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.