Logic terms 2

simple statement

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

inclusive disjunction

An _________ ___________ is where both disjuncts can be true at the same time

Symmetrical translation

If A is married to to B, then B is married to A. (x) (y) (Mxy > Myx)

transitive translation

If A is taller than B, and B is taller than C, then A is taller than C. (x) (y) (z) [(Txy . Tyz) > Txz]

intransitive translation

If A is the mother of B, and B is the mother of C, then A is not the mother of C. (x) (y) (z) [(Mxy . Myz) > ~Mxz]

nontransitive translation

If A loves B and B loves C, then A may or may not love C

predicate logic

Integrates many of the features of categorical and propositional logic. It combines the symbols associated with propositional logic with special symbols that are used to translate predicates

Rule 4 for WFFs

Parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement

Order of operations

The order of handling the logical operators within a proposition. Step-by-step method of generating a truth table. Start with operator that affects least amount of statements, end with main operator

Rule 3 for WFFs

The tilde cannot, by itself, go between two statements

Truth function

The truth value of 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 _______ ________

Statement variable

a _________ __________ can stand for any statement, simple or complex

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 _______________ statement. True when both statements have the same truth value

conjunction

a compound statement that has two distinct statements (called conjuncts) connected by the dot symbol (.) "A and B." Only true when both conjuncts are true

disjunction

a compound statement that has two distinct statements (called disjuncts) connected by the wedge symbol (v). "A or B." Only false when both disjuncts are false

conditional proof

a method that starts by assuming the antecedent of a conditional statement on a separate line an then proceeds to validly derive the consequent on a separate line

indirect proof

a method that starts by assuming the negation of the required statement (the conclusion) and then validly deriving a contradiction on a subsequent line

monadic predicates

a one-place predicate that assigns a characteristic to an individual thing. ex. - Gx

statement form

a pattern of statement variables and logical operators

natural deduction

a proof procedure by which the conclusion of an argument is validly derived from the premises through the use of rules of inference

constructive dilemma (CD)

a rule of inference - implication rule (p > q) . (r > s) p v r ------------- q v s

addition (add)

a rule of inference - implication rule p --- p v q

conjunction (conj)

a rule of inference - implication rule p q ----- p . q

simplification (Simp)

a rule of inference - implication rule p . q ----- p

hypothetical syllogism (HS)

a rule of inference - implication rule p > q q > r ------ p > r

Disjunctive syllogism (DS)

a rule of inference - implication rule p v q ~p ------ q

exportation (exp)

a rule of inference - replacement rule (p . q) > r :: p > (q > r)

material implication (impl)

a rule of inference - replacement rule (p > q) :: (~p v q)

transposition (trans)

a rule of inference - replacement rule (p > q) :: (~q > ~p)

material equivalence (equiv)

a rule of inference - replacement rule (p _= q) :: (p > q) . (q > p) (p _= q) :: (p . q) v (~p . ~q)

commutation (comm)

a rule of inference - replacement rule (p v q) :: (q v p) (p . q) :: (q . p)

distribution (dist)

a rule of inference - replacement rule [p . (q v r)] :: [(p . q) v (p . r)] [p v (q . r)] :: [(p v q) . (p v r)]

association (assoc)

a rule of inference - replacement rule [p v (q v r)] :: [(p v q) v r] [p . (q . r)] :: [(p . q) . r]

Tautology (taut)

a rule of inference - replacement rule p :: (p v p) p :: (p . p)

double negation (DN)

a rule of inference - replacement rule p :: ~~p

De Morgan (DM)

a rule of inference - replacement rule ~(p . q) :: (~p v ~q) or ~(p v q) :: (~p . ~q)

proof

a sequence of steps in which each step is either a premise or follows from earlier steps in the sequence according to the rules of inference

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)

modus tollens (MT)

a valid argument form. a rule of inference - implication rule p > q ~q ------ ~p

Modus ponens (MP)

a valid argument form. rule of inference -implication rule p > q p ------- q

exclusive disjunction

an ________ __________ is where both disjuncts cannot be true at the same time

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.

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

statement function

an expression that does not make any universal or particular assertion about anything, and therefore has no truth value. _________ ________ are simply patterns for a statement

fallacy of affirming the consequent

an invalid argument form. a formal fallacy p > q q ------- p

fallacy of denying the antecedent

an invalid argument form. a formal fallacy p > q ~p ------- ~q

well-formed formulas (WFFs)

compound statement forms that are grammatically correct

relational predicate

establishes a connection between individuals in predicate logic. ex. - Gxy

existential quantifier

formed by putting a backward E in front of an variable, and then placing them both in parentheses. (particular)

predicate

fundamental unit of predicate logic

asymmetrical translation

if A is the father of B, then B is not the father of A. (x) (y) (Fxy > ~Fyx)

nonsymmetrical translation

if A loves B, then B may or may not love A.

conditional

in ordinary language, the word "if" typically precedes the antecedent (first statement) of a _____________. The other statement is called the consequent. The horseshoe symbol (>) is used to translate a __________ statement. Only false when the antecedent is true, and the consequent is false

Principle of replacement

logically equivalent expressions may replace each other within the context of a proof.

replacement rules

pairs of logically equivalent statement forms. Can be used on entire lines, or segments of lines

strategy

referring to a greater, overall goal

justification

refers to the rule of inference that is applied to every validly derived step in a proof

logical operators

special symbols that can be used as part of ordinary language statement translations ( . , v , > , ~ , etc.)

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 and sometimes false

propositional logic

the basic components in _______________ ______ are statements

Rule 1 for WFFs

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

rules of inference

the function of _____ __ __________ is to justify the steps of a proof

main operator

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

Domain of discourse

the set of individuals over which a quantifier ranges

individual constants

the subject of a singular statement is translated using lowercase letters. Can represent names

universal quantifiers

the symbol used to capture the idea that universal statements assert something about every member of the subject class. (x)

Rule 2 for WFFs

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

tactics

the use of small-scale maneuvers or device

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 __________. Symbolized as a tilde (~). Always has the opposite truth value of the statement it is negating

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 both 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 _________ _____________

predicate symbols

uppercase letters are used to symbolize predicates (X,Y,Z)

implication rules

valid argument forms that are validly applied only to an entire line. Order of premises doesn't matter. Do not apply to segments of a line, but only the whole line

bound variables

variables governed by a quantifier

free variables

variables that are not governed by any quantifier

individual variables

x, y, z. universal quantifier is formed by placing one of these inside parentheses.