Introduction to Logic

1. Introduction to Logic • Logical Form: general rules • All logical comparisons must be done with complete statements • A statement is an expression that is true or false but not both • If p or q then r • If I arrive early or I work hard then I will be promoted • Tautologies and Contradictions • A Tautology (t) is a statement that is always true • A Contradiction (c) is a statement that is always false

2. Symbolism • The use of symbols • ~ denotes negation (Not) • If p = true, ~p = false • ^ denotes conjunction (And) • p^q = true iff (if and only if) p = true and q = true • v denotes disjunction (Or) • p vq = true iff p = true or q = true or p^q = true • XOR: exclusive or • P XOR q = (p vq) ^ ~(p^q), “p or q but not both” • Order of operations • ~ is first, ^ and v are co-equal • P^q v r is ambiguous, so parenthesis need to be used: (p^q) v r • ~p^q = (~p) ^ q

3. Connection to Mathematics • Inequalities • x ≤ a means x < a or x = a: (x < a) v (x = a) • Same for x ≥ a • a ≤ x ≤ b means (a ≤ x) ^ (x ≤ b) • a (NOT)> x = a ≤ x • Same for opposite • a (NOT) ≤ x = a > x • Same for opposite

4. Truth Tables • Truth Tables • Every expression has a truth table • This table represents all the possible evaluations of the expression • To build a truth table, construct a table with cells corresponding to every possible value of the variables and the resulting value of the expression

5. Equivalence • Logical equivalence • Two statement forms are logically equivalent iff their truth tables are entirely the same • Ex: p^q = q^p • P = ~(~p) • Showing non-equivalence • Two methods: • Use truth tables: this takes a long time • Use an example statement like “0 < 1”

6. Common Logical Forms • The following are known as axioms. Use these to simplify logical forms easily • Commutative Laws: p^q = q^p , pvq = qvp • Associative Laws: (p^q)^r = p^(q^r), (pvq)vr = pv(qvr) • Distributive Laws: p^(qvr) = (p^q)v(p^r) p v(q^r) = (pvq)^(pvr) _ Identity Laws: p^t = p, pvc = p _ Negation Laws: pv~p = t, p^~p = c _ Double Negative Law: ~(~p) = p _ Idempotent Laws: p^p = p, pvp = p _ Universal Bound Laws: pvt = t, p^c = c _ De Morgan’s Laws: ~(p^q) = ~pv~q, ~(pvq) = ~p^~q _ Absorption Laws: p√(p^q) = p, p^(pvq) = p _ Negations of t and c: ~t = c, ~c = t

7. Conditional Statements • If Structures • Statement form: “if p then q” • Noted: p→q, p is Hypothesis, q is conclusion • Truth Values: p→q is false iff p = true and q = false • In statement forms, “→” is evaluated last • Division Into Cases: Show pvq→r=(p→r)^(q→r) • Build truth table and evaluate each term separately • Then fill in each side of the equation and compare the values

8. Equivalence of If • An If statement can be translated into an Or • p→q = ~pvq • People often use this equivalence in everyday language. • By De Morgan’s Law • ~(p →q) = p^~q • Caution: The negation of an If does not start with “if”

9. Transformations of If • The Contrapositive of an If • The contrapositive of p →q is ~q →~p • A contrapositive is always logically equivalent to the original statement, so it can be used to solve equations • A contrapositive is both the converse and the inverse of a statement • The Converse and Inverse • The Converse of p →q is q →p • The Inverse of p →q is ~p →~q • Neither is logically equivalent to the original statement • If tomorrow is Easter then tomorrow is Sunday • If tomorrow is Sunday then tomorrow is Easter?

10. Other Forms of If • Only If • “p only if q” means that p may occur only if q occurs • Equivalent to: ~q →~p • Equivalent to: p →q • This does not mean “p if q”, which says that if q is true, p must be true

11. Valid and Invalid Arguments • An argument is a sequence of statements and an argument form is a sequence of statement forms. • A basic argument is: p→q p :q _ All statements except the final one are the premises _ The final is the conclusion _ This is read: “If p then q; p occurs, therefore q follows _ The argument is valid iff the conclusion is true when all of the premises are true

12. Testing an Argument • Testing an argument for validity • Identify the premises and conclusion • Construct a truth table showing the possible truth values for each statement and statement form • If a situation exists in which all of the premises are true but the conclusion is false, the argument form is invalid • To simplify, fill in all rows where all premises are true

13. Common Argument Forms • Modus Ponens: A famous argument form • p→q: p:: q • If p occurs then q occurs: p occurs:: therefore q occurs • Modus Tollens • p →q: ~q:: ~p • If q doesn’t occur, p can’t occur • A rule of inference is an argument form that is valid. • There are infinitely many of them • Modus Ponens and Tollens are rules of inference

14. More Common Forms • Generalization • p::pvq and q::pvq • p occurs, therefore either p or q occurred • Used to classify events into larger groups • Specialization • p^q::p and p^q::q • Both p and q occur, therefore p occurred • Used to put events into smaller groups • Elimination • Pvq: ~q::p and pvq:~p::q • P or Q can occur: Q doesn’t:: p must • you can choose one by ruling the other out • Transitivity • p →q:q →r::p →r • If p then q: if q then r:: therefore if p then r • Contradiction Rule: • ~p →c::p • If the negation of p leads to a contradiction, p must be true.

15. A Simple Proof • Proof by Division Into Cases • pvq: p →r:q →r:: r • p or q will occur: if p then r: if q then r:: r occurs • You may only know one thing or another. You must simply show that in either case, the result is the same

16. Iff Defined • The Biconditional (iff) • This is: “p if, and only if q” • Denoted: p↔q and is coequal with → • p iff q = (p→q) ^ (q→p) • If p has the same truth value as q, p↔q is true

17. Fallacies • An error in reasoning that results in an invalid argument • Three kinds • Using ambiguous premises (Statements that are not T/F) • Begging the Question: assuming the conclusion without deriving it from the premises • Jumping to a Conclusion: verifying the conclusion without adequate grounds

18. Common Errors • Converse Error: • p →q: q:: p – FALSE • If p then q: q occurs:: p must occur – FALSE • Inverse Error • p →q: ~p:: ~q - FALSE • If p then q: p doesn’t occur:: q can’t occur - FALSE