Classical Logic and Logical Consequence
Classical logic is the standard formal system of deductive reasoning, and the relation of logical consequence — what follows from what — is the central object it tries to capture.
Definition
Logical consequence is the relation that holds between premises and a conclusion when the truth of the premises guarantees the truth of the conclusion in virtue of logical form; classical logic is the bivalent, truth-functional system in which every sentence is determinately true or false.
Scope
This area covers the standard apparatus of classical deductive logic and the philosophical analysis of the consequence relation it formalizes. It treats both the formal systems (propositional and first-order logic) and the conceptual questions about why an argument is valid: what makes a conclusion follow necessarily from premises, which expressions count as logical, and how formal model-theoretic and proof-theoretic accounts relate to the intuitive notion of 'following from'.
Sub-topics
Core questions
- What does it mean for a conclusion to follow logically from a set of premises?
- Should logical consequence be analyzed model-theoretically (truth-preservation across interpretations) or proof-theoretically (derivability in a deductive system)?
- Which expressions are the 'logical constants', and what marks them off from non-logical vocabulary?
- Is there a single correct logic, or are there multiple equally legitimate consequence relations?
Key concepts
- validity and soundness
- truth-preservation
- logical form
- model-theoretic vs. proof-theoretic consequence
- bivalence and the law of excluded middle
- logical constants
Key theories
- Model-theoretic (Tarskian) consequence
- A conclusion is a logical consequence of premises when there is no interpretation (model) of the non-logical vocabulary on which the premises are true and the conclusion false; validity is truth-preservation across all reinterpretations.
- Logical pluralism
- There is more than one genuine relation of logical consequence, because the notion of a 'case' in which premises hold can be made precise in several admissible ways (e.g., classical, constructive, relevant), each yielding a legitimate logic.
History
The modern conception of classical logic descends from Frege's Begriffsschrift (1879) and was given a precise semantic foundation by Tarski in the 1930s, who defined logical consequence model-theoretically. Quine consolidated the orthodox view of logic as topic-neutral and truth-functional in the mid-twentieth century, while later debates (Etchemendy's critique of Tarski, and logical pluralism) reopened the question of whether the formal definition fully captures the intuitive relation.
Debates
- Does the model-theoretic definition capture genuine consequence?
- Etchemendy argued that Tarski's model-theoretic analysis at best extensionally coincides with, but does not explain, the modal and epistemic features of real logical consequence, since it reduces necessity to mere generalization over interpretations.
- Monism vs. pluralism about logic
- Whether there is exactly one correct logic, or whether classical, intuitionistic, and relevant logics each capture an equally legitimate consequence relation relative to a different but admissible precisification of 'case'.
Key figures
- Alfred Tarski
- W. V. O. Quine
- John Etchemendy
- Gottlob Frege
- JC Beall
- Greg Restall
Related topics
Seminal works
- tarski1936
- quine1986
- etchemendy1990
Frequently asked questions
- What is the difference between validity and truth?
- Validity is a property of arguments: an argument is valid when its conclusion follows from its premises regardless of whether those premises are actually true. Truth is a property of individual sentences. A valid argument can have false premises, and a sound argument is one that is both valid and has true premises.
- Why is it called 'classical' logic?
- The label distinguishes the orthodox bivalent, truth-functional system — in which every statement is either true or false and the law of excluded middle holds — from later non-classical alternatives such as intuitionistic, relevant, or many-valued logics that reject one or more of those assumptions.