Chapter 3: Introduction to Logic

Logic Main 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 ConnectiveSymbol Type 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 StatementNegation All do Some do not Not all do Some do None 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 2 n rows

Conjunction pq p  qp  q TTT TFF FTF FFF

Disjunction pq p  qp  q TTT TFT FTT FFF

Negation p~p~p TF FT

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 pqp → qp → q TTT TFF FTT FFT

Useful results for the Conditional Equivalent to a disjunction: p → q ≡ ~p  q Negation: ~(p → q) ≡ p  ~q

Related Conditional Statements Direct statement p → qp → q If p, then q Converse q → pq → 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 qp is sufficient for q If p, qq is necessary for p p implies qAll p’s are q’s p only if qq 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