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