Logic Chapter 6

Statement Variables

A lowercase letter (p, q, r, s) that represents a any statement (ex: 'p' could represent 'A', 'AvB', 'A≡B', and so on.

Tautology

A statement that is necessarily true (impossible to be false); in a categorical proposition, the main operator that determines the truth value of its simple statements in the truth table will read all "true"; the compound proposition is true regardless of the actual truth value of its components.

Well-Formed Formulas (WFF's)

A syntactically correct arrangement of of symbols. Rules that qualify a formula as well-formed: 1: A capital letter by itself (ex: A) 2: Any compound statement (ex: ~A) 3: Any compound statement enclosed by a set of parenthesis and joined together by an operator. Examples of WWF's: ~((~A • B) ⊃ ~~C); ((~A • B) ⊃ ~~C)

Operator '≡' (Triple Bar)

Also called a biconditional statement; logical function: equivalence; translates: "If and only if"; expresses material equivalence (logical equivalence between two statements); ex: "JFK's tightening security is a sufficient and necessary condition for O'Hare's doing so." becomes J≡O, where JFK=J, and O'Hare=O.

Operator '⊃' (Horseshoe)

Also called a conditional statement; logical function: implication; translates: "If... then", and "only if"; expresses a relation of material implication, and only used if (1) the statement is a truth-functional conditional, and (2) is used if the antecedent is affirmed and not negated; ex: "If Purdue raises tuition, then so does Notre Dame." becomes "P⊃N", where Purdue=P, and Notre Dame=N"; *to translate statements*: the statement that indicates the sufficient condition is the antecedent of the conditional, and the statement that indicates the necessary condition as the consequent.

Operators

Also known as "connectives"; symbols that connect or negate statements; includes: '~' (tilde), ' •' (dot), 'v' (wedge), '⊃' (horseshoe), and '≡' (triple bar).

Consistent Statements

If two or more propositions are consistent if one-all of the lines in the truth table have a value of "true"; the statement does not necessarily have to be true, it rather asserts that it is possible for it to be true (ex: "There is a book on the table" and "There is a cup on the table" can both be true at the same time without either statement actually being true).

Inconsistent Statements

If two or more propositions lack at least one line in the truth table table that have a value of "true".

Self-Contradictory Statement

In a categorical proposition, a statement that is necessarily false (impossible to be true); the main operator that determines the truth value of its simple statement will read all "false"; the compound proposition is false regardless of the actual truth value of its components.

Contingent Statement

In a categorical proposition, a statement whose truth value varies based on the values of its components; if the truth table of the main operators reads both "true" and "false" (at least one of each), then the statement could be either true of false depending on the truth value of its simple statement components.

Contradictory Statements (Propositional Logic)

In a categorical proposition, contradictory if the truth table of the main operators in two categorical proposition have different truth values for EVERY line; ex: 1 TFTF & 2 TFFT=NOT contradictory; 1 TFTF & 2 TFTF=contradictory).

Simple Statement

In propositional logic, a statement that contain any other statements, and therefore, does not use operators when translating such statements.

Conditional Statement

In propositional logic; proposes a hypothetical supposition (the relation of material implication) that is translated into logical notation with the horseshoe (⊃) as the main operator; the statement before the ⊃ is the antecedent (and *sufficient condition*), and the statement following the ⊃ is the consequent (and *necessary condition*); *the statement following "if" is the antecedent regardless of its syntactical position*. Ex: [Kv(S•~T)]⊃[~Fv(M•O)]

Operator '•' (Dot)

Logical function: conjunction; translates: "and", "also", "moreover", "both", "however", etc.; affirms both statements; ex: "It is spring and pleasant" becomes "S•P", where "It is spring"=S, and "pleasant"=P; used medially to both statements.

Operator 'v' (Wedge)

Logical function: disjunction; translates: "or", "unless", "either...or"; states that either both statements are not true, or at least one statement is false, and the other is true; used medially to both statements; ex: "Either is it Saturday, or it is Friday" becomes "S v F", where "Saturday"=S, and "Friday"=F.

Operator '~' (Tilde)

Logical function: negation; translates: "not", "is not", "it is not the case", "it is false", etc.; always to the left of a negated statement letter (ex: "I do not enjoy hot temperature" becomes "~H", where H represents the statement "hot temperature"); precedes a statement, but can never directly precede another operator (valid is separated by brackets or parentheses).

Logically Equivalent Statements (Propositional Logic)

Logically equivalent only if the truth table of the main operators in two categorical propositions have the same values.

Truth Function

Refers to a compound proposition whose truth value originates and is determined from the truth value of its components; if said truth values from the components are known, then the truth value can be determined by the definitions of logical operators, therefore, the truth value of a compound proposition that is expressed as logical operators is a function of the truth values of its components.

Biconditional Statement

Relates material equivalence and uses the phrase or reiteration thereof of "if and only if" by truth-functional statements; translated into logical notation with the triple bar (≡) as the main operator;

Statement Form

Symbolic representation of a compound statement that may or may not contain statement variables and operators, which results in a statement.

Propositional Logic

The fundamental components are statement themselves, and are represented by capital letters; statements are connected by operators; can be a simple statement or a compound statement.

Main Operator

The operator that relates to all statements in a compound statement; ex: "~[(A≡F)•(C≡G)], where '~' is the main operator.