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Converse
The statement q → p, which is NOT logically equivalent to the original conditional p → q.
Inverse
The statement ~p → ~q, which is NOT logically equivalent to the original conditional p → q but is equivalent to the converse.
Modus Tollens
A valid argument form: If p → q and ~q are true, then ~p must be true.
Universal Modus Ponens
A valid argument form combining universal instantiation with modus ponens: ∀x, if P(x) then Q(x); P(a) for a particular a; therefore Q(a).
Universal Modus Tollens
A valid argument form combining universal instantiation with modus tollens: ∀x, if P(x) then Q(x); ~Q(a) for a particular a; therefore ~P(a).
Converse Error
An invalid conclusion resulting from asserting that if p → q and q, then p is true.
Transitivity
A valid argument form: If p → q and q → r, then p → r.
Necessary Condition
"q is a necessary condition for p" means p → q (p cannot be true without q also being true).
Sufficient Condition
"q is a sufficient condition for p" means q → p (if q is true, then p must be true).
Only If
"p only if q" means p → q.
Truth Set
The set of all values that make a predicate true.
Negation of Existential Statement
~(∃x such that P(x)) ≡ ∀x, ~P(x), meaning "there does not exist any x such that P(x)" is equivalent to "for all x, P(x) is false."
Negation of Universal Conditional
~(∀x, if P(x) then Q(x)) ≡ ∃x such that P(x) ∧ ~Q(x)
Conditional Equivalence
p → q ≡ ~p ∨ q (a conditional statement is equivalent to the disjunction of the negation of its hypothesis and its conclusion)
Multiple Quantifier Negation
The negation of ∀x, ∃y such that P(x,y) is ∃x, ∀y such that ~P(x,y)
Proof by Division into Cases
A valid argument form: p ∨ q, p → r, q → r; therefore r.
Contradiction Rule
If ~p leads to a contradiction, then p must be true.
Explicit vs. Implicit Quantification
Explicit quantification uses quantifier symbols; implicit quantification is understood from context (e.g., "If a number is even, then it is divisible by 2" implicitly means "For all numbers, if a number is even, then it is divisible by 2").
Vacuous Truth of Universal Statements
A universal statement ∀x ∈ D, if P(x) then Q(x) is vacuously true if P(x) is false for every x in D.
Order of Operations in Logic
Negation (~), 2) Conjunction (∧) and Disjunction (∨), 3) Conditional (→) and Biconditional (↔)