .

n− 1 n− 1th premise φn−1 is true

n nth premise ¬φn is true.

Now all other derivation lines in this proof are obtained from the earlier linestherein by means of the primitive derivation rules. Note that each such rulepreserves truth, i.e., if its premises are true, its conclusion must be true. Youcan ascertain this fact by focusing your left-hand side eye on the rules andyour right-hand side eye on the relevant truth tables. This fact is of crucialimportance because it implies that all derivation lines must be true since theyare (eventually) obtained from true premises by means of the truth-preservingderivation rules. But this is inconsistent with another property of the proof ofφ namely that of containing two inconsistent derivation lines. If the proof atstake contains both ψ and ¬ψ, then exactly one of these formulas must be trueand the other false.

Let’s move to the second step in the proof of theorem 1. Now we drop ourinitial assumption and include also formulas that require in their indirect proofspreviously proven theses. This time we also have the proof that starts with truepremises, but now the proof may contain derivation lines that have not beenobtained from the earlier lines, but had been proven earlier in separate proofs.Nonetheless, the first part guarantees that each such thesis is a tautology, so it isalways true. Consequently, again we arrive at the conclusion that all derivationlines in the proof of φ are true because they come either from true premises orfrom true theses already proven. And again this is inconsistent.

So we are done because we have shown that the claim that φ is a thesis butnot a tautology leads into inconsistency, which means that this claim is false.

3.5. AXIOMATIC METHOD 43

3.5 Axiomatic method

In logic there exists a different notion of proof, which is related to the notion ofaxiom. The term “axiom” originates in the ancient Greek philosophy, where itdenoted obvious claims as “Either it rains or it does not rain.” In the contextof contemporary formal logic axioms are theses that do not require proofs. Theidea is that when we build a logical theory, we select certain truths as axioms andtry to prove all other truth from those axioms. Usually, we do not impose thecondition that axioms need to be obvious although usually one selects obvioustruths as axioms for the sake of convenience.

There is an important difference between the notion of proof within thenatural deduction method and the notion of proof within the axiomatic method.In both cases proofs are series of derivation lines, but only in the latter case eachderivation lines is a thesis (theorem).

Definition 22. An axiomatic proof of formula φ is a series of derivation linessuch that it starts with of

1. axioms

2. previously proven theses.

Each other line in the proof is derived from some earlier derivation lines ac-cording to one of the axiomatic rules of derivation. Each line in the axiomaticproof is a thesis.

A set of axioms and axiomatic rules of derivation constitutes an axiomaticsystem. There are various axiomatic systems for propositional logic. The onepresented below comes from the work of two mathematicians: Paul Bernays andDavid Hilbert.

First come the axioms.

p→ (q → p). (3.3)

(p→ (q → r))→ ((p→ q)→ (p→ r)). (3.4)

(p→ q)→ (¬q → ¬p). (3.5)

¬¬p→ p. (3.6)

p→ ¬¬p. (3.7)

p ∧ q → p. (3.8)

p ∧ q → q. (3.9)

(p→ q)→ ((p→ r)→ (p→ q ∧ r)). (3.10)

p→ p ∨ q. (3.11)

q → p ∨ q. (3.12)

(p→ r)→ ((q → r)→ (p ∨ q → r)). (3.13)

(p ≡ q)→ (p→ q). (3.14)

(p ≡ q)→ (q → p). (3.15)

(p→ q)→ ((q → p)→ (p ≡ q)). (3.16)


Then come the axiomatic rules of derivation.

E→ ` φ→ ψ` φ` ψ

Note that I set up a trap for you. This rule looks very similar to one of therules in the natural deduction method, but in fact there are slightly different.Find three differences therebetween!

The second rule involves the notion of substitution. The simplest way toexplain this operation goes, as usual, through examples.

Consider first the formula:p→ q.

Now the task is to substitute r for p in this formula. The result is

r → q.

Now a more complex example. Consider the formula:

p→ p ∨ q.

The task is as before: to substitute r for p in this formula. The result is

r → r ∨ q.

Finally, consider the formula

p→ p ∨ r ∧ q.

Now the task is different: to substitute r∧ q for p in this formula. The result ofthis substitution is

r ∧ q → r ∧ q ∨ r ∧ q.

These examples summerise all you should know about substitution. If we substi-tutes formula φ for variable α in formula ψ, we write the result of this operationas ψ(α/φ). This allows us to state the second rule of the Hilbert-Bernays ax-iomatic system for propositional logic.

S ` ψ` ψ(α/φ).

Now you will have an opportunity to appreciate why the natural deductionmethod is called natural. In short, this is supposed to mean that proofs thereare relatively easy to build, at least if you compare them to the axiomatic proofs.Consider the following example of the axiomatic proof. We will try to prove:

(q → r)→ ((p→ q)→ (p→ r)).

So brace yourself!

3.5. AXIOMATIC METHOD 451.

((p→

(q→r)

)→

((p→q)→

(p→r)

))→

((q→r)→

((p→

(q→r)

)→

((p→q)→

(p→r)

)))

S[3.

3:p/3.4,q/(q→r)

]

2.(q→r)→

((p→

(q→r)

)→

((p→q)→

(p→r)

))E →

[1,3.3

]

3.((q→

r)→

((p→

(q→

r))→

((p→

q)→

(p→

r)))→

(((q→

r)→

(p→

(q→

r)))→

((q→

r)→

((p→

q)→

(p→

r)))

)S[

3.4

:p/(q→r),q/(p→

(q→r)

),r/

((p→q)→

(p→r)

)]

4.((q→r)→

(p→

(q→r)

))→

((q→r)→

((p→q)→

(p→r)

))E →

[3,2

]

5.(q→r)→

(p→

(q→r)

)S[

3.3

:p/(q→r),q/p

]

6.(q→r)→

((p→q)→

(p→r)

)E →

[4,5

]

Chapter 4

Predicate Logic

Propositional logic is easy, but also logically powerless, i.e., it cannot be usedfor a number of valid arguments. Let me remind you the example 2.1 from page10.



————————————————————–


I showed earlier that this is a perfectly valid argument, but using proposi-tional logic we will get the following schema thereof:

p1 ∧ p2.

¬q.————————————————————–

¬r.(4.1)

Since

(p1 ∧ p2) ∧ ¬q → ¬r

is not a law of logic, we get an incorrect conclusion to the effect that 2.1 isnot valid. The reason for this failure is the logical weakness of propositionallogic that originates from what we may dub as “shallow parsing”. Namely,propositional logic does not go very deep into the structure of our languageand thought. In particular, it does not analyse simple sentences such as “Allpuddings are nice”. This is the main task of predicate logic.

47

48 CHAPTER 4. PREDICATE LOGIC

4.1 Language

What are predicates? Consider the following set of sentences.

• Ann loves John.

• Ann loves Thomas.

• Ann loves Ann.

• . . .

All these sentences have something in common. And this shared part is apredicate “loves”.

Definition 23. A propositional connective is a predicate if all its argumentsare singular names.

Thus, some predicates have require two arguments (e.g., the verb “to love”),others require only one (e.g., the verb “to shine”), still other require more thantwo.

Another salient notion of predicate logic is the notion of quantifier. As usual,it is best to start with an example. Consider the following sentence:

Ann loves everybody.

As we already know, this sentence consists of:

1. the (singular) name: “Ann”,

2. the (binary) predicate: “loves”.

What is left is the quantifier or better the expression whose meaning and func-tion is equivalent to that of one of the quantifiers. You can easily imagineanother quantifier for “sombody”.

In the strict sense of the word only formal languages contain quantifiers. Inmost cases we use only two of them:

1. universal quantifier:

(a) symbol: “∀”(b) reading: “for all”

(c) usage:∀x x2 ≤ 0.

2. existential quantifier:

(a) symbol: “∃”(b) reading: “for some” or “there exists”

(c) usage:∃x x2 = 0.

4.1. LANGUAGE 49

Nonetheless, logicians usually assume that natural language also involvesquantifiers or at least expressions that are semantically equivalent to quantifiers.Here are some examples:

• Everybody loves somebody somehow.

• Some politicians are not trust-worthy.

• Every whale is a mammal.

• No human being is immortal.

• . . .

Even given this optimistic assumptions of the logicians there is some workto be done when one wants to make the function of those quantifiers explicit.The exercise at stake concerns a sort of translation from natural language to thelanguage of predicate logic. This translation may be split into two phases. Inthe first phase we translate the natural language expressions to a semi-formalmodified natural language, in which the structure of sentences is made explicit.The second phase translates those semi-formal expressions to the formal lan-guage of logic. Let me now show the examples of the first phase translations -cf. table 4.1. The second phase is illustrated in table 4.3 below.

Natural language Semi-formal translationSome politicians are not trust-worthy. There is x such that (x is a politician and x is not trust-worthy).

Every whale is a mammal. For every x, if x is a whale, then x is a mammal.No human being is immortal. For every x, if x is a human being, then x is not immortal.

Table 4.1: Examples of semi-formal translations

Now we can get acquainted with the definition of the language of predicatelogic. As before (consult definitions 13 and 14 above), the definition consists oftwo parts.

Definition 24. The alphabet of predicate logic is the set that contains thefollowing elements:

1. (an infinite number of) predicate letters: P , Q, R, S, T , P1, Q1, R1, S1,T1, P2, R2, . . . ,

2. (an infinite number of) nominal variables: x, y, z, x1, y1, . . . ,


4. (two) quantifiers: ∀, ∃,


Each finite string of the aforementioned symbols is called an inscription.


Well-formed Ill-formedP (x) P (P )

P (x, y, z) x(x, y, z)∀x∃y P (x, y) ∀∃y P (x, y)∀x∃x P (y, y) ∀x x→ x

Table 4.2: Inscriptions in predicate logic

Definition 25. The language of predicate logic is the smallest subset of the setof all inscriptions that satisfies all of the following conditions:

1. If ∆ is a predicate letter and α1, α2, . . . , αn (n > 0) are nominal variables,“∆(α1, α2, . . . , αn)” belongs to the language of predicate logic.

2. If an inscription φ belongs to the language of predicate logic, then inscrip-tion ¬φ belongs to this language as well.

3. If inscriptions φ and ψ belong to the language of predicate logic, then allof the following inscriptions (φ ∧ ψ), (φ ∨ ψ), (φ→ ψ), (φ ≡ ψ) belong tothis language as well.

4. If an inscription φ belongs to the language of predicate logic and α isa nominal variable, then “∀α φ” and “∃α φ” belong to the language ofpredicate logic.

As in the case of propositional logic, we will use convention 15 to reduce thenumber of parentheses assuming that quantifiers precede every sentential con-nectives. Table 4.2 shows some examples of the well- and ill-formed inscriptionsin predicate logic.

Now we are in a position to complete the aforementioned translation fromnatural language to predicate logic.

Semi-formal translation Formal translationThere is x such that (x is a politician and x is not trust-worthy). ∃x (P1(x) ∧ ¬Q1(x)).

For every x, if x is a whale, then x is a mammal. ∀x (P2(x)→ Q2(x)).For every x, if x is a human being, then x is not immortal. ∀x (P3(x)→ ¬Q3(x)).

Table 4.3: Examples of formal translations

Where:

1. P1(x) stands for “x is a politician”.

2. Q1(x) stands for “x is a trust-worthy”.

3. P2(x) stands for “x is a whale”.

4. Q2(x) stands for “x is a mammal”


5. P3(x) stands for “x is a human being”.

6. Q3(x) stands for “x is immortal”.

4.2 Tableaux technique

Let me remind you that the semantic tableaux method for propositional logic(see section 3.3, which starts at p. 32 above) consists in constructing semantictableaux or trees for propositional formulas according to the strictly defined setof rules. That whether the formula for which the semantic tableaux is built isor is not a law of propositional logic depends on whether the tableaux has acertain property, namely whether it is live (open) or dead (closed).

The semantic tableaux method for propositional logic is a simple extensionof the method for propositional logic. In particular,

1. we start building a tableaux in exactly the same way as in propositionallogic, i.e., we take the negation of the formula we check as the root of thetableaux,

2. we build the tableaux using the rules defined for propositional logic andthe following four rules for quantifiers:

DUQ: ∀α φ

φ(α/ξ)where: ξ is any singular name.

DEQ: ∃α φ

φ(α/ξ)where: ξ is a singular name that is different from all other singular namesin the tableaux.

DNUQ: ¬∀α φ

¬φ(α/ξ)where: ξ is a singular name that is different from all other singular namesin the tableaux.

DNEQ: ¬∃α φ

¬φ(α/ξ)where: ξ is any singular name.

3. we check whether the formula at stake is a law of logic in exactly thesame way as in propositional logic, i.e., we check whether the tableaux weconstructed is live or dead.


Our definition of the “predicate logic” rules employs the notion of singularname. Although singular names do not belong to the language of propositionallogic, we use them when we construct semantic tableaux. In principle, there areno additional restrictions on those names, but for the sake of simplicity we willuse the so-called nominal letters: “a”, “b”, “c”, “a1”, “b1”, etc. instead of thefully feathered names, say, from natural language.

Consider now the following example of the use of the semantic tableauxmethod in predicate logic. We are asked to check whether formula ∀x P (x)→ ∃x P (x)is a law of logic.

We start by placing the negation of this formula at the root of a tableaux:

¬(∀x P (x)→ ∃x P (x))

Then we use the rule DNI from propositional logic:

¬(∀x P (x)→ ∃x P (x))

∀x P (x)

¬∃x P (x))

Now we apply the new rule DUQ to the formula in the bold face:

¬(∀x P (x)→ ∃x P (x))

∀x P(x)

¬∃x P (x))

P(a)

Finally, we apply the rule DNEQ:

¬(∀x P (x)→ ∃x P (x))

∀x P (x)

¬∃x P(x))

P (a)

¬P(a)

Since the resulting tableaux is dead (closed), we conclude that ∀x P (x)→ ∃x P (x)is a law of logic.

Now consider a more complex case. Is ∀x (P (x)∨Q(x))→ ∀x P (x)∨∀x Q(x)a law of logic? Let’s check that out:


¬(∀x P (x) ∨ ∀x P (x))

¬∀x P (x)

¬∀x Q(x)

¬P (a)

¬Q(b)

P (a) ∨Q(a)

P (b) ∨Q(b)

P (a)

P (b) Q(b)

Q(a)

P (b) Q(b)

Note that we use rule DNUQ twice, so we were forced to introduce twosingular names: a and b. Since this tree has a live branch - see below - therefore∀x (P (x) ∨Q(x))→ ∀x P (x) ∨ ∀x Q(x) is not a law of logic.

¬(∀x P (x) ∨ ∀x P (x))

¬∀x P (x)

¬∀x Q(x)

¬P (a)

¬Q(b)

P (a) ∨Q(a)

P (b) ∨Q(b)

...

... ...

Q(a)

P (b) ...

Finally consider the following formula: ∀y∃x P (x, y)→ ∃x∀y P (x, y).

Let’s check this out:


¬(∀y∃x P (x, y)→ ∃x∀y P (x, y)

∀y∃x P (x, y)

¬∃x∀y P (x, y)

∃x P (x, a)

P (b, a)

¬∀y P (a, y)

¬∀y P (b, y)

¬P (a, c)

¬P (b, d)

∃x P (x, d)

. . .

Of course, this tableaux is not finished as we should use rule DEQ to formula∃x P (x, d). But note that we applied this very rule to formula ∃x P (x, a), whichis above ∃x P (x, d) in the tableaux. The latter use yielded a series of formulasthat terminates at ∃x P (x, d). So if we applied rule DEQ to ∃x P (x, d), wewould get another series of formulas that would terminate at, say, ∃x P (x, g),which again requires the use of DEQ, etc. Consequently, our table will neverbe finished, i.e., it is infinite.

This shows an important phenomenon of predicate logic and the semantictableaux method. Namely, there are infinite semantic tableux in predicate logicand the latter table is an example thereof. This means that we are not in aposition to check in an easy way for all predicate logic formulas whether theyare or are not laws of logic. This fact is a corollary to the important metalogicaltheorem to the effect that predicate logic is not decidable, i.e., roughly speak-ing, that there are no “mechanical” methods that find laws of predicate logic.This theorem was proved by Alonzo Church in 1936. More precisely speaking,predicate logic is semi-decidable, which means that

1. if a formula φ is a law of logic, then there is a finite dead tree for φ,

2. for some formulas that are not laws of logic, there exist only infinite livetrees.

So, in a sense, the tableaux method, and any other “mechanical” method,allows us to find all laws of logic but does not allow us to find all non-laws.


4.3 Natural deduction

Let me remind you that the natural deduction method for propositional logicconsists in (either direct or indirect) proofs that show that certain formulasare laws of logic (see section 3.4 that starts at page 36). Those proofs aresimply sequences of formuals, which sequences are built according to the recipesdescribed in definitions 19 and 20. Those definitions require that any new lineof proof is added as prescribed by the so-called primitive derivation rules, whichare defined on pages 38 - 39.

As in the case of the tableaux method, natural deduction for predicate logicis simply an extension of the method used in propositional logic. This is tomean that:

1. direct and indirect proofs are built according to the same defintions asbefore (i.e., according to 19 and 20),

2. all rules of derivation defined for propositional logic are valid in predicatelogic (i.e., you may use E→, I∧, E∧, I∨, E∨, I≡, E≡).

Besides you are allowed to use four new rules that introduce and eliminatequantifiers. Since these rules reach the internal structure of sentences, theirdefinitions are pretty complex and require some auxiliary notions. So let’s beginour journey.

First, consider formula “∀x (P (x)→ Q(x))”, which represents, say,“∀x (x is a catholic priest then x is male)”. We will say that “(P (x)→ Q(x))”is the range of the universal quantifier ∀x in formula “∀x (P (x) → Q(x))”. Ingeneral, φ is the range of universal quantifier ∀ α in “∀αφ” and ψ is the rangeof existential quantifier “∃α in ∃α ψ”.

Definition 26. A variable α is unbound ( free) in a formula φ if

1. α occurs in φ,

2. α does not stand directly after any quantifier, i.e. φ contains neither ∀αnor ∃α inscriptions,

3. α is not within the range of any quantifier that occurs in φ.

Needless to say, if a variable is not free in a formula, then it is bound there(provided, of course, that it occurs there at all!). Table 4.4 shows you someexamples of free and bound variables.

The metaphor of freedom has to do with the fact that we may substitutecertain formulas for free variables and we must not substitute anything forbound variables. Consider first a mathematical formula

x2 > 1.

Note that because x is free there, we can obtain the following equations fromthis formula:


formula free variables bound variablesP (x)→ Q(x) x -∃x(P (x)→ Q(x)) − x∀x∃y P (x, y, z) z x, y

∀x∃y P (x, y, z)→ Q(x) x, z y

Table 4.4: Examples of bound and unbound variables

1. 22 > 1,

2. 32 > 1,

3. etc.but also

4. y2 > 1,

5. z2 > 1.

Consider now a formula with no free variable, e.g.,

∀y (y2 + 1 > 0).

Note that it makes little sense to substitute anything for y there. For example,

∀3 (32 + 1 > 0)

makes no (mathematical) sense.Although the operation of substitution is fairly intuitive, it involves some

tricky issues that may lead to the trap of inconsistency. Therefore, the logiciansusually define it quite meticulously.

Definition 27. Assume that variable α is free in φ. φ(α/ξ) is the result ofsubstitution of ξ for α provided that the following two conditions are satisfied:

1. wherever α is free in φ, we insert ξ,

2. if it happens that ξ has its own free variables, then no such variable becomesbound after any such insertion.

Again table 4.5 may clarify this formal gobbledegook.

4.3.1 New rules

The natural deduction method for predicate logic has four new rules.

1. rule of introduction of universal quantifier

I∀ φ∀α φ ,

if α is not a free variable in any assumptions of the proof at stake.


formula how we substitute result of substitutionP (x, y)→ R(y, x) x/y P (y, y)→ R(y, y)∀x P (x, y)→ R(y, x) x/y ∀x P (x, y)→ R(y, y)

∃y P (x, y) x/y -∃y P (x, y) x/z ∃y P (z, y)

Table 4.5: Substitution in predicate logic

before I∀ after I∀Man is a sinner. Everybody is a sinner.

P (x) ∀x P (x)P (x) ∀y P (x)

Table 4.6: D∀ - examples

2. rule of elimination of universal quantifier

E∀ ∀α φφ(α/ξ)

before E∀ after E∀Everybody is a sinner. John is a sinner.

∀x P (x) P (a)∀x P (x) P (x)∀x P (x) P (y)

∀x∀y P (x, y) ∀y P (x, y)∀x∃y P (x, y) ∃y P (x, y)∀x∃y P (x, y) ∃y P (z, y)

Table 4.7: E∀ - examples

3. rule of introduction of existential quantifier

I∃ φ(α/ξ)∃α φ

4. rule of elimination of existential quantifier

E∃ ∃α φφ(α/τβ1,β2,...,βn

),

where:

• τ is a constant,

• β1, β2, . . . , βn are all free variables in φ except for α.


before I∃ after I∃John is a sinner. There is someone who is a sinner.

P (a) ∃y P (y)P (x) ∃x P (x)P (y) ∃z P (z)

P (x) ∧Q(x) ∃z (P (z) ∧Q(z))P (x) ∧Q(y) ∃z (P (z) ∧Q(y))P (x) ∧Q(y) ∃z (P (x) ∧Q(z))

Table 4.8: I∃ - examples

You should be warned that using rule E∃ can be dangerous, i.e it may leadyou into the absurd or inconsistency. For instance, consider the following “proof-spoof”.

1. ∃x x+ 1 = 1 ass.2. ∃x x+ 1 = 2 ass.3. a+ 1 = 1 E∃ : 14. a+ 1 = 2 E∃ : 25. 1 = 2 3, 4

So starting from a pair of innocuous assumptions you got a plain absurd.What is wrong with this “proof”? Well, you were careless when using rule E∃for the second time you introduced the same constant, i.e. a. And nothing inthe whole world guarantees that the number that ∃x x+ 1 = 1 refers to is thesame as the number that ∃x x+ 1 = 2 refers to.

This should learn you the lesson: When using E∃ you add constants to yourproof, each use should add a fresh new constant.

If you followed this advice, you would get the following valid derivation:

1. ∃x x+ 1 = 1 ass.2. ∃x x+ 1 = 2 ass.3. a+ 1 = 1 E∃ : 14. b+ 1 = 2 E∃ : 25. ??? . . .

Another trap to be avoided has to do with the possibility that formula φ in∃α φ may contain other free variables than α. If you don’t treat them properly,you will get the following “proof-spoof”:

1. ∀x∃y y < x ass.2. ∃y y < x E∀ : 1.3. b < x E∃ : 24. ∀x a < x I∀ : 35. ∃y∀x y < x I∃ : 4

You can reconstruct this problem also in natural language; however, thistime the use of our rules is, of course, less transparent:


1. Everybody has a head ass.2. x has a head E∀ : 1.3. x has head b E∃ : 24. Everybody has head b I∀ : 35. There is a head that is had by everybody I∃ : 4

4.3.2 Some laws of predicate logic

Now I will show how these rules work proving a number of laws of predicatelogic. Here is a bunch of those that are usually used (and confused).

• ∀x P (x)→ ∃x P (x),

• ∀x P (x) ≡ ∀y P (y),

• ∃x P (x) ≡ ∃y P (y),

• ∀x∀y P (x, y) ≡ ∀y∀x P (x, y),

• ∃x∃y P (x, y) ≡ ∃y∃x P (x, y),

• ∃x∀y P (x, y)→ ∀y∃x P (x, y),

• ¬∀x P (x) ≡ ∃x ¬P (x),

• ¬∃x P (x) ≡ ∀x ¬P (x),

• ∀x (P (x) ∧Q(x)) ≡ ∀x P (x) ∧ ∀x Q(x),

• ∀x P (x) ∨ ∀x Q(x)→ ∀x (P (x) ∨Q(x)),

• ∃x (P (x) ∧Q(x))→ ∃x P (x) ∧ ∃x Q(x),

• ∃x P (x) ∨ ∃x Q(x) ≡ ∃x (P (x) ∨Q(x)),

• ∀x (P (x)→ Q(x))→ (∀x P (x)→ ∀x Q(x)),

• ∀x (P (x)→ Q(x))→ (∃x P (x)→ ∃x Q(x)),

• (∃x P (x)→ ∃x Q(x))→ ∃x (P (x)→ Q(x)),

• ∀x (P (x) ≡ Q(x))→ (∀x P (x) ≡ ∀x Q(x)),

• ∀x (P (x) ≡ Q(x))→ (∃x P (x) ≡ ∃x Q(x)).


4.4 Predicate logic at work

We are now in a position to check whether arguments are valid or not. Perhapsyou still remember the example 2.1 from page 10.



————————————————————–


As you might remember, propositional logic turned out to be too weak toconfirm our intuitions that this is a valid argument. Let us see how predicatelogic will score.

First, we need to rewrite this argument in the language of predicate logic.As before (cf. tables 4.1 and 4.3 above), we will do it in two steps:

For all x, if x is a pudding then x is nice and y is a pudding.

For all x, if x is nice then x is not wholesome.

————————————————————–

y is not wholesome.

(4.2)

∀x(P (x)→ Q(x)) ∧ P (y).

∀x(Q(x)→ ¬R(x)).

————————————————————–

¬R(y).

(4.3)

where

• P (x) stands for: x is a pudding,

• Q(x) stands for: x is nice,

• R(x) stands for: x is wholesome.

Thus, argument 2.1 is valid if and only if the following formula is a law ofpredicate logic:

(∀x(P (x)→ Q(x)) ∧ P (y)) ∧ ∀x(Q(x)→ ¬R(x))→ ¬R(y).

4.4. PREDICATE LOGIC AT WORK 61

Using either of the methods we know, we can easily show that it is a law, so theargument in question is indeed valid. Predicate logic rules!

Let us pick up another example. What do you think about the followingargument?

Friends of my friends are my friends.

————————————————————–

Enemies of my enemies are my friends.

(4.4)

First, we need to supplement it with the definition of enemy: anybody whois not my friend is my enemy. Secondly, we need to undercover the implicitquantifiers in this argument. Thirdly, we need to generalise over the first personqualification. Afterwards out argument looks as below:

Any friend of any of someone’s friends is his or her friend.

Anybody who is not someone’s friend is his or her enemy.

—————————————————————————————–

Any enemy of any of someone’s enemies is his or her friend.

(4.5)

Now we can commence the formalisation procedure.

For all x, y, z, if x is a friend of y and y is a friend of z then x is a friend of z.

For all x, y, x is an enemy of y if and only if x is not a friend of y.

——————————————————————————————————–

For all x, y, z, if x is an enemy of y and y is an enemy of z then x is a friend of z.

(4.6)

∀x, y, z (P (x, y) ∧ P (y, z)→ P (x, z)).

∀x, y (Q(x, y) ≡ ¬P (x, y)).

———————————————————

∀x, y, z (Q(x, y) ∧Q(y, z)→ P (x, z)).

(4.7)

where

• P (x, y) stands for: x is a friend of y,

• Q(x, y) stands for: x is an enemy of y.


So what we need to investigate is whether formula

∀x, y, z(P (x, y)∧P (y, z)→ P (x, z))∧∀x, y(Q(x, y) ≡ ¬P (x, y))→ ∀x, y, z(Q(x, y)∧Q(y, z)→ P (x, z))

is a law of predicate logic. The following truth-tree shows that it is not. Thisimplies that argument 4.4 is not valid.

4.4. PREDICATE LOGIC AT WORK 63¬(∀x,y,z

(P(x,y

)∧P

(y,z

)→P

(x,z

))∧∀x,y

(Q(x,y

)≡¬P

(x,y

))→∀x,y,z

(Q(x,y

)∧Q

(y,z

)→P

(x,z

)))

∀x,y,z

(P(x,y

)∧P

(y,z

)→P

(x,z

))∧∀x,y

(Q(x,y

)≡¬P

(x,y

))

¬(∀x,y,z

(Q(x,y

)∧Q

(y,z

)→P

(x,z

))))

∀x,y,z

(P(x,y

)∧P

(y,z

)→P

(x,z

))

∀x,y

(Q(x,y

)≡¬P

(x,y

))


¬(Q

(a,b

)∧Q

(b,c

)→P

(a,c

))

Q(a,b

)∧Q

(b,c

)

¬P(a,c

)

Q(a,b

)

Q(b,c

)

P(a,b

)∧P

(b,c

)→P

(a,c

)

Q(a,b

)≡¬P

(a,b

)

Q(b,c

)≡¬P

(b,c

)

Q(a,c

)≡¬P

(a,c

)

¬(P

(a,b

)∧P

(b,c

))

¬P(a,b

)

Q(a,b

)

¬P(a,b

)

Q(b,c

)

¬P(b,c

)

Q(a,c

)

¬P(a,c

)

¬Q(a,c

)

P(a,c

)

¬Q(b,c

)

P(b,c

)

¬Q(a,b

)

P(a,b

)

¬P(b,c

)

P(a,c

)

]

4.5. AXIOMATIC METHOD 65

4.5 Axiomatic method

As in the case of the natural deduction method, the axiomatic method in pred-icate logic is an extension of the same method in propositional logic. Thus, weare able to use the same axioms and rules as before and, of course, we havesome more formal machinery to properly handle quantifiers.

Strictly speaking, for the reasons that should not bother us right now, weneed to modify the propositional logic axioms. That is, instead of axioms re-ported on page 43, we will use the following ones:

φ→ (ψ → φ). (4.8)

(φ→ (ψ → χ))→ ((φ→ ψ)→ (φ→ χ)). (4.9)

(φ→ ψ)→ (¬ψ → ¬φ). (4.10)

¬¬φ→ φ. (4.11)

φ→ ¬¬φ. (4.12)

φ ∧ ψ → φ. (4.13)

φ ∧ ψ → ψ. (4.14)

(φ→ ψ)→ ((φ→ χ)→ (φ→ ψ ∧ χ)). (4.15)

φ→ φ ∨ ψ. (4.16)

ψ → φ ∨ ψ. (4.17)

(φ→ χ)→ ((ψ → χ)→ (φ ∨ ψ → χ)). (4.18)

(φ ≡ ψ)→ (φ→ ψ). (4.19)

(φ ≡ ψ)→ (ψ → φ). (4.20)

(φ→ ψ)→ ((ψ → φ)→ (φ ≡ ψ)). (4.21)

As you can see the only difference between this and the previous set of axiomsconcerns the following change in variables:

1. instead of p, we have now φ

2. instead of q, we have now ψ

3. instead of r, we have now χ

The difference between these two kinds of variables has to do with the idea oflevels of language (see section 2.2.2 on page 19). The variables such as p, q,and r belong to an (object) language and the variables φ, ψ, and χ belong toits meta-language. Using those meta-linguistic variables we are able to handlethose tautologies of predicate logic that have their roots in propositional logic.It is worth to emphasise that each axiom on page 65 is an axiom schema, i.e.,it encapsulates an infinite number of axioms. Logic is mighty! For example,axiom 4.8 contains all of the following formulas:

• P (x)→ (Q(x)→ P (x)),


• P (y)→ (Q(y)→ P (y)),

• P (x)→ (R(x)→ P (x)),

• R(x)→ (Q(x)→ R(x)),

• P (x)→ (Q(x)→ P (x)),

• (P (x)→ (Q(x)→ P (x))→ (R(z)→ (P (x)→ (Q(x)→ P (x))),

• etc.

So the axiomatic method for predicate logic uses the above meta-linguisticaxioms. It also needs the axiomatic version of rule E→ (see page 44). Moreover,the axiomatic method for predicate logic requires four new rules that concernquantifiers:

IUQA ` φ→ ψ` (∀α φ)→ ψ

IUQC ` φ→ ψ` φ→ (∀α ψ)provided that α is not free in φ.

IEQA ` φ→ ψ` (∃α φ)→ ψprovided that α is not free in ψ.

IEQC ` φ→ ψ` φ→ (∃α ψ)

And, finally, we need one more rule for the operation of substitution fornominal variables because we cannot use its counterpart from propositionallogic (S): after all, we have no propositional letters in predicate logic!

SUBST ` φ` φ(α/β)

4.6 “Truth-tables” for unary predicate logic

Although there are no truth-table method for the whole predicate logic, thereis still some hope for a simple tool there. Namely, there are various parts ofpredicate logic for which we can define specific generalisations of the truth-tablemethod we found so easy in propositional logic. One of these parts is unarypredicate logic defined below.

In the simple terms, unary predicate logic is this part of predicate logicthat is build out unary predicates, i.e., out of those predicates that have oneargument. Thus, formula

∀x P (x)→ ∃x P (x)

4.6. “TRUTH-TABLES” FOR UNARY PREDICATE LOGIC 67

does belong to unary predicate logic and formula

∀x∀y P (x, y)→ ∃x∃y P (x, y)

does not. Incidentally, note that unary predicate logic is too weak to analysesuch arguments as 4.4 above (page 61), however, it is expressive enough tohandle such cases as argument 2.1.

Of course, in logic we need to be more precise. Consequently, we need aformal definition of the language of unary predicate logic. Here it goes.

Definition 28. The alphabet of unary predicate logic is the set that containsthe following elements:

1. (an infinite number of) unary predicate letters: P , Q, R, S, T , P1, Q1,R1, S1, T1, P2, R2, . . . ,

2. (an infinite number of) nominal variables: x, y, z, x1, y1, . . . ,


4. (two) quantifiers: ∀, ∃,


Each finite string of the aforementioned symbols is called an inscription.

Definition 29. The full language of unary predicate logic is the smallest subsetof the set of all inscriptions that satisfies all of the following conditions:

1. If ∆ is an unary predicate letter and α is a nominal variable, “∆(α)”belongs to the full language of predicate logic.

2. If an inscription φ belongs to the full language of predicate logic, theninscription ¬φ belongs to this language as well.

3. If inscriptions φ and ψ belong to the full language of predicate logic, thenall of the following inscriptions (φ∧ψ), (φ∨ψ), (φ→ ψ), (φ ≡ ψ) belongto this language as well.

4. If an inscription φ belongs to the full language of predicate logic and α isa nominal variable, then “∀α φ” and “∃α φ” belong to the full languageof predicate logic.

For the sake of simplicity, we will consider only the so-called closed formulas,i.e., only formulas in which all variables are bound. For instance,

∀x P (x)→ ∃x P (x)

is closed and

∀x P (x)→ P (x)


is not because x is free therein. Moreover, we exclude all closed formulas thatcontain nested quantifiers. Thus, we will not consider

∀x∃y[P (x) ∧Q(y)].

Now we will proceed towards the truth-table method for the closed fragmentof unary predicate logic. Consider first the following two arguments:

Every man is a sinner.

————————————–

Some men are sinners. (4.22)

It is false that every man is a sinner.

—————————————————–

No man is a sinner. (4.23)

The following schemas show the structure of these arguments:

1. argument 4.22:

∀x (P (x)→ Q(x)).

—————————

∃x (P (x) ∧Q(x)). (4.24)

2. argument 4.23:

¬[∀x (P (x)→ Q(x))].

—————————

∀x (P (x)→ ¬Q(x)). (4.25)

Then the following two formulas are the basis for these inferences:

∀x (P (x)→ Q(x)) ∧ ∃x P (x)→ ∃x (P (x) ∧Q(x)).

¬[∀x (P (x)→ Q(x))]→ ∀x (P (x)→ ¬Q(x)).

Are they laws of logic? They are if for all predicates that we are able tosubstitute for predicate letters “P” and ”Q”, we obtain therefrom true sentences.Do we always obtain truths therefrom? Table 4.9 contains some examples ofsuch substitutions and their results.


Form

ula

Su

bst

itu

tion

forP

(x)

Su

bst

itu

tion

forQ

(x)

Tru

th-v

alu

e

∀x(P

(x)→Q

(x))∧∃x

P(x

)→∃x

(P(x

)∧Q

(x))

xis

ala

wye

rx

isa

hu

man

bei

ng

tru

e∀x

(P(x

)→Q

(x))∧∃x

P(x

)→∃x

(P(x

)∧Q

(x))

xis

ahu

man

bei

ng

xis

ala

wye

rtr

ue

∀x(P

(x)→Q

(x))∧∃x

P(x

)→∃x

(P(x

)∧Q

(x))

xis

au

nic

orn

xis

an

an

imal

tru

e∀x

(P(x

)→Q

(x))∧∃x

P(x

)→∃x

(P(x

)∧Q

(x))

...

...

tru

e¬[∀x

(P(x

)→Q

(x))

]→∀x

(P(x

)→¬Q

(x))

xis

am

an

xis

aw

om

an

tru

e¬[∀x

(P(x

)→Q

(x))

]→∀x

(P(x

)→¬Q

(x))

xis

ahu

man

bei

ng

xis

ala

wye

rfa

lse

Tab

le4.9

:S

ub

stit

uti

on

s-

inqu

est

for

law

sof

logic


Of course, this method is not foolproof since in general you need to be quitea genius to find the right substitutions for more complex formulas. But at thisstage the truth-table method for unary predicate logic offers its help.

First, the truth-table method generalises from the substitutions and insteadof imagining what predicates we should substitute for predicate letters it rec-ommends that we should focus on the extensions of these predicates, i.e., onthe sets of objects satisfying the respective predicates. So instead of consideringthe predicate “is a lawyer”, it considers the set of all lawyers. And instead ofconsidering the predicate “is red”, it considers the set of all red objects. Andso on.

Assume that we have a formula φ that is built out of n predicate letters:P1, P2, . . . , Pn. The truth-table method associates with each of P1, P2, . . . , Pnits extension. Now the method does not “know” what these sets really are. Itonly “cares” whether they are empty or not. Or, more precisely speaking, it is“interested in” that whether they and all of their combinations are empty ornot.

Suppose that we deal with a formula that has only two predicates: P andQ. Figure 4.1 shows all combinations of the respective distribution classes:

1. region one in the figure represents all objects that belong to the extensionof P but do not belong to the extension of Q, i.e., all objects that are P ,but are not Q,

2. region two in the figure represents all objects that are P and Q,

3. region three in the figure represents all objects that are not P but are Q,

4. region four in the figure represents all objects that are neither P nor Q.

Thus, each region represents a certain set of object, which is called here adistribution class.

Now the truth-table method picks up each of these classes and investigateswhether it is empty or not.1 We have thus 16 cases - see table 4.10 - which wewill call distributions (over the four distribution classes).

If we had three predicates, we would need 8 regions - see figure 4.2 - whichwould determine 256 distributions. In general, if a formula has n predicateletters, it yields 2n distribution classes and 22n

distributions over these classes.There is a special notation to name each distribution class and mark whether

it is empty or not. Suppose that your formula is built of n predicate letters:P1, P2, . . . Pn. For some reason, you are interested in the distribution class, inwhich each object is such that

1. it is P1,

2. it is P2,

3. . . .

1“Non-empty” means here that there is at least one object within the class.


Figure 4.1: Two distribution classes

Region One Region Two Region Three Region Four

empty empty empty emptyempty empty empty non-emptyempty empty non-empty emptyempty empty non-empty non-emptyempty non-empty empty emptyempty non-empty empty non-emptyempty non-empty non-empty emptyempty non-empty non-empty non-empty

non-empty empty empty emptynon-empty empty empty non-emptynon-empty empty non-empty emptynon-empty empty non-empty non-emptynon-empty non-empty empty emptynon-empty non-empty empty non-emptynon-empty non-empty non-empty emptynon-empty non-empty non-empty non-empty

Table 4.10: Distributions for two predicates

4. it is Pn.

The name of your class is then 11 . . . 1︸︷︷︸n times

. But if you are interested in the distri-


Figure 4.2: Three distribution classes


P1 P2 . . . Pn Name

yes yes . . . yes 11 . . . 1︸︷︷︸n times

no yes . . . yes 01 . . . 1︸︷︷︸n times

yes no . . . yes 10 11 . . . 1︸︷︷︸n−2 times

no no . . . yes 00 11 . . . 1︸︷︷︸n−2 times

. . . . . . . . . . . . . . .no no . . . no 00 . . . 0︸︷︷︸

n times

Table 4.11: Naming convention for distribution classes

bution class, in which each object is such that

1. it is not P1,

2. it is P2,

3. . . .

4. it is Pn,

then the name of your class is 0 1 . . . 1︸︷︷︸n−1 times

. The general principle for naming of

distribution classes is explained by table 4.11.If you want to say that a distribution class is empty, you simply strike it

through. And if you want to mark it as non-empty, you underline it. Thus,

11 . . . 1

means that the class in which all objects are P1, P2, . . . , and Pn is non-empty.Now in principle we should check the truth-value of a formula for each of

these distribution classes. However, because it will take a very long time even forsimple formulas with three predicate letters, the truth-table method presentedhere is in fact a simplified version, a sort of reductio ad absurdum. Our methodis based on two assumptions:

Assumption 2. No distribution class is both empty and non-empty.

Assumption 3. It is not the case that all distribution classes are empty.

Note that assumption 3 always excludes one distribution from the set of allpossible distributions. For instance, in the example referred to in table 4.10 thefirst row will not be considered.

Here is the algorithm for the truth-table method for unary predicate logic.


1. Is φ a law of logic?

2. Write down all distribution classes for φ!

3. If φ contains subformulas that are built out of predicate letters only withthe help of truth-functional connectives, calculate the truth-values of thesesubformulas using the truth-tables for propositional logic (section 3.2 onpage 29) in each of the distribution classes!

4. Assume that φ is false!

5. Derive all possible consequences from assumption 4 until you are able todetermine the status of all distribution classes, i.e. whether they are emptyor not! Deriving these consequences use the following rules:

(a) rules “from formulas to distribution classes”

i. if a subformula (of φ) ∀α ψ is true, then mark all distributionclasses in which ψ is false as empty, i.e., strike them through!

ii. if a subformula (of φ) ∀α ψ is false, then mark at least onedistribution class in which ψ is false as non-empty, i.e., underlineit!

iii. if a subformula (of φ) ∃α ψ is true, then mark at least one dis-tribution class in which ψ is true as non-empty, i.e., underlineit!

iv. if a subformula (of φ) ∃α ψ is false, then mark all distributionclasses in which ψ is true as empty, i.e., strike them through!

(b) rules “from distribution classes to formulas”

i. if all distribution classes in which ψ is false are empty, then marka subformula (of φ) ∀α ψ as true!

ii. if some distribution class in which ψ is false is non-empty, thenmark a subformula (of φ) ∀α ψ as false!

iii. if all distribution classes in which ψ is true are empty, then marka subformula (of φ) ∃α ψ as false!

iv. if some distribution class in which ψ is true is non-empty, thenmark a subformula (of φ) ∃α ψ as true!

6. Check whether the distributions you obtained in step 5 are in accordancewith assumptions 1 (on page 20), 2, and 3!

(a) If all of them are consistent with all assumptions, then φ is not a lawof logic.

(b) If some of them is not consistent with at least one assumption, thenφ is a law of logic.


Let us experience the power of this method!Is the following formula a logical law?

∀x P (x) ∨ ∀x Q(x)→ ∀x (P (x) ∨Q(x))

We start writing down all distribution classes for this formula:

∀x P (x) ∨ ∀x Q(x) → ∀x (P (x) ∨ Q(x))1 1 1 11 0 1 00 1 0 10 0 0 0

Then we move forward to step 3 of the algorithm and do our calculations:

∀x P (x) ∨ ∀x Q(x) → ∀x (P (x) ∨ Q(x))1 1 1 1 11 0 1 1 00 1 0 1 10 0 0 0 0

Now we will assume that our formula is not a law of logic, i.e., we will assumethat it is false:

∀x P (x) ∨ ∀x Q(x) → ∀x (P (x) ∨ Q(x))1 1 1 1 11 0 1 1 0

10 1 0 1 10 0 0 0 0

Using only the truth-tables for propositional connectives we arrive at:

∀x P (x) ∨ ∀x Q(x) → ∀x (P (x) ∨ Q(x))1 1 1 1 11 0 1 1 0

1 1 00 1 0 1 10 0 0 0 0

Now we apply rule 5(a)ii from the algorithm:

∀x P (x) ∨ ∀x Q(x) → ∀x (P (x) ∨ Q(x))1 1 1 1 11 0 1 1 0

1 1 00 1 0 1 10 0 0 0 0


Rule 5(b)ii gives us then:

∀x P (x) ∨ ∀x Q(x) → ∀x (P (x) ∨ Q(x))1 1 1 1 11 0 1 1 0

0 1 0 1 00 1 0 1 10 0 0 0 0

So finally we arrive at the following distribution:

∀x P (x) ∨ ∀x Q(x) → ∀x (P (x) ∨ Q(x))1 1 1 1 11 0 1 1 0

0 1 0 1 00

0 1 0 1 10 0 0 0 0

This distribution is inconsistent with assumption 1, so we conclude that

∀x P (x) ∨ ∀x Q(x)→ ∀x (P (x) ∨Q(x))

is a law of logic.

Chapter 5

Definitions

One of the main motivations for logic has to do with precision. Logic can teachyou to think and speak precisely. One of the methods it applies to achieve thisend are definitions. They work more or less as follows.

1. Assume that a certain term or expression is not precise enough.

2. Consult the logical body of knowledge on definitions and try to define it!

3. if you succeed, then the term/expression is question should be more precisethan before provided that you understand and use it in accordance withthe definition you built.

5.1 Definition of definition and other funny stuff

A definition is a short explication of a given object or word.This definition is obviously circular, but it is not viciously circular as it gives

you some understanding what definitions are or how they work. Moreover, itdirectly leads to a important distinction between the so-called nominal and realdefinitions - see below. On the other hand, the definition does not tell you much.

The word “definition” may refer to three kinds of entities:

1. defining activities; e.g.,

• John’s saying “A bachelor is an unmarried man” to his son who doesnot know who a bachelor is;

2. linguistic results of those activities; e.g.,

• the sentence “A bachelor is an unmarried man” written by John ona sheet of paper;

3. semantic meanings of those results; e.g.,

• the meaning of the noun “bachelor”;

77

78 CHAPTER 5. DEFINITIONS

In what follows I will use the term “definition” in the second sense.The term “definition” comes from the Latin verb “definire”, which means

to set up a boundary. We could say that every definition sets up the boundaryaround the required meaning of a certain word or phrase. Thus, what is beingdefined in a definition is called the definition’s definiendum and the means bywhich the definition does its job is called the definition’s definiens.

Except for making our thought and speech more precise, we can use defini-tions for the following three purposes:

1. to better understand what we define;

2. to introduce new words and expressions to the language;

3. to shorten old words and expressions.

If your definition is to achieve its agenda, it needs to satisfy at least thefollowing two conditions:

1. consistencyA definition is consistent if it does not yield inconsistency (within a contextof any consistent theory).

2. eliminabilityA definition is eliminable if its definiendum can always be entirely replacedwith its definiens.

As for the former condition, consider the following definition of multiplica-tion:

x =y

z≡ y = x ∗ z (5.1)

If we were to add this definition to the standard arithmetic, we would get thefollowing inconsistency:

1. 0 = 00 ≡ 0 = 0 ∗ 0 from 5.1

2. 0 = 0 ∗ 0 law of arithmetics3. 0 = 0

0 1, 34. 1 = 0

0 ≡ 0 = 1 ∗ 0 from 5.15. 0 = 1 ∗ 0 law of arithmetics6. 1 = 0

0 5, 65. 0 = 1 3, 66. 0 6= 1 law of arithmetcs

This means that definition 5.1 is inconsistent.As for the condition of eliminability, consider the following example:

The 2007 Merriam-Webster dictionary defines a “hill” and a “moun-tain” this way:Hill - “1: a usually rounded natural elevation of land lower than amountain”Mountain - “1a: a landmass that projects conspicuously above itssurroundings and is higher than a hill”. 1

1From http://en.wikipedia.org/wiki/Circular_definition

5.2. TYPOLOGY OF DEFINITIONS 79

Take any landmass. Is it a hill or a mountain? If it was to be a hill, then it shouldbe lower than a mountain. But what is a mountain? Well, this is everythingthat is higher than a hill. However, we do not know what it is because we havenot yet settled what is a mountain. This is an example of a circular definition,which does not meet the eliminability constraint.

5.2 Typology of definitions

There are various typologies of definitions and we consider just two of them. Thefirst one takes into account the definition’s functions and the second concernsits form.

5.2.1 Nominal and real definitions

A definition is nominal if it defines a meaning of a certain word. A definition ifreal if it defines the nature of a certain thing.

• nominal: The noun “bachelor” denotes the class of all unmarried men.

• real: A bachelor is an unmarried man.

For the practical purposes, this distinction is negligible, but it is importantfor those who, as logicians are, are slightly obsessed with the distinction betweenlanguages and meta-languages (cf. section 2.2.2 on page 19 above).

An important subcategory of real definitions are ostensive (or deictic) defi-nitions, which are defining activities that amounts to pointing to something andsaying “This is . . . ”.

Nominal definitions are usualy divided into:

1. descriptiveA nominal definition is descriptive if it aims at describing the actual mean-ing of its definiens.

• The word “water” denotes the chemical substance by the name of“H2O”.

• “George Washington” is the name of the first President of the UnitedStates.

2. stipulativeA nominal definition is stipulative establishes a new meaning for thedefiniens.

• For the purposes of argument, we will define a “student” to be aperson under 18 enrolled in a local school.

• I suggest using “apatheist” to refer to people who disbelieve in gods(atheists), but who also dont really care about whether any godsexist or not (that is to say, they are apathetic about the question).


3. regulatoryA nominal definition is regulatory if it establishes a new meaning for thedefiniens on the basis of the actual meaning.

• The word “pipeline” or “pipeline system” means all parts of a pipelinefacility through which a hazardous liquid or carbon dioxide moves intransportation, including, but not limited to, line pipe, valves, andother appurtenances connected to line pipe, pumping units, fabri-cated assemblies associated with pumping units, metering and deliv-ery stations and fabricated assemblies therein, and breakout tanks.

Note that all descriptive definitions are either true or false because they are,in fact, claims about certain words. On the other hand, no stipulative definitionhas a truth-value because it just suggests or recommends that a defined wordshould have a new meaning.

5.2.2 Definitions in normal form and other forms of defi-nitions

A definition has a normal form if it is either identity or equivalence. That is tosay, each normal definition has one of the following forms:

α = η(β1, β2, . . . , βn) (5.2)

δ(α1, α2, . . . , αn) ≡ γ(α1, α2, . . . , αn) (5.3)

α = η(β1, β2, . . . , βn) ≡ γ(α, β1, β2, . . . , βn) (5.4)

Here are some examples of normal definitions:

tanx =sinx

cosx. (5.5)

x ≤ y ≡ x < y ∨ x = y. (5.6)

x =y

z≡ z 6= 0 ∧ y = x ∗ z. (5.7)

There are a number of definitions that do not have normal forms:

1. axiomatic definitions

2. definitions by abstraction

3. conditional definitions

4. inductive definitions

An axiomatic definition is established by a set of axioms that, if taken astrue, determine the meanings of the words and phrases occurring therein. Forexample, the set of axioms for propositional logic (see 43 above) determine themeaning of the five truth-functional connectives.

5.3. NORMAL DEFINITIONS AND THEIR SOUNDNESS 81

A definition by abstraction is based on the so-called abstraction principle.The rough idea is that for a special class of relations, called equivalence relations,it is possible to define a certain abstract notion by means of a relation selectedfrom this class that, so to speak, links together less abstract objects. Considerthe following example:

direction of line x = direction of line y ≡ x||y.

This formula can be interpreted as a definition of the notion of line direction:line directions are such entities that the direction of one line is identical to thedirection of another line if and only if these lines are parallel. In other words,the direction of a line x is the set of all lines that are parallel to x. Take anotherexample: a blood group is the set of all human beings such that their bloodagglutinate with one another within this set.

A conditional definition is in fact a conjunction of conditional clauses suchthat their consequents are normal definitions and their antecedents express con-ditions under which these normal definitions are applicable.

|x| =

−x if x < 00 if x = 0x if x > 0

(5.8)

An inductive definition consists of three parts:

1. clause that says that certain objects unconditionally have a certain prop-erty;

2. clause that says that certain objects have a certain property provided thatother objects have this property;

3. no other object has this property.

Consider the following example of Fibocciani numbers. A natural numberis a Fibocciani number if and only if

1. it is either 0 or 1;

2. if n and m are the last two previously recognised Fibocciani numbers, thenn+m is a Fibocciani number;

3. no other number is a Fibocciani number;

5.3 Normal definitions and their soundness

Each definition should be consistent and eliminable. We can formally spec-ify certain formal conditions that guarantee that for normal definitions theserequirements are satisified:

1. most general definiendum condition


2. no vicious circle

3. homogeneity

4. existence and uniqueness

• for definitions of the form: α = η(β1, β2, . . . , βn) ≡ γ(α, β1, β2, . . . , βn))

Definition 30. A normal definition “φ ≡ ψ” satisfies the most general definien-dum condition if no variables occurs in its definiendum, i.e., in φ, more thanonce.

Definition 5.9 does not satisfy this condition and definition 5.10 does.

p→ p ≡ ¬(p ∧ ¬p). (5.9)

p→ q ≡ ¬(p ∧ ¬q). (5.10)

Definition 31. A normal definition “φ ≡ ψ” satisfies no vicious circle conditionif

1. no term defined in its definiens, i.e., in φ, occurs in its definiendum, i.e.,in ψ;

2. no term defined by means of any terms defined in its definiens occurs inits definiendum.

Definition 32. A normal definition “φ ≡ ψ” satisfies homogeneity condition,i.e., is homogenous, if each free variable that occurs in its definiens, i.e., in φ,occurs as free in its definiendum, i.e., in ψ.

Consider the following definition of a prince: x is a prince iff x is a son ofy and y is a king. This definition does not satisfy the homogeneity condition.Why should it bother us? Assume now that John is a son of William and Annand that William is a king and Ann is not. Then:

• John is a prince because John is a son of William and William is a king.

• John is not a prince because John is a son of Ann and Ann is not a king.

So, our innocent definition led us to inconsistency!

Definition 33. A normal definition α = η(β1, β2, . . . , βn) ≡ γ(α, β1, β2, . . . , βn)satisfies existence and uniqueness condition if it is the case that

∀β1, β2, . . . , βn∃!α γ(α, β1, β2, . . . , βn).

The symbol ∃! denotes the uniqueness quantifier :

∃!α φ(α) ≡ ∃α φ ∧ ∀β1, β2(φ(β1) ∧ φ(β2)→ β1 = β2). (5.11)

Check by yourseld whether the following definition is all acceptable!

x = fiancee of y ≡ x proposed to y ∧ y accepted his proposal.

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