Logic Chapter 7

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

Inclusive Disjunction

A compound statement in which both disjuncts can be true at the same time

Exclusive Disjunction

A compound statement in which both disjuncts cannot be true at the same time

Disjunction

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

Conjunction

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

Compound Statement

A statement that has at least on 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 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 variable results in a statement. A substitution instance of an argument occurs when a uniform substitution of statements for the variable results in an argument

Modus Ponens

A valid argument form (affirming the antecedent)

Modus Tollens

A valid argument form (denying the consequent)

Valid

An argument in which assuming the premises are true, it is impossible for the conclusion to be false. In other words, the conclusion follows necessarily from the premises

Well-Formed Formula

An arrangement of operator symbols such that the resulting symbolic expressions are grammatically correct

Truth table

An arrangement of truth values for a truth functional compound proposition that displays for every possible case how the truth value of the proposition is determined by the truth values of its simple components

Fallacy of affirming the consequent

An invalid argument form (formal fallacy)

Fallacy of denying the antecedent

An invalid argument form; it is a formal fallacy

Conditional Statements

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

Statement Form

In propositional logic, an arrangement of logical operators and statement variables such that a uniform substitution of statements for the variable results in a statement

Simple Statement

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

Argument Form

Refers to the structure of an argument, not to its content. In propositional logic, an argument form is an arrangement of logical operators and statement variables

Logical Operators

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

Non contingent 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

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

Scope

The statement or statements that a logical operator connects

Truth-Functional Proposition

The truth value of a compound proposition that uses one of the five logical operators can be determined solely on the basis of the truth value of its components

Negation

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

Inconsistent Statements

Two (or more) statements that do not have even on 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

Contradictory Statement

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

Logically Equivalent

Two truth functional statements that have identical truth tables under the main operator. When this occurs, they are logically equivalent

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