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# Chapter 7 Propositional and Predicate Logic - PowerPoint PPT Presentation

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

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Propositional and Predicate Logic

• What is Logic?

• Logical Operators

• Translating between English and Logic

• Truth Tables

• Complex Truth Tables

• Tautology

• Equivalence

• Propositional Logic

• Deduction

• Predicate Calculus

• Quantifiers  and 

• Properties of logical systems

• Abduction and inductive reasoning

• Modal 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.

• And Λ

• Or V

• Not ¬

• Implies → (if… then…)

• Iff ↔ (if and only if)

• 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.

• 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”.

• 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.

• 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.

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

• 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

• 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.

• 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 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).

• The following are wff’s:

• P, Q, R…

• true, false

• (A)

• ¬A

• A Λ B

• A v B

• A → B

• A ↔ B

• 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

• 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 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].

• Soundness: Is every theorem valid?

• Completeness: Is every tautology a theorem?

• Decidability: Does an algorithm exist that will determine if a wff is valid?

• 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 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.”

• 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)

• 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

• 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

• 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

• 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

• 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".

• 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 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

• 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

• 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 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

• 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.

• Draw a truth table for the following expressions:

• 1. ¬AΛ(AVB)Λ(BVC)

• 2. ¬AΛ(AVB)Λ(BVC)Λ¬D