Founded by Aristotle as a science, logic establishes the rules that govern reason in its hypothetico-deductive operations.
Paradoxes (insoluble) reveal the limits of any formalism, or rather the impossibility of an absolute formalism.
More precisely
Logic is the art or science of necessary relations between propositions.
It formalizes the conditions under which a conclusion follows validly from given premises.
In the Aristotelian tradition, logic studies three fundamental operations of the mind:
— simple apprehension,
— judgment,
— reasoning,
and is deployed in the analysis of syllogisms, elementary models of deductive inference.
With the development of symbolic logic (Boole, Frege, Russell), logic becomes a mathematical formalism aimed at representing the structure of reasoning with precision.
Contemporary logic explores various extensions, such as:
— modal logics,
— intuitionistic logics,
— paraconsistent logics,
— deontic logics, etc.
Yet all logic rests on undemonstrable axioms, and paradoxes — Russell’s paradox, the Liar paradox, the barber, etc. — reveal the impossibility of constructing a formal system that is both complete and wholly coherent.
Gödel’s incompleteness theorems demonstrate that no sufficiently rich formal system can prove its own consistency.
Thus, although indispensable, logic cannot claim absolute self-sufficiency; it depends on a higher principle that guarantees its validity.
This principle is intelligence, the intuitive faculty of meaning and being.
Logic is only the instrument of discursive reason; it is not the ultimate foundation of truth.
Classical metaphysics teaches that logic — as the science of formal relations — is subordinate to ontology, since the validity of reasoning ultimately depends on what is.
Logical truth thus rests ultimately on ontological truth.
See “Metaphysics of Paradox”.
For further reading
- Aristotle, Organon — Foundation of logic as a science of reasoning.
- Boole, An Investigation of the Laws of Thought — Early algebraic formalization.
- Frege, Begriffsschrift — Birth of modern logic.
- Russell & Whitehead, Principia Mathematica — Logicist program.
- Gödel, Incompleteness Theorems — Intrinsic limits of formal systems.
- Bruno Bérard, Métaphysique du paradoxe, vol. 1 Paradoxes et limites du savoir ;
vol. 2 La connaissance paradoxale (Paris, L’Harmattan, 2019).