Philosophy of logic examines the fundamental questions underlying logical reasoning itself: What makes an argument valid? What are logical laws, and why do they hold? Is logic discovered (like mathematical truths) or invented (like formal systems)? How many logics are there, and how do we choose between them? These questions, addressed by both philosophers and mathematicians, underlie debates about the foundations of mathematics, the nature of truth, and the limits of formal reasoning.
When someone makes an argument — "all humans are mortal, Socrates is human, therefore Socrates is mortal" — something makes that argument valid in a way that "all cats are orange, Socrates is orange, therefore Socrates is a cat" is not. But what exactly is that something? Is it a feature of the words used? Of the world? Of how minds work? Of some abstract logical structure existing independently of all these things?
Most people implicitly assume there is one logic — the classical logic taught in mathematics and philosophy courses, which includes the law of excluded middle (every statement is either true or false) and the law of non-contradiction (no statement is both true and false). But alternatives exist and are seriously studied: intuitionistic logic rejects excluded middle; paraconsistent logics allow limited contradiction without explosion; fuzzy logics allow degrees of truth between 0 and 1. Choosing between these is partly a philosophical question — which logic is "correct"? — and partly a pragmatic one — which is most useful for the purposes at hand?
Logicism — the position that mathematics is reducible to logic — was championed by Gottlob Frege (1848–1925) and Bertrand Russell. Frege attempted to derive all of arithmetic from pure logical axioms in his Begriffsschrift (1879) and Grundgesetze der Arithmetik (1893–1903). Russell's discovery of his paradox (1901) showed Frege's specific system was inconsistent; Russell and Whitehead's Principia Mathematica (1910–1913) attempted to salvage the programme using type theory. Most philosophers now consider logicism in its original strong form untenable — Gödel's incompleteness theorems suggest mathematics cannot be fully reduced to any fixed formal system — but the programme's influence on mathematical logic was decisive and permanent.
Formalism, associated primarily with David Hilbert, holds that mathematics is the study of formal symbol-manipulation systems — the symbols need not refer to anything real, and the question of their "meaning" is separate from the question of whether the formal rules are consistent and productive. On this view, logic and mathematics are entirely human constructions. Hilbert's formalist programme — finding a complete, consistent formal system for all of mathematics and proving it consistent by elementary ("finitary") means — was the ambition whose impossibility Gödel demonstrated in 1931.
Intuitionism, developed by L.E.J. Brouwer (1881–1966), holds that mathematics is a mental construction — mathematical objects exist only insofar as they can be explicitly constructed by the human mind, and a mathematical statement is true only if there is a constructive proof of it. Brouwer rejected the law of excluded middle (you cannot assert "P or not-P" without constructively establishing which holds) and consequently rejected all non-constructive proofs, including most proofs by contradiction that merely show a non-constructive object must exist. Intuitionism has a corresponding logic (intuitionistic logic) and is the philosophical precursor to constructive type theory (see the Type Theory entry).
Mathematical Platonism holds that mathematical objects (numbers, sets, functions) exist independently of human minds or formal systems — they are discovered, not invented. On this view, the statement "there are infinitely many prime numbers" was true before anyone proved it, and would be true even if no intelligent beings existed. Most working mathematicians report being de facto Platonists in their everyday practice (they feel they are exploring a pre-existing mathematical landscape), even if they hold more nuanced philosophical views when pressed. Platonism about logic specifically holds that logical laws are features of mind-independent reality, not merely convenient conventions.
Logical pluralism — the view that there is more than one correct logic — has been defended by JC Beall and Greg Restall (2000s). On this view, classical logic, intuitionistic logic, and relevant logic are all "correct" in the sense that each captures a genuine consequence relation, defined by a different notion of what it means for a conclusion to follow necessarily from premises. The choice between them depends on the context and purpose, not on which one is the uniquely "right" one. This contrasts with logical monism (there is exactly one correct logic) and logical nihilism (there is no correct logic at all — an extreme position with few defenders).
A central question in philosophy of logic concerns the logical constants — the expressions like "and," "or," "not," "if...then," "all," "some" that appear in logical arguments. What makes these expressions logical rather than non-logical? One influential answer (associated with Tarski) is that logical constants are the expressions whose extensions are invariant under all permutations of the universe — they don't discriminate among individual objects. Another approach (proof-theoretic) identifies logical constants by their introduction and elimination rules in natural deduction systems. Neither characterisation is universally accepted.
The Liar paradox — "This sentence is false" — poses a fundamental challenge. If the sentence is true, it is false; if false, it is true. Classical logic with a naive truth predicate ("Tr(⌜P⌝) ↔ P" for all sentences P) is inconsistent in the presence of self-reference, as Alfred Tarski showed rigorously (1935). Tarski's solution was to hierarchically stratify truth predicates — truth for sentences of level n is definable at level n+1 but not at level n. More recent approaches include revision theories of truth (Gupta, Herzberger), paraconsistent treatments that allow the liar to be both true and false without trivialising the system, and restricting the comprehension schema that generates the paradox. The liar paradox remains an active area of research in formal philosophy of language and philosophical logic.