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Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other than its logical constants). Thus, logical truths such as "if p, then p" can be considered tautologies. Logical truths are thought to be the simplest case of statements which are analytically true (or in other words, true by definition). All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.

Logical truths are generally considered to be necessarily true. This is to say that they are such that no situation could arise in which they could fail to be true. The view that logical statements are necessarily true is sometimes treated as equivalent to saying that logical truths are true in all possible worlds. However, the question of which statements are necessarily true remains the subject of continued debate.

Treating logical truths, analytic truths, and necessary truths as equivalent, logical truths can be contrasted with facts (which can also be called contingent claims or synthetic claims). Contingent truths are true in this world, but could have turned out otherwise (in other words, they are false in at least one possible world). Logically true propositions such as "If p and q, then p" and "All married people are married" are logical truths because they are true due to their internal structure and not because of any facts of the world (whereas "All married people are happy", even if it were true, could not be true solely in virtue of its logical structure).

Rationalist philosophers have suggested that the existence of logical truths cannot be explained by empiricism, because they hold that it is impossible to account for our knowledge of logical truths on empiricist grounds. Empiricists commonly respond to this objection by arguing that logical truths (which they usually deem to be mere tautologies), are analytic and thus do not purport to describe the world. The latter view was notably defended by the logical positivists in the early 20th century.

Logical truths and analytic truths

Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a second class of analytic statements, typified by "no bachelor is married". The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate. "No bachelor is married" can be turned into "no unmarried man is married" by substituting "unmarried man" for its synonym "bachelor".

In his essay Two Dogmas of Empiricism, the philosopher W. V. O. Quine called into question the distinction between analytic and synthetic statements. It was this second class of analytic statements that caused him to note that the concept of analyticity itself stands in need of clarification, because it seems to depend on the concept of synonymy, which stands in need of clarification. In his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, given a re-evaluation of the truth-values of every other statement in one's complete theory.

Truth values and tautologies

Considering different interpretations of the same statement leads to the notion of truth value. The simplest approach to truth values means that the statement may be "true" in one case, but "false" in another. In one sense of the term tautology, it is any type of formula or proposition which turns out to be true under any possible interpretation of its terms (may also be called a valuation or assignment depending upon the context). This is synonymous to logical truth.

However, the term tautology is also commonly used to refer to what could more specifically be called truth-functional tautologies. Whereas a tautology or logical truth is true solely because of the logical terms it contains in general (e.g. "every", "some", and "is"), a truth-functional tautology is true because of the logical terms it contains which are logical connectives (e.g. "or", "and", and "nor"). Not all logical truths are tautologies of such a kind.

Logical truth and logical constants

Logical constants, including logical connectives and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible if, and only if their conjunction is logically false. One statement logically implies another when it is logically incompatible with the negation of the other. A statement is logically true if, and only if its opposite is logically false. The opposite statements must contradict one another. In this way all logical connectives can be expressed in terms of preserving logical truth. The logical form of a sentence is determined by its semantic or syntactic structure and by the placement of logical constants. Logical constants determine whether a statement is a logical truth when they are combined with a language that limits its meaning. Therefore, until it is determined how to make a distinction between all logical constants regardless of their language, it is impossible to know the complete truth of a statement or argument.

Logical truth and rules of inference

The concept of logical truth is closely connected to the concept of a rule of inference.

Logical truth and logical positivism

Logical positivism was a movement in the early 20th century that tried to reduce the reasoning processes of science to pure logic. Among other things, the logical positivists claimed that any proposition that is not empirically verifiable is neither true nor false, but nonsense.

Non-classical logics

Non-classical logic is the name given to formal systems which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.

See also

• Contradiction
• False (logic)
• Logical truth table, a mathematical table used in logic
• Satisfiability
• Tautology (logic) (for symbolism of logical truth)
• Theorem
• Validity

References

1. Quine, Willard Van Orman, Philosophy of logic
2. MacFarlane, J. (May 16, 2005). Logical Constants. Metaphysics Research Lab, Stanford University.
3. Alfred Ayer, Language, Truth, and Logic
4. Theodore Sider, (2010). Logic for philosophy

External links

• Zalta, Edward N. (ed.). "Logical Truth". Stanford Encyclopedia of Philosophy.
• Logical truth at the Indiana Philosophy Ontology Project
• Logical truth at PhilPapers

v t e Mathematical logic
General	Axiom list Cardinality First-order logic Formal proof Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory
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
Proof theory	Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence (from ZFC) Proof of impossibility Ordinal analysis Reverse mathematics Self-verifying theories
Model theory	Interpretation function of models Model equivalence finite saturated spectrum submodel Non-standard model of arithmetic Diagram elementary Categorical theory Model complete theory Satisfiability Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity
Computability theory	Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory
Related	Abstract logic Algebraic logic Automated theorem proving Category theory Concrete/Abstract category Category of sets History of logic History of mathematical logic timeline Logicism Mathematical object Philosophy of mathematics Supertask
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v t e Common logical connectives
Tautology/True ${\displaystyle \top }$
Alternative denial (NAND gate) ${\displaystyle \uparrow }$ Converse implication ${\displaystyle \leftarrow }$ Implication (IMPLY gate) ${\displaystyle \rightarrow }$ Disjunction (OR gate) ${\displaystyle \lor }$
Negation (NOT gate) ${\displaystyle \neg }$ Exclusive or (XOR gate) ${\displaystyle \not \leftrightarrow }$ Biconditional (XNOR gate) ${\displaystyle \leftrightarrow }$ Statement (Digital buffer)
Joint denial (NOR gate) ${\displaystyle \downarrow }$ Nonimplication (NIMPLY gate) ${\displaystyle \nrightarrow }$ Converse nonimplication ${\displaystyle \nleftarrow }$ Conjunction (AND gate) ${\displaystyle \land }$
Contradiction/False ${\displaystyle \bot }$
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v t e ‌Logical truth ⊤
Functional:	truth value truth function ⊨ tautology
Formal:	theory formal proof theorem
Negation	⊥ false contradiction inconsistency

v t e Common logical symbols
∧  or  & and ∨ or ¬  or  ~ not → implies ⊃ implies, superset ↔  or  ≡ iff | nand ∀ universal quantification ∃ existential quantification ⊤ true, tautology ⊥ false, contradiction ⊢ entails, proves ⊨ entails, therefore ∴ therefore ∵ because
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