Propositional Logic Definitions (Ch. 7)

Argument Form

(refers to the structure of an argument, not its content) an arrangement of logical operators and statement variables such that a substitution instance of statement variables results in an argument

Biconditional

A compound statement consisting of two conditionals--one indicated by the word "if" and the other indicated by the phrase "only if." The triple bar symbol is used to translate a biconditional statement

Fallacy of affirming the consequent

An invalid argument form: If A, then B; B; so, A. (it is a formal fallacy)

Modus Ponens

If A then B; A, so B-- valid argument form (also referred to as affirming the antecedent) ('modus' = method, 'ponens' = affirming)

Modus Tollens

If A then B; not B, so not A-- valid argument form (also referred to as denying the consequent)

Rule 4 (WFFs)

Parenthesis, brackets, and braces must be used to eliminate ambiguity in a compound/complex statement

Rule 3 (WFFs)

The tilde (~) cannot, by itself, go between two statements

Inconsistent Statements

Two (or more) statements that do not have even one line on their respective truth tables where the main operators are true (but they can be false) at the same time

Consistent Statements

Two (or more) statements that have at least one line on their respective truth tables where the main operators are true

Inclusive Disjunction

When we assert that at least one disjunct is true, and possibly both disjuncts are true. Given this, an inclusive disjunction is false when both disjuncts are false, otherwise it is true.

Sufficient Condition

Whenever one event ensures that another event is realized. In other words, the truth of the antecedent guarantees the truth of the consequent.

Necessary Condition

Whenever one thing is essential, mandatory, or required in order for another thing to be realized. In other words, the falsity of the consequent ensures the falsity of the antecedent.

Conjunction

a compound statement that has two distinct statements (called conjuncts) connected by the dot symbol

Disjunction

a compound statement that has two distinct statements (called disjuncts) connected by the wedge symbol

Compound Statement

a statement that has at least one simple statement and at least one logical operator as components

Self-Contradiction

a statement that is necessarily false

Tautology

a statement that is necessarily true

Statement Variable

a statement variable (p, q, r, s) can stand for any statement, simple or compound

Substitution Instance

a substitution instance of a statement occurs when a uniform substitution of statements for the variables results in a statement. A substitution instance of an argument occurs when a uniform substitution of statements for the variables results in an argument.

Statement Form

an arrangement of logical operators and statement variables such that a uniform substitution of statements for the variables results in a statement

Fallacy of denying the antecedent

an invalid argument form: If A, then B; not A; so, not B. (it is a formal fallacy)

Well-Formed Formula

any statement letter standing alone, or a compound statement such that an arrangement of operator symbols and statement letters results in a grammatically correct symbolic expression

Conditional Statement

in ordinary language, the word "if" typically precedes the antecedent of a conditional, and the statement that follows the word "then" is referred to as the consequent

Simple Statement

one that does not have any other statement or logical operator as a component

Truth Table

shows every possible truth value for compound propositions, provides definitions of logical operators

Logical Operators

special symbols that can be used as part of ordinary language statement translations

Noncontingent Statements

statements such that the truth values in the main operator column do not depend on the truth values of the component parts

Contingent Statements

statements that are neither necessarily true nor necessarily false (they are sometimes true, sometimes false)

Propositional Logic

the basic components in propositional logic are statements

Rule 1 (WFFs)

the dot, wedge, horseshoe, and triple bar symbols must go between two statements (either simple or compound)

Main Operator

the operator that has the entire well-formed formula in its scope

Order of Operations

the order of handling the logical operators within a proposition; it is a step-by-step method of generating a complete truth table

Rule 2 (WFFs)

the tilde (~) goes in front of statements it is meant to negate

Truth-Function

the truth value of a compound proposition is a function of the truth values of its component statements and the logical operators

Negation

the word "not" and the phrase "it is not the case that" are used to deny the statement that follows them, and we refer to their use as negation

Contradictory Statements

two statements that have opposite truth values under the main operator on every line of their respective truth tables

Logically Equivalent Statements

two truth-functional statements that have identical truth tables under the main operator

Exclusive Disjunction

when we assert that at least one disjunct is true, but not both. In other words, we assert that the truth of one excludes the truth of the other. Given this, an exclusive disjunction is true when only one of the disjuncts is true, otherwise it is false