Chapter 3: Introduction to Logic. Logic Main goal: use logic to analyze arguments (claims) to see if...

Chapter 3: Introduction to Logic

LogicMain goal: use logic to analyze arguments (claims) to see if they are valid or invalid. This is useful for math theory, but also in the real world any time someone is trying to convince you of something.

To analyze an argument, we break it down into smaller pieces: statements, logical connectives and quantifiers.

Statements

• A statement is a declarative sentence that is either true or false (but not both at the same time)

• A compound statement consists of simple statements combined using logical connectives like and, or, not, if…then.

• The negation of a statement must have the opposite truth value to the original statement

Connectives and Symbols

Connective SymbolType of

Statement

and conjunction

or disjunction

not ~ negation

Quantifiers

• Universal quantifier: all, every, each. Statement is true if the claim is true for every object it is referring to.

• Existential quantifier: some, there exists, for at least one. Statement is true if the claim is true for al least one object it is referring to.

Negations of Quantified Statements

Statement Negation

All doSome do not

Not all do

Some doNone do

All do not

Truth Tables

• Shows truth value of a compound statement for all possible truth values of the component statements

• If there are n component statements, then the truth table has 2n rows

Conjunction

p q p q

T T T

T F F

F T F

F F F

Disjunction

p q p q

T T T

T F T

F T T

F F F

Negation

p ~p

T F

F T

Equivalent Statements

• Two statements are equivalent if they have the same truth value for every possible situation, and we write p ≡ q

• De Morgan’s Laws:

~(p q) ≡ ~p ~q

~(p q) ≡ ~p ~q

The Conditional

• If p, then q

• Symbols: p → q

• p is the antecedent, q is the consequent

Truth Table for Conditional

p q p → q

T T T

T F F

F T T

F F T

Useful results for the Conditional

• Equivalent to a disjunction:

p → q ≡ ~p q

• Negation:

~(p → q) ≡ p ~q

Related Conditional Statements

Direct statement p → q If p, then q

Converse q → p If q, then p

Inverse ~p → ~q If not p, then not q

Contrapositive ~q → ~p If not q, then not p

Equivalences

Direct statement and contrapositive are equivalent:

p → q ≡ ~q → ~p

Converse and Inverse are equivalent:

q → p ≡ ~p → ~q

Common wording for p → q

If p, then q p is sufficient for q

If p, q q is necessary for p

p implies q All p’s are q’s

p only if q q if p

Analyzing Arguments• We will use deductive reasoning to

determine whether logical arguments are valid or invalid.

• A logical argument is made up of premises (assumptions, statements assumed to be true) and a conclusion.

• An argument is valid if the fact that all the premises are true forces the conclusion to be true.

• An argument that is not valid is invalid, or a fallacy.

Techniques to Analyze Arguments

• Using truth values (assume premises are true and see if this forces conclusion to be true)

• Comparing with known valid or invalid arguments

Valid Argument Forms• Modus Ponens

[(p → q) p] → q• Modus Tollens

[(p → q) ~q] → ~p• Disjunctive Syllogism

[(p q) ~p] → q• Transitivity

[(p → q) (q → r)] → (p → r)

Fallacies

• Fallacy of the Converse

[(p → q) q] → p• Fallacy of the Inverse

[(p → q) ~p] → ~q