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Propositional Logic

A branch of formal, deductive Logic in which the basic unit of thought is the proposition

Truth-functional

A proposition is_____ when it’s truth value depends upon the truth values of its component parts

Simple Proposition

A proposition with only one component part

Compound Proposition

A proposition with more than one component part ( or modified in another way)

Logical Operators

Words that combine or modify simple propositions to make compound proportions

Proportional Constant

An uppercase letter that represents a single, given proposition

Propositional Variable

A lowercase letter that represents any proposition

Negation

The logical operator that denies or contradicts a proposition

Truth Table

A listing of the possible truth values for a set of one or more propositions

Defining Truth Table

A____ displays the truth values produced by a logical operator modifying a minimum number of variables

Conjunction

A logical operator that joins two propositions and is true if and only if both the propositions are true

Disjunction

A logical operator that joins two propositions and is true if and only if one or both of the propositions is true

Conditional

The_______ operator asserts that one component implies the other. It is false if and only if the antecedent is true and the consequent is false

Antecedent

The proposition following the “if”

Consequent

The proposition following the “then”

Logically Equivalent

Two propositions are______ if and only if they have identical truth values in a truth table

Tautology

A proposition that is always true due to its logical structure

Self-contradiction

A proposition that is false by logical structure

Valid Argument

In a_____, if the premises are true, the conclusion must be true

Invalid Argument

If the premises can be true and the conclusion false, it is a_______

Consistent

When propositions can be true at the same time

Dilemma

A valid argument which presents a choice between two conditionals

Constructive Dilemma

________ work like modus ponens

Destructive Dilemma

_______work like modus tollens

Formal Proofs of Validity

A step-by-step deduction of a conclusion from a set of premises, each step being justified by an appropriate basic rule

Rules of Inference

Valid argument forms which can be used to justify steps in proof

De Morgan’s Theorems

(~p v q) = (p•q)

Commutation

(p v q) = (q v p)

Association

[(p v q) v r) = [p v (q v r)] and [(p•q)•r]=[p•(q•r)]

Distribution

[p• (q v r)]= [(p•q) v (p•r)] and [p v (q•r)]=[(p v q) • (p v r)]

Double Negation

~~p = p

Transposition

( if p then q) = ( if ~q then ~p)

Material Implication

( if p then q) = (~p v q)

Material Equivalence

[(p=q) = (if p then q) • (if q then p)]

Exportation

[if (p•q) then r] = [ if p then (if q then r)]

Tautology

p = (p v p) and p = (p•p)

Conditional Proof

A special rule in a formal proof which allows us to assume the antecedent of a conditional and, once we delude the consequent, to conclude the entire conditional

Reductio and Absurdum

A special rule which allows us to assume the negation of a proposition, device a self-contradiction, then conclude the proposition

NOR

The logical operator_________ is equivalent to ~(p v q). It is truth-functionally complete

Truth-functionally Complete

A set of logical operators is________ if and only if all possible combinations of true and false for two variables are derivable using only those logical operators

Truth Tree

A diagram that shows a set of propositions being decomposed into their literals

Recover the Truth Values

Determining the truth values of the simple propositions for which the propositions in the set would all be true

Open Branch

A path on a truth tree which includes no contradictions

Closed Branch

A path on a truth tree for which a contradiction has been found

v

or

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and

~

not

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if and only if