Chapter 7 Propositional and Predicate Logic

Chapter 7 Propositional and Predicate Logic

Chapter 7 Contents (1) • What is Logic? • Logical Operators • Translating between English and Logic • Truth Tables • Complex Truth Tables • Tautology • Equivalence • Propositional Logic

Chapter 7 Contents (2) • Deduction • Predicate Calculus • Quantifiers  and  • Properties of logical systems • Abduction and inductive reasoning • Modal logic

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.

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

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.

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”.

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.

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.

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:

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

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.

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

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).

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

Deduction • The process of deriving a conclusion from a set of assumptions. • Use a set of rules, such as: A A → B B If 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

Deduction - Example

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.

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

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.

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”.

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.

 • 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].

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?

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.

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.”

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)

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

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

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

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

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

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".

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).

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

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

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!

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

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.

Class Exercise • Draw a truth table for the following expressions: • 1. ¬AΛ(AVB)Λ(BVC) • 2. ¬AΛ(AVB)Λ(BVC)Λ¬D

Chapter 7 Propositional and Predicate Logic