1. Symbolic Logic

Conjunction

The compound statement "p ^ q" whose truth value is "true" if both p and q are true and "false" otherwise

Conditional

The compound statement "p → q" whose truth value is "false" if p is true and q is false and "true" otherwise

Bi-conditional

The compound statement "p ↔ q" that is "true" if p and q have the same truth value and "false" if they have differing truth values

Compound Statement

A statement formed from combining statements. The truth value of a compound statement is dependent on the truth values of the constituent parts.

Contradiction

A statement that is always "false", symbolized as c

Tautology

A statement that is always "true", symbolized as t

Combinations of Statements

Given n independent statements, there are 2^n possible combinations for the truth values of those statements

Associative Laws

(p ∧ q)∧ r ≡ p ∧ (q ∧ r) and (p V q) V r ≡ p V (q V r). Because of this law, we often just write these as p ∧ q ∧ r and p V q V r

Statement

A declarative sentence that is either true or false, but not both

Predicate

A sentence containing one or more variables that becomes a statement when those variables are given values

Truth Table

A way to tabulate the truth value of a compound statement. Each of the 2^n possible combinations of the truth values of the constituent parts forms a row of the table. The first n columns are the base statements, and the remaining columns then build up each slightly larger part of the full compound statement so that the analysis of its truth value is clearly explained

Valid Argument Form

An argument form is said to be valid if it is a tautology (statement that is always true)

Syllogism

An argument form with 2 or fewer premises

Conditional Language

Each of the following are ways of saying p → q : (1) "if p, then q", (2) "q, if p", (3) "p implies q", (4) "p only if q", (5) "p is a sufficient condition for q", (6) "q is a necessary condition for p", (7) "q unless ¬p" .

Universal Statement

Given a predicate P(x), the statement "For all x in the set D, P(x) holds", written as "∀x ∈ D,P(x)" is called a universal statement

Uniqueness

Given a predicate P(x), the statement "There is exactly one x in the set D for which P(x) holds", written as "∃!x ∈ D:P(x)" is called a uniqueness statement

Existential Statement

Given a predicate P(x), the statement "There is some x in the set D for which P(x) holds", written as "∃x ∈ D:P(x)" is called an existential statement

Universal Instantiation

If the predicate P(x) is true for all x ∈ D, then P(x0) is true for any particular x0 ∈ D.

Inverse Error

Note that the argument form ((¬p) ^ (p → q)) → (¬q) is NOT valid

Converse Error

Note that the argument form(q ∧ (p → q)) → p is NOT valid

Negation

The compound statement "not p" whose truth value is the opposite of p

Dis-junction

The compound statement "p V q" whose truth value is "false" if both p and q are false, and "true" otherwise

Argument Form

The compound statement (p1∧p2∧···∧pn) → q, which is often stylized by simply listing each of the statements p1,p2,...,pn on its own line, and then writing ∴q. The statements p1,p2,...,p n are called the premises and the statement q is called the conclusion

Division into Cases

The valid argument form ((p1 V p2) ∧ (p1 → q) ∧ (p2 → q)) → q

Transitivity

The valid argument form ((p1 → p2) ∧ (p2 → q)) → (p1 → q)

Modus Tollens

The valid argument form ((¬q) ∧ (p → q)) → (¬p)

Contradition

The valid argument form (p → c) → (¬p)

Modus Ponens

The valid argument form (p ∧ (p → q)) → q

Statement Equivalence

Two compound statements are called equivalent if they share identical truth value in all circumstances. One way to demonstrate equivalence is to demonstrate that when the two compound statements are in the same truth table, they have identical values in each row.

Inverse Fallacy

p → q /= (¬p) → (¬q)

Converse Fallacy

p → q /= q → p

Contrapositive Law

p → q ≡ (¬q) → (¬p)

Distributive Laws

p ∧ (q V r) ≡ (p ∧ q) V (p ∧ r) and p V (q ∧ r) ≡ (p V q)∧ (p V r).

Idempotent Laws

p ∧ p ≡ p and p V p ≡ p

Commutative Laws

p ∧ q ≡ q ∧ p and p V q ≡ q V p .

Identity Laws

p ∧ t ≡ p and p V c ≡ p

Absorption Laws

p V (p ∧ q) ≡ p and p ∧ (p V q) ≡ p

Negation Laws (p,t,c)

p V (¬p) ≡ t and p ∧ (¬p) ≡ c

Universal Bound Laws

p V t ≡ t and p ∧ c ≡ c

Conditional Negation

¬(p → q) ≡ p ∧ (¬q)

De Morgan's Laws

¬(p ∧ q) ≡ (¬p) V (¬q) and ¬(p V q) ≡ (¬p) ∧ (¬q)

Double Negatives

¬(¬p) ≡ p

Negation Laws (t,c)

¬t ≡ c and ¬c ≡ t