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Chapter 7Propositional and Predicate Logic

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Chapter 7 Contents (1)

What is Logic? Logical Operators Translating between English and Logic Truth Tables Complex Truth Tables Tautology Equivalence Propositional Logic

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Chapter 7 Contents (2)

Deduction Predicate Calculus Quantifiers and Properties of logical systems Abduction and inductive reasoning Modal logic

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What is Logic?

Reasoning about the validity of arguments. An argument is valid if its conclusions

follow logically from its premises – even if the argument doesn’t actually reflect the real world: All lemons are blue Mary is a lemon Therefore, Mary is blue.

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Logical Operators

And Λ Or V Not ¬ Implies → (if… then…) Iff ↔ (if and only if)

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What is a Logic? What is a Logic? _ A logic consists of three components:

1. Syntax: A language for stating propositions/sentences.

2. Semantics: A way of determining whether a given proposition/sentence is true or false. (Model theory) 3. Inference system: Rules for inferring/deducing theorems from other theorems.

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Translating between English and Logic

Facts and rules need to be translated into logical notation.

For example: It is Raining and it is Thursday: R Λ T R means “It is Raining”, T means “it is

Thursday”.

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Translating between English and Logic

More complex sentences need predicates. E.g.: It is raining in New York: R(N) Could also be written N(R), or even just R.

It is important to select the correct level of detail for the concepts you want to reason about.

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Truth Tables Tables that show truth values for all

possible inputs to a logical operator. For example:

A truth table shows the semantics of a logical operator.

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Complex Truth Tables We can produce

truth tables for complex logical expressions, which show the overall value of the expression for all possible combinations of variables:

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Tautology The expression A v ¬A is a tautology. This means it is always true, regardless of the value of

A. A is a tautology: this is written

╞ A A tautology is true under any interpretation. Example: A A A V ¬A An expression which is false under any interpretation is

contradictory. Example: A Λ ¬ A

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Equivalence

Two expressions are equivalent if they always have the same logical value under any interpretation: A Λ B B Λ A

Equivalences can be proven by examining truth tables.

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Some Useful Equivalences A v A A A Λ A A A Λ (B Λ C) (A Λ B) Λ C A v (B v C) (A v B) v C A Λ (B v C) (A Λ B) v (A Λ C) A Λ (A v B) A A v (A Λ B) A

A Λ true A A Λ false false A v true true A v false A

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Propositional Logic

Propositional logic is a logical system. It deals with propositions. Propositional Calculus is the language

we use to reason about propositional logic.

A sentence in propositional logic is called a well-formed formula (wff).

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Propositional Logic The following are wff’s: P, Q, R… true, false (A) ¬A A Λ B A v B A → B A ↔ B

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Deduction The process of deriving a conclusion from a set of

assumptions. Use a set of rules, such as:

A A → B BIf A is true, and A implies B is true, then we

know B is true. (Modus Ponens) If we deduce a conclusion C from a set of

assumptions, we write: {A1, A2, …, An} ├ C

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Deduction - Example

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Predicate Logic

The first of these, predicate logic, involves using standard forms of logical symbolism which have been familiar to philosophers and mathematicians for many decades.

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Most simple sentences, for example, ``Peter is generous'' or

``Jane gives a painting to Sam,'' can be represented in terms of logical

formulae in which a predicate is applied to one or more arguments

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Predicate Calculus

Predicate Calculus extends the syntax of propositional calculus with predicates and quantifiers: P(X) – P is a predicate.

First Order Predicate Calculus (FOPC) allows predicates to apply to objects or terms, but not functions or predicates.

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Quantifiers and - For all:

xP(x) is read “For all x’es, P (x) is true”. - There Exists:

x P(x) is read “there exists an x such that P(x) is true”.

Relationship between the quantifiers: xP(x) ¬(x)¬P(x) “If There exists an x for which P holds, then it is not

true that for all x P does not hold”.

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Existential Quantifier -”there exists”

There are times when, rather than claim that something is true about all things, we only want to claim that it is true about at least one thing.

For example, we might want to make the claim that "some politicians are honest," but we would probably not want to claim this universally.

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A way that mathematicians often phrase

this is "there exists a politician who is honest."

Our abbreviation for "there exists" is " ", which is called the existential quantifier because it claims the existence of something.

If we use P for the predicate "is a politician" and H for the predicate "is honest," we can write "some politicians are honest" as:

x[Px Hx].

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Properties of Logical Systems

Soundness: Is every theorem valid? Completeness: Is every tautology a

theorem? Decidability: Does an algorithm exist

that will determine if a wff is valid? Monotonicity: Can a valid logical

proof be made invalid by adding additional premises or assumptions?

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Abduction and Inductive Reasoning Abduction:

B A → B A Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the

time: Every non-flying bird I’ve seen before has been a

penguin; hence that non-flying bird must be a penguin. Not valid reasoning, but likely to work in many

situations.

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Inductive Reasoning

Inductive Reasoning enable us to make predictions based on what has happened in the past.

Example: “The Sun came up yesterday and the day before, and everyday I know before that, so it will come up again tomorrow.”

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Three Kinds of Reasoning

Broadly speaking there are 3 kinds of reasoning:

deductive – Based on the use of modus ponens and other deductive rules and reasoning.

abductive – Based on common fallacy. inductive – Based on history (what has

happened in the past)

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Examples A deductive argument consists of n

premisses and a conclusion. If the argument is valid, then if the

premisses are true the conclusion must be true:

Premiss 1: If it's raining then the streets are wet Premiss 2: It's raining ----------------- Therefore the streets are wet

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All horses have brains Herman is a horse -------------- Therefore Herman has a brain

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When Conclusion Does Not Follow From the Premisses

The following are invalid: If it's raining then the streets are wet

The streets are wet --------------- Therefore it's raining

All horses have brains Herman has a brain --------------- Therefore Herman is a horse

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Examples of Invalid Arguments

The following two arguments are invalid: If it's raining then the streets are wet

The streets are wet -------------- Therefore it's raining

All horses have brains Herman has a brain -------------- Therefore Herman is a horse

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More on Deductive Reasoning

An argument can have any number of premisses:

If p then q If q then r If r then s If s then t p -------

Therefore t

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Abductive reasoning Abduction is "reasoning backwards". We

start with some facts and reason back to a hypothesis. E.g.

If someone has measles they have spots and a sore throat Jimmy has spots and a sore throat ------------------------ Therefore Jimmy has measles

This isn't formally valid, of course. In fact it is a famous fallacy, called "confirming the consequent".

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An Earlier Example

If it's raining then the streets are wet The streets are wet -------------- Therefore it's raining

Nevertheless this does seem to be how doctors work.

They use abduction to generate hypotheses, which they then test (for instance, by doing a blood test).

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Inductive reasoning

Inductive reasoning is reasoning from particular cases or facts to a general conclusion:

raven 1 is black raven 2 is black . . raven n is black ----------- Therefore all ravens are black

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More Examples horse 1 has a brain

horse 2 has a brain . . horse n has a brain ------------- Therefore all horses have brains

These go from SOME to ALL: All observed (i.e. some) Xs have property P

------------------------------- Therefore all Xs have P

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Limitations

This isn't formally valid. The conclusion does not formally follow

from the observed facts. At one time people believed that all

observed swans are white, therefore all swans are white.

This is false, of course, because there are black swans in Western Australia!

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Modal logic Modal logic is a higher order logic. Allows us to reason about certainties, and

possible worlds. If a statement A is contingent then we say that

A is possibly true, which is written:◊A

If A is non-contingent, then it is necessarily true, which is written:

A

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Reasoning in Modus Logic The following rules are examples of the

axioms that can be used to reason in modus logic:

A ◊A ¬A ¬◊A ◊A ¬A We cannot draw truth tables to prove them;

however, you can reason by your understanding of the meaning of the operators.

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Class Exercise

Draw a truth table for the following expressions:

1. ¬AΛ(AVB)Λ(BVC)

2. ¬AΛ(AVB)Λ(BVC)Λ¬D