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Propositional Logic

Proposition2

Propositions can be divided into simple propositions and compound propositions.

A simple (or basic) proposition is a proposition that contains no connectives (e.g., not, and, or, if, etc.).

Compound propositions are composed of simple propositions and logical connectives.

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In propositional logic, we deal with propositions as the basic units of meaning. i.e. We are not concerned about the structure within a

simple proposition.

Simple/basic propositions4

John is a man.Mary is beautiful.Paul stole the watch.

Compound propositions5

It is not the case that John is clever.If you give me the money, then I will be rich.Either Mary is a liar or John misunderstands

Mary.

Symbols6

We use capitalized letters to represent propositions.

So if the proposition “John is clever” is A and the proposition “Mary is happy” is B, then the compound proposition “John is clever and Mary is happy” is A and B.

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We use the following symbols (called “logical operators” to translate connectives:1) ~ negation (not, it is not the case that)2) • conjunction (and, but, also)3) v disjunction (or, unless)4) implication (if …then, only if)5) ≡ equivalence (if and only if)

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In some books, ‘&’ or ‘’ is used for and; ‘’ is used for not; ‘’ is used for imply; ‘’ is used for equivalent.

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Example:It is not the case that E ~EB and C B • CEither C or D C v DIf A then B A BH if and only if D H ≡ D

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Only ~ is placed in front of an expression.Other operators are placed between

expressions.

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A combination of letters by the legitimate uses of the logical operators and parentheses is a well-formed formula (WFF).

Which of the following are WFFs? ~ v B CA • B • C~P v ~Q

Main Operator12

The main operator is the operator in a compound proposition that governs the largest component(s) in the proposition. It is also the last operator to be dealt with in finding out the truth value (T or F) of a compound proposition.

Compare to our use of +, , , .In 4 (3 + 2), “” is the main mathematical

operator.

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The main operator of the following propositions is “~”

~B~(A B)~((A≡ F) • (D ≡ E))

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When a proposition has negation as its main operator, the proposition is also called “a negation”.

This also applies to other operators.

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Main operator is “•”.K • ~L(E v F) • (G v H)((R T) v (S U)) • ((W ≡ X) v (Y ≡ Z))

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The main operator is “v”.~C v ~D(F • H) v (~K • ~L)(S • (T U)) v (X • (Y ≡ Z))

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The main operator is “”.H ~J(A v C) (D • E)(K v (S • ~T)) (~F v (M • O))

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The main operator is “≡”.M ≡ ~T(B v D) ≡ (A • C)(K v (F I)) ≡ (~L • (G v H))

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Translation

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In order to represent the structure of a compound proposition or an argument, we need to translate it into symbols.

Two different sentences expressing the same proposition should be translated into the same capital letter.

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Sometimes, two different connectives (e.g., “or” and “unless”) should also be translated into the same logical operator.

The translations should have the same meaning as the original sentences.

Negation22

Negation is not a problematic symbol.

Conjunction23

Conjunction means the obtaining of truth for both conjuncts. Therefore, “but” in English is also read as a conjunction.

“John left early but Mary stayed” means it is true that “John left early” and “Mary stayed”.

(J • M)

Disjunction24

Disjunction should be read as the inclusive “or”. That means both disjuncts can be true or either one of the disjuncts is true. The only impossible case is where both disjuncts are false.

“Unless” is also considered a disjunction. “Unless you work hard, you will fail” is

equivalent to “Either you work hard, or you will fail”.

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Parentheses are sometimes required to avoid ambiguity.

For example, (A • B) v C has different meaning from A • (B v C).

Similar to the use of brackets in mathematics.

Implication26

In an implication or conditional proposition, “If A, then B,” A is called the “antecedent,” and B is called the “consequent.”

The antecedent always follows “If”, “provided that”, “on condition that”, etc.

The consequent always follows “then”, “only if”, and “implies that”, etc.

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Given that A ⊃ B,A is a sufficient condition of B.B is a necessary condition of A.

Necessary condition28

The necessary condition B is one that must occur in order for the condition A to occur.

In other words, without B, A cannot occur.E.g.:

Having GPA 1.7 or above is necessary for graduation. Being vulnerable is necessary for being courageous.

Sufficient condition29

The sufficient condition A is one with which that B can occur.

But note that there may be many other ways for B to occur.

E.g.: Being a father is sufficient for being a parent. Cleaning hands is sufficient for removing bacteria

from them.

Different relations30

• Given any two conditions, X and Y, there are 4 ways in which X may be related to Y:• X is both necessary and sufficient for Y.• E.g.: Being a father is necessary and sufficient for

being a male parent. • X is necessary but not sufficient for Y.• E.g.: Having water is necessary for making soup.

• X is sufficient but not necessary for Y.• E.g.: Having a son is sufficient for being a parent.

• X is neither necessary nor sufficient for Y.• E.g.: Getting a university degree is neither necessary

nor sufficient for success

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The fact that X is neither necessary nor sufficient for Y does not imply that X is not important to Y.

E.g.: Democracy is not necessary nor sufficient for having a good government. But it might still be more likely for democracy to produce a good government.

Possibility32

To say that X is necessary for Y is to say that it is not possible for Y to occur without X.

To say that X is sufficient for Y is to say that it is not possible for X to occur without Y.

Given that there are different kinds of possibility (technological, physical, logical), there are different kinds of necessary conditions and sufficient conditions.

Equivalence33

Given two conditions, X and Y, to say that X is equivalent to Y means that X is the necessary and sufficient for Y.

That means “X ≡ Y” is logically equivalent to “(X Y) • (Y X)”.

We can show that they are logically equivalent by using truth table (discussed in the next topic).

Translation Exercise34

John cooks dinner and Dick reads novels, or Mary drinks wine.

John cooks dinner, and Dick reads novels or Mary drinks wine.

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Either John cooks dinner and Dick reads novels or Mary drinks wine.

John cooks dinner and either Dick reads novels or Mary drinks wine.

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John cooks dinner or both Dick reads novels and Mary drinks wine.

John cooks dinner or Dick and Mary drink wine.

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If John cooks dinner, then if Dick reads novels, then Mary drinks wine.

If John’s cooking dinner implies that Dick reads novels, then Mary drinks wine.

If either John cooks dinner and Dick reads novels or Billy sleeps late, then Mary drinks wine.

Translation of categorical statements

There are 4 types of categorical statements: All apples are plants.

This statement is equivalent to “If a is an apple, a is a plant.”

No apples are vegetables. This statement is equivalent to “If a is an apple, a is not a

vegetable.” Some apples are red.

This statement is equivalent to “a is an apple and a is red.” Some apples are not red.

This statement is equivalent to “a is an apple and a is not red.”

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Definitions of Logical Operators

Truth Tables40

The 5 logical operators are defined by truth tables.

A truth table shows how the truth value of a compound proposition depends on the truth value of its components

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P ~P

T F

F T

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P Q P Q

T T T

T F F

F T F

F F F

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P Q P v Q

T T T

T F T

F T T

F F F

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P Q P Q

T T T

T F F

F T T

F F T

Question45

Each one of the 4 cards below has a letter on one side and a number on the other. Tell me which card(s) you definitely need to turn over, and only that (those) card(s), in order to determine whether the cards are following the rule to the effect that: if a card has vowel on one side, it has an even number on the other side.

E T 4 7

Conditional Statement46

Please note that when P is F, whether Q is T or F, P Q is T.

E.g.: The proposition “If Paul is bald, then he is rich” is true when Paul is not bald.

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P Q P Q

T T T

T F F

F T F

F F T

Applications48

We use the five tables to compute the truth-values of any compound propositions.

Steps:1) Assign letters to basic propositions (avoid

duplication).2) Translate by joining letters with the logical

operators.3) Insert parentheses as required.

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4) Assign truth-values to individual letters.5) Compute the truth-values of the next-

smallest compound proposition, and so on.6) Compute the truth-value under the main

operator.

Example 150

(A v D) EGiven A is true and D, E false, then(T v F) F T F F

Example 251

Given A, B, C true and D, E, F false:(~(D v F) • (B v ~A)) ~(F ~C)(~(F v F) • (T v ~T)) ~(F ~T)(~( F ) • ( T v F)) ~(F F)(T • ( T )) ~(T) T F F

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Truth Tables

Truth tables for propositions53

The previous examples work because the truth values of individual letters are given in advance. That corresponds to how normal people understand a complicated saying by checking the truth and falsity of the basic propositions first.

Now we want to exhaust all the possible combinations of truth-values in order to know the properties of the compound propositions. This may save us time to check the truth and falsity of the individual propositions.

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We have a way to know the number of rows of a complete truth table for any given propositions.

It depends on the number of letters involved.If there are 2 letters, there are 4 rows; 3

letters, 8 rows and so on.Therefore, for n Letters, there are 2n rows.

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For example, we can draw the truth-table for (A v ~B) B by constructing 4 rows.

Then we assign the truth values to individual letters.

The rule is to alternate the truth-values for the rightmost column, then alternate pairs for truth-values for the next column and so on. In this way, we need not worry about omission or repetition.

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(A v ~B) B

T T

T F

F T

F F

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Then we put in the truth value of the consequent.

(A v ~B) B

T T T

T F F

F T T

F F F

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Then we compute the truth-value of the shortest compound statement.

(A v ~B) B

T F T T

T T F F

F F T T

F T F F

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Then we move on to do the same for v and finally we compute the truth-values under the main connective .

(A v ~ B) B

T T F T T T

T T T F F F

F F F T T T

F T T F F F

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In the above example, we notice that the truth-values of the compound proposition only depend on the truth-values of B.

That means the compound proposition is a needlessly complicated way of saying B, if we are only interested in the truth and falsity of a proposition in all possible circumstances.

As a summary, drawing the truth-tables can enable us to see certain important features of a compound proposition.

Types of Propositions61

Propositions can be divided into 3 types based on their logical features: Tautology

A compound proposition is a tautology or logically true if it is true regardless of the truth-values of its components.

Self-contradiction A compound proposition is said to be self-contradictory or

logically false if it is false regardless of the truth-values of its components.

Contingency A compound proposition is said to be contingent if its

truth-values depend on the truth-values of its components.

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Proposition type Column under main operator

Tautology All true

Self-contradictory All false

Contingent At least one true, at least one false

Relations between Propositions 63

Two propositions are (logically) equivalent if they have the same truth-values regardless of the truth-values of the components.

Two propositions are (logically) contradictory if they have opposite truth-values regardless of the truth-values of their components.

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Two propositions are (logically) consistent if there is at least one line in the truth table on which both of them are true.

Two propositions are (logically) inconsistent if there is no line on which both of them are true.

Logically equivalent65

K L ~ L ~ KT TT T F T TT F TT FF F T F FF F TF TT T F T TT T FF TT F T F TT T F

Logically contradictory66

K L K • ~ LT T T T F F TT F F T T T F F T T F F F TF T F F F T F

Logically consistent67

K V L K • LT T T T T TT T F T F FF T T F F T F F F F F F

Logically inconsistent68

K ≡ L K • ~ LT T T T F F TT F F T T T FF F T F F F TF T F F F T F

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Any pair of propositions can only be either consistent or inconsistent.

If two propositions are logically contradictory, they are also logically inconsistent.

But if two propositions are logically equivalent, they may not be logically consistent (two self-contradictions are both logically equivalent and logically inconsistent).

Consistency & Validity70

An argument is invalid if and only if the premises and the falsehood of the conclusion are logically consistent. I.e., an argument is invalid if and only if there is a

logical possibility that the premises are true and the conclusion is false.

An argument is valid if and only if the premises and the falsehood of the conclusion are logically inconsistent. I.e., an argument is valid if and only if there is no

logical possibility that the premises are true and the conclusion is false.

Truth tables for arguments71

1) Follow the previous rules.2) Write out the symbolized argument with

single slash between premises and double slash between the last premise and the conclusion.

3) Look for a line in the table in which the main operators of all of the premises are true and the main operator of the conclusion false. If there is, the argument is invalid. Otherwise, the argument is valid.

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4) Any argument having mutually inconsistent premises is valid regardless of what the conclusion may be.

5) Any argument having a tautologous conclusion is valid regardless of what its premises may be.

Example73

If Bin Laden is responsible for the 911 attack, then bombing Afghanistan is a right action.

Bin Laden is not responsible for the 911 attack.Therefore, bombing Afghanistan is not a right

action.

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Use letter L, B to symbolize the two propositions.

L B~L ---------~B

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L B L B / ~ L // ~ BT T T T T F T F T T F T F F F T T FF T F TT T TT F FF TF F F T F T F T F

Indirect truth tables76

Drawing a large table is troublesome.We need a faster way to spot invalidity.We do it by assuming that the argument is

invalid.Then we work backward to obtain truth-

values for all the components. If no contradiction is seen, then the argument

is indeed invalid.

Testing for Invalidity77

~A (B v C)~B------------------C A

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~A (B v C) / ~B // C AThen assign false to the conclusion and truth

to the premises. ~ A (B v C) / ~ B // C A T T F F FT F T F T T T F T F F

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The previous slide shows how each step is obtained. But in practice we need not solve a problem by showing each step with different lines. The below is a one-line proof of the argument’s invalidity, the red T/F highlighting the assumptions.

~ A (B v C) / ~ B // C A T F T F T T T F T F F

Testing for consistency80

Similar to before except that we have no conclusion.

Instead we assume that all the propositions are consistent by assigning T to the main operator of each.

If there is a contradiction, then the propositions are inconsistent, otherwise consistent.

Example81

A v B B (C v A)

C ~B~A

A v B / B (C v A) / C ~ B / ~ A F T T T T T T F T T F T T F

contradiction