Logic Chapter 7

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 single components

exclusive disjunction

an exclusive disjunction is where both disjuncts cannot be true at the same time

inclusive disjunction

an inclusive disjunction is where both disjuncts can be true at the same time

main operator

the operator that has in its range the largest component or components in a compound statement

Rule 1 of a WFF

1) the dot, wedge, horseshoe, and triple bar symbols must go between two statements

Rule 2 of a WFF

2) the tilde goes in front of the statement it is meant to negate

biconditional

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

modus ponens

a valid argument form (also referred to as affirming the antecedent)

simple statement

one that does not have any other statement as a component

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)

order of operations

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

rule 3 of a WFF

3) the tilde cannot, by itself, go between two statements

rule 4 of a WFF

4) parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement

conditional

in ordinary language, the word "if" typically precedes the antecedent of a conditional. the horseshoe symbol is used to translate a conditional statement. if=antecedent. only if=consequent

fallacy of denying the antecedent

invalid argument form, it is a formal fallacy

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

conjunction

a compound statement that has two distinct statements (called conjuncts) connected by the dot (.) symbol. and, but still, moreover, while, however, also, although, yet.

disjunction

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

statement form

a pattern of statement variables and logical operators

compound statement

a statement that has at least one simple statement as a component

self-contradiction

a statement that is necessarily false (p . ~p)

tautology

a statement that is necessarily true (p v ~p)

statement variable

a statement variable can stand for any statement, simple or complex

modus tollens

a valid argument form (denying the consequent)

argument form

an arrangement of logical operators and statement variables in which a consistent replacement of the statement variables by statements results in an argument

fallacy of affirming the consequent

an invalid argument form; it is a formal fallacy

well-formed formulas

compound statements forms that are grammatically correct

propositional logic

the basic components in propositional logic are statements

truth function

the truth value os a truth-functional compound proposition is determined by the truth values of its components and the definitions of the logical operators involved. any truth-functional compound proposition that can be determined in this manner is said to be a truth function.

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

contradictory statements

two statements that have opposite truth values on every line of their respective truth tables

logically equivalent

two truth-functional statements may appear different but have identical truth tables. when this occurs, they are logically equivalent