Logical consequence

https://en.wikipedia.org/wiki/Logical_consequence

If an argument with premises $\phi_1,\dots,\phi_n$ and conclusion $\psi$ is valid, then $\psi$ is a logical consequence of $\phi_1,\dots,\phi_n$.

We can also define logical consequence directly in terms of truth-value assignments: $\psi$ is a logical consequence of $\phi_1,\dots,\phi_n$ iff every truth-value assignment that makes $\phi_1,\dots,\phi_n$ true also makes $\psi$ true.

Logical consequence (entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when a statement logically follows from one or more statements.

A valid logical argument is an argument in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises.

The philosophical analysis of logical consequence involves the questions:

• In what sense does a conclusion follow from its premises?

• What does it mean for a conclusion to be a consequence of premises?

All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.

Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation.

A statement is said to be a logical consequence of a set of statements, for a given formal language, iff, using only logic, the statement must be true if every statement in the set is true.

Logicians make precise accounts of logical consequence regarding a given language either by constructing a deductive system for that language, or by formal intended semantics for that language.

The Polish logician Alfred Tarski identified 3 features of an adequate characterization of entailment: 1. The logical consequence relation relies on the logical form of the sentences 2. The relation is a priori, i.e. it can be determined with or without regard to empirical evidence (sense experience) 3. The logical consequence relation has a modal component.

The most widely prevailing view on how to best account for logical consequence is to appeal to formality. This is to say that whether statements follow from one another logically depends on the structure or logical form of the statements without regard to the contents of that form.

If you know that Q follows logically from P, then no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a logical consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori. However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a priori property of logical consequence is considered to be independent of formality.

The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs and via models. The study of the syntactic consequence of a logic is called its proof theory whereas the study of its semantic consequence is called its model theory.

A formula A is a syntactic consequence within some formal system Ψ of a set Γ of formulas if there is a formal proof in Ψ of A from the set Γ, Γ ⊢ᵩ A. Syntactic consequence does not depend on any interpretation of the formal system.

A formula A is a semantic consequence within some formal system Ψ of a set of statements Γ, Γ ⊨ᵩ A, iff there is no model in which all members of Γ are true and A is false. In other words, the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true.