- Antecedent
- The left-hand operand of a conditional; the X is X \rightarrow Y.
- Axiom
- Something we accept as true without requiring a proof.
- Associative
- Of an operator, meaning the order of operations when several of that operator are applied in a row does not matter. To say some operator \cdot is associative means that (P \cdot Q) \cdot R \equiv P \cdot (Q \cdot R) for all propositions P, Q, and R.
- Biconditional
- See Iff.
- Bi-implication
- See Iff.
- Commutative
- Of an operator, meaning the order of its operands does not matter. To say some operator \cdot is commutative means that P \cdot Q \equiv Q \cdot P for all propositions P and Q.
- Conjunction
- Logical AND (\land).
- Consequent
- The right-hand operand of a conditional; the Y is X \rightarrow Y.
- Contradiction
- A logical expression that is equivalent to FALSE (\bot).
- De Morgan’s laws
Two specific, related logical equivalences; see our list of equivalences for their form.
Because of their related structure, it is not uncommon to refer to both together in the singular (i.e. De Morgan’s law
).
- Disjunction
- Logical OR (\lor).
- Domain
- The possible values a variable could take under a quantifier; for example, if the domain is
all animals
then \forall x \;.\; F(x) means F is true for all animals
.
- Equivalent
- Two logical expressions P and Q are equivalent if and only if the expression P \leftrightarrow Q is a tautology.
- Formula
- see Logical Expression.
- Iff
- A contraction of
if and only if
, a name for the operator \leftrightarrow.
- Logical Expression
- One or more propositions or predicates, combined with operators so that the whole is a predicate or proposition.
- Necessary
Cannot happen without. If A is a necessary condition for B, then we know both
- Without A, no B is possible. \lnot A \rightarrow \lnot B
- If you see B, A must also be. B \rightarrow A
Often used to suggest partial causation or a requirement. Compare Sufficient.
- Predicate
A single word for two related concepts:
- In logic, an incomplete proposition, where one or more component has been replaced by a Variable.
- In programming, a subroutine that (a) has no side-effects and (b) always returns a Boolean value.
- Proposition
- A statement that, by construction, must either be true or false.
- Quantifier
- One of \forall or \exists; some people also include \nexists while others think of that as being shorthand for \lnot \exists.
- Satisfiable
- A satisfiable expression is not a contradiction.
- Sentence
- see Logical Expression.
- Sufficient
Always happens if. If A is a sufficient condition for B, then we know both
- If you see A, B must also be. A \rightarrow B
- If you don’t see B, A can’t be. \lnot B \rightarrow \lnot A
Often used to suggest causation. Compare Necessary.
- Tautology
- A logical expression that is equivalent to TRUE (\top).
- Universe of Discourse
- see Domain.
- Variable
A single word for (at least) three concepts with similar but non-identical meaning:
- In algebra, a place-holder for a single numeric value.
- In logic, a place-holder for a single element from the Domain, generally used with Quantifiers and Predicates.
- In programming, a named region of memory that may take different values at different times.