Logic Glossary

A form of reasoning that seeks the best explanation for a set of observed facts or evidence. Unlike deduction, abduction does not guarantee the truth of its conclusion, but it generates plausible hypotheses.

The 'if' part of a conditional statement 'If P then Q.' P is the antecedent, and its truth or falsehood determines conditions under which the conditional may be evaluated.

A compound statement of the form 'P if and only if Q' that is true when both components share the same truth value (both true or both false) and false otherwise.

A property of a logical system meaning that every semantically valid formula can be derived (proved) within the system. Godel's completeness theorem shows that first-order logic is complete.

A compound statement of the form 'If P then Q' (also called material implication). It is false only when P is true and Q is false; in all other cases it is considered true.

A logical connective (AND) that produces a compound statement which is true only when both component statements are true. Symbolized as P AND Q or P & Q.

The 'then' part of a conditional statement 'If P then Q.' Q is the consequent, representing the result or outcome that is asserted to follow when the antecedent holds.

A compound proposition that is neither a tautology nor a contradiction, meaning it is true under some truth-value assignments and false under others. Most everyday statements are contingencies.

A compound proposition that evaluates to false under every possible assignment of truth values to its component propositions. No interpretation can make a contradiction true.

A property of a logical system or problem indicating that there exists an algorithm that can determine, in a finite number of steps, whether any given statement is provable. Propositional logic is decidable; first-order logic is not.

A logical connective (OR) that produces a compound statement which is true when at least one component statement is true. In logic, this is inclusive OR, meaning it is also true when both components are true.

The set of all objects or entities over which variables in a logical expression can range. The domain must be specified or understood in order to determine the truth of quantified statements.

An error in reasoning that arises from a flaw in the logical structure or form of an argument, making it invalid regardless of its content. Examples include affirming the consequent and denying the antecedent.

A logical rule that justifies the derivation of a conclusion from premises. Standard inference rules include modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism.

An error in reasoning that arises from problems with the content, context, or delivery of an argument rather than its formal structure. Examples include ad hominem, straw man, and appeal to authority.

Two statements are logically equivalent if they have the same truth value in every possible interpretation. Logical equivalence is used to transform and simplify expressions, as in De Morgan's Laws.

An extension of classical logic that includes operators for necessity and possibility. Modal logic is used to reason about what must be true, what could be true, and related concepts in philosophy, linguistics, and computer science.

A proof system in logic that mirrors ordinary human reasoning patterns by introducing and eliminating logical connectives through inference rules, without relying on axioms.

A logical operation that reverses the truth value of a proposition. If P is true, NOT P is false, and vice versa. It is the simplest logical connective, taking only one operand.

A statement template containing one or more variables that becomes a proposition when specific values are substituted for the variables. Predicates express properties or relationships of objects.

A logical symbol that specifies the extent to which a predicate applies to elements in a domain. The two primary quantifiers are the universal quantifier (for all) and the existential quantifier (there exists).

A relationship between premises and a conclusion such that every interpretation making all the premises true also makes the conclusion true. It is the semantic counterpart of syntactic derivability.

A property of a logical system meaning that every formula derivable within the system is semantically valid. In a sound system, proofs can never lead to false conclusions from true premises.

A form of deductive argument with two premises and a conclusion, originally formalized by Aristotle. Categorical syllogisms involve statements about classes of objects using terms like 'all,' 'some,' and 'no.'

A compound proposition that evaluates to true under every possible assignment of truth values to its component propositions. Tautologies represent logical truths that hold universally.

A string of symbols that is grammatically correct according to the formation rules of a formal logical language. Only well-formed formulas can be meaningfully evaluated for truth or falsehood.