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

Introduction to Logic

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

Symbolism

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

Connection to Mathematics

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

Truth Tables

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”

Equivalence

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

Common Logical Forms

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

Conditional Statements

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”

Equivalence 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?

Transformations 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

Other Forms of If

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

Valid and Invalid Arguments

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

Testing an Argument

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

Common Argument 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.

More Common Forms

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

A Simple Proof

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

Iff Defined

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

Fallacies

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

Common Errors