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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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Chapter 7 Propositional and Predicate Logic

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Chapter 7 propositional and predicate logic

Chapter 7

Propositional and Predicate Logic


Chapter 7 contents 1

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

Chapter 7 Contents (2)

  • Deduction

  • Predicate Calculus

  • Quantifiers  and 

  • Properties of logical systems

  • Abduction and inductive reasoning

  • Modal logic


What is 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

Logical Operators

  • AndΛ

  • OrV

  • Not¬

  • Implies→(if… then…)

  • Iff↔(if and only if)


What is a logic

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

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 logic1

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

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

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

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

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

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  AA Λ false  false

  • A v true  trueA v false  A


Propositional logic

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 logic1

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

Deduction

  • The process of deriving a conclusion from a set of assumptions.

  • Use a set of rules, such as:

    AA → 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

Deduction - Example


Predicate logic

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.


Chapter 7 propositional and predicate logic

  • 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

  • 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

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

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.


Chapter 7 propositional and predicate logic

  • 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

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 and Inductive Reasoning

  • Abduction:

    BA → 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

  • 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

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

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


Chapter 7 propositional and predicate logic

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


When conclusion does not follow from the premisses

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

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

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

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

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 reasoning1

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

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

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

  • 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

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

Class Exercise

  • Draw a truth table for the following expressions:

  • 1. ¬AΛ(AVB)Λ(BVC)

  • 2. ¬AΛ(AVB)Λ(BVC)Λ¬D


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