Chapter 7: Logic definitions

Biconditioinal

A compound statement consisting of two conditionals and put in English as "if and only iff" it will be true where the two sides of the biconditional are both true and where they are both false. It is a sharthand way of writing the conjunction of two conditionals. The relationship the biconditional shows is called "material equivalence"

Conditional

A compound statement which contains an antecedent followed by a horseshoe followed by a consequent, the former of which purports to give a sufficient condition for the latter; the only time a conditional is false is when the antecedent is true and the consequent is false.

Conjunction

A compound statement which contains two simple statements (conjuncts) connected by the dot operator; both conjuncts must be true for the conjunction to be true.

Disjunction

A compound statement which contains two simple statements (disjuncts) connected by a wedge; both disjuncts must be false for the disjunction to be false.

Statement Form

A patter of statement variables and logical operators

Simple statement

A statement that does not have any other statement as a component

Compound statement

A statement that either has more than one simple statement as components or contains a simple statement and negation

Self contradiction

A statement that is necessarily false, that is, always false no matter what; the truth value under its main operator are all false

Tautology

A statement that is necessarily true, that is, always true no matter what; the truth values under its main operator are all true

Contingent statement

A statement that is sometimes true, sometimes false (has both true and false results in the truth table) i.e it is neither necessarily true nor necessarily false

Statement variable

A statement variable can stand for any statement whether simple or complex. We use lower case letters for statement variables, traditionally starting with p. (the upper case letter we have been using to stand for statements only stand for simple statements, so technically they are not statement variables, although we might call them that casually)

Non-contingent statement

A statement whose truth value does not depend on the truth values of teh component parts - i.e tautologies and self contradictions; such a statement is either necessarily true or necessarily false (has either all Ts or all Fs in the truth table)

Truth-function

A truth function is a truth-functional compound proposition which can have its truth value determined by analyzing the truth values of its components

Modus Tollens

A valid argument form that goes like this 1. if p, then q. 2. Not q. Therefore not p.

Modus Ponens

A valid argument form that goes like this: 1. if p, then q. 2. p. Therefore, q

Logical operator/logical connective

A word or symbol used to connect two statements and or "operate" on their meaning; part of the toolbox needed to translate ordinary language sentences

Argument form

An arrangement of logical operators and statement variables in which a uniform replacement of the statement variables by statements results in an argument; more informally, an argument form is a certain structure of argument wehre replacing the statements with other statements still results in the same kind of argument (see modus ponens for example)

Truth table

An arrangement of possible truth-values (we can have truth tables for variables, operators, statements, arguments... we can also have complete truth tables, showing every possible combo of truth values or incomplete shorthand ones)

Fallacy of affirming the consequent

An invalid argument form related to modus ponens, where instead of affirming the antecedent of the conditional, one affirms the consequent of the conditional. (so: 1. if p, then q. 2. q Therefore, p) this is a formal fallacy

Fallacy of denying the atecedent

An invalid argument form related to modus tollens, where instead of denying the consequent, you deny the antecedent. (so: 1. if p, then q 2. Not p. Therefore, not q) formal fallacy

Well-formed formula

Compound statement forms that are grammatically correct; they have the dot, wedge, horseshoe, or triple bar between two statements, the tilde to the left of whatever is supposed to be negated AND NOT by itself between two statements, and they use parentheses, brackets, and braces to disambiguate statements

Propositional Logic

Logic which takes the statement as its primary subject matter; logic about statements

Negation

The denial of the statement that the negation is attached to; what is meant by words like "not", "it is not the case that", "it is false that"

Inclusive Disjunction

The disjunction of logic; where both disjuncts can be true at the same time

Exclusive disjunction

The disjunction sometimes used on natural language, where both disjuncts cannot be true at the same time (Either clean your room or you're in trouble)

Main operator

The operator with the widest scope; the operator that governs the largest number of components in a compound statement

Order of operations

The order of handling logical operators in a statement when doing a truth table; we do the operator with the smallest scope first and then move on to bigger and bigger pieces until we get to the main connective

Inconsistent statements

Two (or more) statements that do not have at least one line on their truth tables where the main operators are true for each statement

Consistent Statements

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

Contradictory statements

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

Logically equivalent

When two truth-functional statements have identical truth tables; two statements are logically equivalent when they are always going to have the same truth value