Truth-value semantics

In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus,^[1] H. Leblanc, and J. Michael Dunn and Nuel Belnap.^[2] It is also called the substitution interpretation (of the quantifiers) or substitutional quantification.

The idea of these semantics is that a universal (respectively, existential) quantifier may be read as a conjunction (respectively, disjunction) of formulas in which constants replace the variables in the scope of the quantifier. For example, $\forall xPx$ may be read ( $Pa\land Pb\land Pc\land \dots$ ) where $a,b,c$ are individual constants replacing all occurrences of $x$ in $Px$ .

The main difference between truth-value semantics and the standard semantics for predicate logic is that there are no domains for truth-value semantics. Only the truth clauses for atomic and for quantificational formulas differ from those of the standard semantics. Whereas in standard semantics atomic formulas like $Pb$ or $Rca$ are true if and only if (the referent of) $b$ is a member of the extension of the predicate $P$ , respectively, if and only if the pair $(c,a)$ is a member of the extension of $R$ , in truth-value semantics the truth-values of atomic formulas are basic. A universal (existential) formula is true if and only if all (some) ground substitution instances of the unquantified subformula are true. Compare this with the standard semantics, which says that a universal (existential) formula is true if and only if for all (some) members of the domain, the formula holds for all (some) of them; for example, $\forall xA$ is true (under an interpretation) if and only if for all $k$ in the domain $D$ , $A(k/x)$ is true (where $A(k/x)$ is the result of substituting $k$ for all occurrences of $x$ in $A$ ). (Here we are assuming that constants are names for themselves—i.e. they are also members of the domain.)

Truth-value semantics is not without its problems. First, the strong completeness theorem and compactness fail. To see this consider the set $\{F(1),F(2),\dots \}$ . Clearly the formula $\forall xF(x)$ is a logical consequence of the set, but it is not a consequence of any finite subset of it (and hence it is not deducible from it). It follows immediately that both compactness and the strong completeness theorem fail for truth-value semantics. This is rectified by a modified definition of logical consequence as given in Dunn and Belnap 1968.^[2]

Another problem occurs in free logic. Consider a language with one individual constant $c$ that is nondesignating and a predicate $F$ standing for 'does not exist'. Then $\exists xFx$ is false even though a substitution instance (in fact every such instance under this interpretation) of it is true. To solve this problem we simply add the proviso that an existentially quantified statement is true under an interpretation for at least one substitution instance in which the constant designates something that exists.

References

^ Marcus, Ruth Barcan (1962). "Interpreting quantification". Inquiry. 5 (1–4): 252–259. doi:10.1080/00201746208601353. ISSN 0020-174X.
^ ^a ^b Dunn, J. Michael; Belnap, Nuel D. (1968). "The Substitution Interpretation of the Quantifiers". Noûs. 2 (2): 177. CiteSeerX 10.1.1.148.1804. doi:10.2307/2214704. ISSN 0029-4624. JSTOR 2214704.

Mathematical logic

General

Theorems (list)
and paradoxes

Gödel's completeness and incompleteness theorems
Tarski's undefinability
Banach–Tarski paradox
Cantor's theorem, paradox and diagonal argument
Compactness
Halting problem
Lindström's
Löwenheim–Skolem
Russell's paradox

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types of sets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps and cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Set theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive

Formal systems (list),
language and syntax

Alphabet Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant non-logical variable Term Theory list
Example axiomatic systems (list)	of arithmetic: Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry: Euclidean: Elements Hilbert's Tarski's non-Euclidean Principia Mathematica