##  [Validity](/validity-0) 

 Definition

Property of a sentence (formula closed under free variables) that it is true under every admissible interpretation or valuation in the logic; a sentence that holds in all models is valid (a logical truth).

 

 

 

 

 

 





## Principle

Principle

Validity requires universal truth across the class of models specified by the semantics: a valid sentence cannot be falsified by any admissible structure or assignment.

 

 

 

 

 





## Demonstration

Demonstration

In propositional logic, p ∨ ¬p is valid because every truth assignment makes it true; in first-order logic, ∀x (P(x) → P(x)) is valid in standard semantics because it is true in every possible structure.

 

 

 

 

## Misapplication

Misapplication

Calling a sentence valid because it is derivable in a particular proof system without checking soundness: a proof system may derive sentences that are not semantically valid if it is unsound, or may fail to derive some validities if it is incomplete.

 

 

 

 

 





## Consequence

Consequence

When a sentence is valid it is a logical certainty and can be used as a premise without introducing semantic contingent risk; validity supports abstraction and theorem generalization.

 

 

 

 

## Reversal

Reversal

The opposite of validity is falsifiability or contingency: a contingent sentence is true in some models and false in others; satisfiability is weaker—validity implies that its negation is unsatisfiable.

 

 

 

 

 





## Boundary

Boundary

Validity is relative to the semantics and the signature (language) — adding nonlogical axioms or changing domain conditions (e.g., finite models) can change whether a sentence is valid in that theory.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Validity vs theoremhood: validity is semantic (truth in all models), theoremhood is syntactic (derivable in a proof system); completeness links them but practical differences matter for automated reasoning.

 

 

 

 

 





## Synthesis

Synthesis

Validity designates sentences that are universally true across all admissible models; it is the semantic notion of logical truth that underwrites general theorems and safe inference steps.