PAR230 Logic Test #2
statement form
(1) In categorical logic, a statement form is an arrangement of logical vocabulary and letters that stand for class terms such that a uniform substitution of class terms for the letters results in a statement. (2) In propositional logic, an arrangement of logical operators and statement variables such that a uniform substitution of statements for the variables results in a statement.
substitution instance
(1) In categorical logic, a substitution instance of a statement occurs when a uniform substitution of class terms for the letters results in a statement. A substitution instance of an argument occurs when a uniform substitution of class terms for the letters results in an argument. (2) In propositional logic, a substitution instance of a statement occurs when a uniform substitution of statements for the variables results in a statement. A substitution instance of an argument occurs when a uniform substitution of statements for the variables results in an argument.
argument form
(1) In categorical logic, an argument form is an arrangement of logical vocabulary and letters that stand for class terms such that a uniform substitution of class terms for the letters results in an argument. (2) In propositional logic, an argument form is an arrangement of logical operators and statement variables.
identity relation
A binary relation that holds between a thing and itself.
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 biconditional statement.
inclusive disjunction
A compound statement in which both disjuncts can be true at the same time.
exclusive disjunction
A compound statement in which both disjuncts cannot be true at the same time.
conjunction
A compound statement that has two distinct statements (called conjuncts) connected by the dot symbol.
disjunction
A compound statement that has two distinct statements (called disjuncts) connected by the wedge symbol.
conditional proof (CP)
A method that starts by assuming the antecedent of a conditional statement on a separate line and then proceeds to validly derive the consequent on a separate line.
indirect proof (IP)
A method that starts by assuming the negation of the required statement and then validly deriving a contradiction on a subsequent line.
monadic predicate
A one-place predicate that assigns a characteristic to an individual thing.
statement function
A pattern for a statement. It does not make any universal or particular assertion about anything, and it has no truth value.
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.
universal generalization (UG)
A rule by which we can validly deduce the universal quantification of a statement function from a substitution instance with respect to the name of any arbitrarily selected individual (subject to restrictions).
addition (Add)
A rule of inference (implication rule).
conjunction (Conj)
A rule of inference (implication rule).
constructive dilemma (CD)
A rule of inference (implication rule).
disjunctive syllogism (DS)
A rule of inference (implication rule).
hypothetical syllogism (HS)
A rule of inference (implication rule).
simplification (Simp)
A rule of inference (implication rule).
modus ponens(MP)
A rule of inference (implication rule). A valid argument form (also referred to as affirming the antecedent).
modus tollens (MT)
A rule of inference (implication rule). A valid argument form (also referred to as denying the consequent).
De Morgan (DM)
A rule of inference (replacement rule).
association (Assoc)
A rule of inference (replacement rule).
commutation (Com)
A rule of inference (replacement rule).
distribution (Dist)
A rule of inference (replacement rule).
double negation (DN)
A rule of inference (replacement rule).
exportation (Exp)
A rule of inference (replacement rule).
material equivalence (Equiv)
A rule of inference (replacement rule).
tautology (Taut)
A rule of inference (replacement rule).
transposition (Trans)
A rule of inference (replacement rule).
existential instantiation (EI)
A rule that permits giving a name to a thing that exists. The name can then be represented by a constant.
existential generalization (EG)
A rule that permits the valid introduction of an existential quantifier from either a constant or a variable.
proof
A sequence of steps (also called a deduction or a derivation) in which each step either is 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 and at least one logical operator as components
self-contradiction
A statement that is necessarily false.
tautology
A statement that is necessarily true.
logical truth
A statement that is necessarily true; a tautology.
statement variable
A statement variable can stand for any statement, simple or compound.
well-formed formula
An arrangement of operator symbols such that the resulting symbolic expressions are grammatically correct.
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.
irreflexive relationship
An example of an irreflexive relationship is expressed by the statement "Nothing can be taller than itself."
fallacy of affirming the consequent
An invalid argument form; it is a formal fallacy.
fallacy of denying the antecedent
An invalid argument form; it is a formal fallacy.
definite description
Describes an individual person, place, or thing.
relational predicate
Establishes a connection between individuals.
existential quantifier
Formed by putting a backward E in front of a variable, and then placing them both in parentheses.
symmetrical relationship
Illustrated by the following: If A is married to B, then B is married to A.
transitive relationship
Illustrated by the following: If A is taller than B, and B is taller than C, then A is taller than C.
asymmetrical relationship
Illustrated by the following: If A is the father of B, then B is not the father of A.
intransitive relationship
Illustrated by the following: If A is the mother of B, and B is the mother of C, then A is not the mother of C.
nontransitive relationship
Illustrated by the following: If Kris loves Morgan and Morgan loves Terry, then Kris may or may not love Terry.
conditional statement
In ordinary language, the word "if " typically precedes the antecedent of a conditional, and the statement that follows the word "then" is referred to as the consequent.
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.
principle of replacement
Logically equivalent expressions may replace each other within the context of a proof.
simple statement
One that does not have any other statement or logical operator as a component.
replacement rules
Pairs of logically equivalent statement forms.
predicate symbols
Predicates are the fundamental units in predicate logic. Uppercase letters are used to symbolize the units.
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.
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).
propositional logic
The basic components in propositional logic are statements.
rules of inference
The function of rules of inference is to justify the steps of a proof.
reflexive property
The idea that anything is identical to itself is expressed by the reflexive property.
instantial letter
The letter (either a variable or a constant) that is introduced by universal instantiaton or existential instantiation.
finite universe method
The method of demonstrating invalidity that assumes a universe, containing at least one individual, to show the possibility of true premises and a false conclusion.
main operator
The operator that has the entire well-formed formula in its scope.
order of operations
The order of handling the logical operators within a proposition; it is a step-by-step method of generating a complete truth table.
change of quantifier (CQ)
The rule allows the removal or introduction of negation signs. (The rule is a set of four logical equivalences.)
universal instantiation (UI)
The rule by which we can validly deduce the substitution instance of a statement function from a universally quantified statement.
domain of discourse
The set of individuals over which a quantifier ranges.
scope
The statement or statements that a logical operator connects.
individual constants
The subject of a singular statement is translated using lowercase letters. The lowercase letters act as names of individuals.
individual variables
The three lowercase letters x, y, and z.
truth-functional proposition
The truth value of a compound proposition that uses one of the five logical operators can be determined solely on the basis of the truth value of its components.
tactics
The use of small-scale maneuvers or devices.
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.
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 under the main operator on every line of their respective truth tables.
logically equivalent
Two truth-functional statements that have identical truth tables under the main operator. When this occurs, they are logically equivalent.
implication rules
Valid argument forms that are validly applied only to an entire line.
bound variables
Variables governed by a quantifier.
free variables
Variables that are not governed by any quantifier.
nonreflexive relationship
When a relationship is neither reflexive nor irreflexive.
nonsymmetrical relationship
When a relationship is neither symmetrical nor asymmetrical, then it is nonsymmetrical. Illustrated by the following: If Kris loves Morgan, then Morgan may or may not love Kris.
instantiation
When instantiation is applied to a quantified statement, the quantifier is removed, and every variable that was bound by the quantifier is replaced by the same instantial letter.
sufficient condition
Whenever one event ensures that another event is realized. In other words, the truth of the antecedent guarantees the truth of the consequent.
necessary condition
Whenever one thing is essential, mandatory, or required in order for another thing to be realized. In other words, the falsity of the consequent ensures the falsity of the antecedent.