Logical validity is a central concept in logic concerning whether a conclusion follows from given premises. In the standard semantic account, an argument is valid if no admissible interpretation makes all its premises true and its conclusion false. The term also applies to a formula that is true under every admissible interpretation. Validity concerns the logical relationship between statements rather than their actual truth, distinguishing deductive reasoning from reasoning that provides only probable support. (plato.stanford.edu)
Validity, truth, and soundness
Truth and validity apply to different things: premises and conclusions may be true or false, while arguments may be valid or invalid. A valid argument need not contain true premises. For example, “All fish are mammals; all mammals are birds; therefore, all fish are birds” has a valid form despite its false premises and conclusion. What validity excludes is the combination of true premises with a false conclusion. (iep.utm.edu)
An argument is sound when it is valid and all its premises are true. Consequently, a sound argument has a true conclusion. Conversely, a true conclusion does not establish validity: an invalid argument may reach a true conclusion accidentally. Soundness therefore combines a logical requirement with a requirement concerning the truth of the starting assumptions. (iep.utm.edu)
Inductive reasoning differs in its intended support. Observations may make a conclusion highly probable without ruling out its falsity. Failure to meet deductive validity does not, by itself, show that an inductive argument lacks evidential value; the standards governing the two kinds of inference differ. (plato.stanford.edu)
Logical form and counterexamples
Validity is commonly understood as truth preservation in virtue of logical form. A standard example is modus ponens:
- If (P), then (Q).
- (P).
- Therefore, (Q).
The letters represent statements, while the conditional structure remains fixed. Under the standard interpretation of the conditional, no substitution produces true premises and a false conclusion. Formal analysis abstracts from particular subject matter while retaining the logical structure responsible for the inference. (plato.stanford.edu)
By contrast, the inference “If (P), then (Q); (Q); therefore (P)” is invalid. Setting (P) false and (Q) true supplies a counterexample: both premises are true, but the conclusion is false. This fallacy, called affirming the consequent, illustrates why a single counterexample disproves validity, whereas numerous favorable examples do not establish it. (logic.stanford.edu)
Semantic definition
In formal semantics, let (\Gamma) be a set of premises and (\varphi) a conclusion. The notation
[ \Gamma \models \varphi ]
means that every interpretation satisfying all members of (\Gamma) also satisfies (\varphi). This relation is called logical consequence or semantic entailment. In model theory, an interpretation that satisfies the premises but not the conclusion is a countermodel. (plato.stanford.edu)
A formula is valid when it holds in every interpretation, written (\models\varphi). In propositional logic, interpretations assign truth values to atomic statements, and a valid formula is a tautology. In first-order logic, interpretations additionally specify a domain and meanings for predicate, function, and constant symbols. For formulas with free variables, validity requires satisfaction under every variable assignment as well. (logical.stanford.edu)
For finitely many premises in classical logic, argument validity is equivalent to validity of
[ (P_1\land\cdots\land P_n)\rightarrow Q. ]
Here material implication is a connective within the language; entailment is a relation between premises and conclusion. A conditional can be true under one interpretation without being valid under all interpretations. (logical.stanford.edu)
Proof and validity
Proof theory approaches inference through derivations governed by rules of inference. The notation (\Gamma\vdash\varphi) states that a formal proof of (\varphi) from (\Gamma) exists in a specified formal system. Unlike semantic entailment, derivability is defined through symbolic rules and syntax. (logical.stanford.edu)
A proof system is sound relative to a semantics if every derivable conclusion is semantically entailed. It is complete if every semantically entailed conclusion is derivable. Gödel’s completeness theorem establishes this correspondence for classical first-order logic with suitable proof calculi. Systems such as natural deduction provide rule-based methods for constructing these derivations. Soundness of a proof system must be distinguished from soundness of an individual argument. (logic.stanford.edu)
Testing validity and computational limits
For a finite propositional argument, a truth table tests every assignment to its atomic statements. The argument is invalid exactly when some row makes every premise true and the conclusion false. Equivalently, (\Gamma\models\varphi) holds when (\Gamma\cup{\neg\varphi}) is unsatisfiable. This connects validity testing with the Boolean satisfiability problem. (logical.stanford.edu)
General first-order validity is undecidable: no algorithm always terminates with a correct yes-or-no answer for every first-order sentence. Nevertheless, validity is semidecidable. Systematic enumeration of proofs eventually discovers a proof of any valid sentence, but may continue indefinitely for an invalid one. Completeness therefore does not imply a terminating decision procedure. (builds.openlogicproject.org)
Dependence on the logical framework
Validity is assessed within a specified logic. Classical logic validates excluded middle, (P\lor\neg P), whereas intuitionistic logic does not validate it unrestrictedly. Classical logic also permits arbitrary conclusions from inconsistent premises; paraconsistent logic rejects this unrestricted explosion principle. These differences concern the accepted consequence relation and interpretation of logical expressions, not simply disagreement about factual premises. (plato.stanford.edu)