Formal Logic & Propositional Structure

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Abstractions about the structure and interpretation of logical statements, including categorical propositions, quantification, logical form and interpretation, plus related inference fallacies and encoding devices like pairing functions.

14 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Categorical proposition — A proposition asserting or denying that all or some members of a subject category belong to a predicate category.
• Ground expression — A formal term or formula containing no free variables because every constituent is a constant, function application or fully closed construction.
• Interpretation (logic) — An assignment of denotations to the nonlogical symbols of a formal language over a domain, determining the truth or satisfaction of its formulas.
• Logical equality — A truth-functional connective that is true exactly when its two propositions have the same truth value.
• Logical form — A precise formal-semantic representation of a statement or argument that exposes the structure relevant to validity and interpretation.
• Non-logical symbol — A constant, function or relation symbol in a formal language whose denotation varies with the chosen interpretation or model.
• Pairing function — A bijection that uniquely encodes an ordered pair of natural numbers as one natural number, with generalizations to higher arity or other infinite sets.
• Prenex normal form — A first-order formula form in which all quantifiers occur in one leading prefix followed by a quantifier-free matrix.
• Proof by example — The invalid inference that a universal claim has been proved merely by exhibiting one or several favorable instances.
• Propositional function — An open sentence containing free variables that becomes true or false when admissible values are substituted.
• Subalternation — In the traditional square of opposition, the immediate inference from a universal categorical proposition to its corresponding particular proposition, and contrapositively from the particular's falsity to the universal's falsity.
• Uniqueness quantification — The logical assertion that exactly one object in a domain satisfies a specified predicate.
• Universal quantification — The logical operation asserting that a predicate holds for every member of a declared domain of discourse.
• Vector logic — An algebraic representation of logical truth values and connectives as vectors and matrices acting on a finite-dimensional space.