Logical Consequence

Logical consequence is the relation between premises and a conclusion that follows from them by logic alone. On the standard truth-preservation account, a conclusion is a logical consequence of its premises when no logically admissible interpretation makes all the premises true and the conclusion false. Two principal approaches make this idea precise: the semantic approach, which examines interpretations or models, and the deductive approach, which examines possible proofs. Whether either approach fully captures the ordinary idea of “following logically” is a philosophical question distinct from its mathematical formalization. (iep.utm.edu)

Consequence, truth, and validity

Logical consequence concerns the connection between statements, not whether those statements are actually true. Consider:

• Every crystal is a plant.
• Every plant is an organism.
• Therefore, every crystal is an organism.

The conclusion follows from the premises despite the first premise being false. By contrast, an argument can have true premises and a true conclusion without the premises logically supporting that conclusion. An argument has logical validity precisely when its conclusion is a logical consequence of its premises; a valid argument with true premises is a sound argument. (logic.stanford.edu)

The relation underlying deductive reasoning is therefore stronger than mere evidential support. Inductive reasoning may make a conclusion plausible while leaving open the possibility that it is false. Deductive consequence, under the truth-preservation account, excludes a true-premise, false-conclusion counterexample. (iep.utm.edu)

Logical consequence must also be distinguished from material implication. The expression $P\to Q$ is a formula within a formal language. The expression $P\models Q$ is a statement about a consequence relation. In classical semantics, $P\to Q$ is true whenever $P$ is false or $Q$ is true, whereas $P\models Q$ requires truth preservation across every admissible interpretation. Thus, a conditional’s truth in one interpretation does not establish entailment. (plato.stanford.edu)

Semantic consequence

The semantic approach belongs to semantics and model theory. Let $\Gamma$ be a set of premises and $\varphi$ a conclusion. The notation

means that every model satisfying all members of $\Gamma$ also satisfies $\varphi$. For sentences, this is expressed as

where $M\models\Gamma$ abbreviates satisfaction of every sentence in $\Gamma$. A model satisfying the premises but not the conclusion is a countermodel. Finding one establishes that the proposed entailment fails. (iep.utm.edu)

In propositional logic, models are truth-value assignments to propositional variables. For example,

No assignment makes both premises true and $Q$ false. A truth table can check every assignment when only finitely many variables are involved. In first-order logic, a model instead supplies a domain of objects and interpretations of names, predicates, and function symbols. (plato.stanford.edu)

The distinction between logical and nonlogical vocabulary is crucial. Logical constants, such as conjunction and universal quantification, retain their stipulated semantic roles while nonlogical expressions vary in interpretation. What counts as logical vocabulary helps determine which consequences hold. (iep.utm.edu)

Deductive consequence

The deductive approach defines consequence through formal proof. Writing

means that $\varphi$ is derivable from $\Gamma$ in a specified deductive system $D$. Such a system determines admissible proof steps through rules of inference and, in some presentations, logical axioms. Standard proof frameworks include natural deduction and sequent calculus. Unlike semantic consequence, derivability directly concerns the syntactic construction of a proof. (iep.utm.edu)

For example, modus ponens permits the derivation of $Q$ from $P\to Q$ and $P$. A proof demonstrates the inferential steps connecting assumptions to a conclusion rather than merely asserting that no countermodel exists. This approach is central to proof theory. (iep.utm.edu)

Two metatheoretic properties connect these approaches:

• Soundness: if $\Gamma\vdash_D\varphi$, then $\Gamma\models\varphi$.
• Completeness: if $\Gamma\models\varphi$, then $\Gamma\vdash_D\varphi$.

Gödel’s completeness theorem establishes the correspondence for classical first-order logic with an appropriate proof calculus. Consequently, semantic entailment and syntactic derivability coincide there, although their definitions remain different. (plato.stanford.edu)

Structural properties

Abstract consequence relations can be studied independently of particular connectives. Classical consequence satisfies three basic structural conditions:

1. Reflexivity: a premise is a consequence of any premise set containing it.

   $$\varphi\in\Gamma\ \Rightarrow\ \Gamma\models\varphi.$$

2. Monotonicity: adding premises cannot invalidate an existing consequence.

   $$\Gamma\models\varphi,\quad\Gamma\subseteq\Delta \ \Rightarrow\ \Delta\models\varphi.$$

3. Cut or transitivity: consequences may be used as intermediate premises. If every member of $\Delta$ follows from $\Gamma$, and $\varphi$ follows from $\Delta$, then $\varphi$ follows from $\Gamma$. These conditions also characterize familiar abstract consequence operators. (plato.stanford.edu)

A further property is compactness: whenever $\Gamma\models\varphi$, some finite subset $\Gamma_0\subseteq\Gamma$ already entails $\varphi$. The compactness theorem gives this property for classical first-order logic. Ordinary finite proofs likewise use only finitely many premises. Full second-order logic is not compact; unrestricted consequences under its standard semantics therefore cannot all be captured by an ordinary finitary proof calculus. Compactness is a property of certain consequence relations, not part of the definition of consequence itself. (iep.utm.edu)

Historical development

Aristotle treated deduction as discourse in which, certain things having been posited, something different follows necessarily because of them. His theory of syllogisms analyzed valid patterns involving quantified statements. This supplied an influential account of necessary inference, although its expressive resources differed substantially from modern logic. (plato.stanford.edu)

A decisive modern formulation was developed by Alfred Tarski in his 1936 paper on logical consequence. Tarski examined the relation’s necessity, formality, and independence from empirical information, and articulated an account in terms of models: a conclusion follows when every model of the premises is a model of the conclusion. His work made the interpretation of nonlogical vocabulary central to the analysis while leaving open questions about the proper boundary of logical vocabulary. (iep.utm.edu)

Consequence in different logics

A consequence claim is evaluated relative to a language, proof system, or semantics. Different logics need not agree about which inferences are valid.

In classical logic, the principle of explosion allows any conclusion to follow from contradictory premises:

Under classical semantics, no interpretation satisfies both premises, so there can be no countermodel to the entailment. Relevance logics challenge forms of inference that allow conclusions without an appropriate connection to their premises. (iep.utm.edu)

Intuitionistic logic does not generally validate double-negation elimination:

It also does not generally validate the law of excluded middle, $P\lor\neg P$. These differences reflect a constructive treatment of logical reasoning rather than a failure to apply classical rules correctly. (plato.stanford.edu)

Nonmonotonic reasoning uses consequence relations in which additional information may withdraw a previously warranted conclusion. For example, a default inference from an object’s being a bird to its flying may be defeated by learning that it is a penguin. Such frameworks formalize defeasible reasoning; they do not preserve the monotonicity characteristic of classical and intuitionistic deduction. (plato.stanford.edu)

Philosophical questions and practical significance

One central question concerns the choice of logical constants. Truth preservation under reinterpretation becomes a determinate test only after specifying which expressions remain fixed. Proposed criteria include invariance under transformations of a domain, but there is no universally accepted boundary separating all logical from nonlogical expressions. (plato.stanford.edu)

Another question concerns logical pluralism: whether more than one consequence relation can legitimately explicate logical following. Pluralists allow multiple appropriate precisifications; opponents maintain that logic should identify a uniquely correct relation. Agreement between models and proofs within a particular system does not by itself resolve this disagreement. (plato.stanford.edu)

Formal accounts also depend on accurately representing an argument. Ordinary-language meanings, context, and suppressed assumptions can affect whether a proposed formalization preserves the original inference. A proof about that formalization establishes a consequence within the selected system, not automatically the adequacy of the translation into it. (plato.stanford.edu)

In mathematical reasoning, consequence identifies what follows from chosen assumptions, while formal derivations provide explicit witnesses to that relationship. Deductive systems thereby support the organization and checking of proofs and have applications in mathematics and computer science. Their use separates the question of whether a conclusion follows from assumptions from the further question of whether those assumptions correctly represent the intended subject matter. (iep.utm.edu)

References

1. Logical Consequenceiep.utm.edu
2. Logical Consequence, Model-Theoretic Conceptionsiep.utm.edu
3. Deductive-Theoretic Conceptions of Logical Consequenceiep.utm.edu
4. Classical Logicplato.stanford.edu
5. Logical Consequenceplato.stanford.edu
6. Logical Consequence (Spring 2024 Edition)plato.stanford.edu
7. Logical Consequence (Summer 2023 Edition)plato.stanford.edu
8. Non-Monotonic Logic (Summer 2003 Edition)plato.stanford.edu
9. Non-monotonic Logic (Winter 2022 Edition)plato.stanford.edu
10. Compactnessiep.utm.edu
11. Intuitionistic Logic (Summer 2024 Edition)plato.stanford.edu