The laws of thought are an obsolete [1] [2] way to refer to three logical principles: the law of identity (LOI), the law of non-contradiction (LNC), and the law of excluded middle (LEM). [3]
In modern logic these are simply some of the class of tautologies, and are not inference rules. [a] There is no system of logic which uses the three "laws" as axioms, and the interpretations of even just the three "laws" varies widely. [1] [b]
The expression "laws of thought" gained prominence through its use by idealist and conceptualist logicians such as George Boole (1815–64). In fact, Boole named his second logic book An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities (1854).
Modern logicians, in almost unanimous disagreement with Boole, take this expression to be false; none of the above propositions classed under "laws of thought" are explicitly about thought per se, a mental phenomenon studied by psychology, nor do they involve explicit reference to a thinker or knower as would be the case in pragmatics or in epistemology. The distinction between psychology (as a study of mental phenomena) and logic (as a study of valid inference) is widely accepted (see psychologism).
The law of identity can be written symbolically as a = a. The law of non-contradiction can be written ¬(p ∧ ¬p). and the law of excluded middle: p ∨ ¬p. The a refers to singular terms; while the p refers to propositions, or whole sentences. In propositional logic, there is no law of identity.
Gottfried Wilhelm Leibniz claimed that the law of identity, which he expresses as "Everything is what it is", is the first primitive truth of reason which is affirmative, and the law of noncontradiction is the first negative truth (Nouv. Ess. IV, 2, § i), arguing that "the statement that a thing is what it is, is prior to the statement that it is not another thing" (Nouv. Ess. IV, 7, § 9). Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A." [5]
Leibniz formulated three additional principles, either or both of which sometimes were counted as a "law of thought":
The latter two constitute Leibniz's Law, which, unlike the law of identity, explains identity.
Several authors have added the principle of sufficient reason as the fourth "law of thought". [6] William Hamilton and Arthur Schopenhauer were among them. [7] [8] Schopenhauer later believed the law of excluded middle and the principle of sufficient reason were the two laws of thought. [9]
George Boole's 1854 treatise on logic, An Investigation on the Laws of Thought is the most famous logical work which thought logic was about "laws of the mind." [10]
Gottlob Frege's attack on psychologism proved highly influential, especially after convincing Husserl. [10] Before Frege, Bolzano also seemed to be against psychologism in logic.
Russell notes that "for no very good reason, three of these principles have been singled out by tradition under the name of 'Laws of Thought'. [c] And these he lists as follows:
Russell and Whitehead's Principia Mathematica derives over a hundred different formula as theorems, among which are the Law of Excluded Middle ❋1.71, and the Law of Non-Contradiction ❋3.24.
In modern so called classical logic propositions and predicate expressions are two-valued, with either the truth value "truth" or "falsity" but not both. [12] The law of excluded middle therefore says each proposition has at least one truth value. There are no truth gaps. The law of non contradiction says no proposition has more than one truth value. There are no truth gluts.
The law of non-contradiction and the law of excluded middle create a dichotomy in a so-called logical space, the points in which are all the consistent combinations of propositions. Each combination would contain exactly one member of each pair of contradictory propositions, so the space would have two parts which are mutually exclusive and jointly exhaustive. The law of non-contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle is an expression of its jointly exhaustive aspect.
There are also many valued logics, with gaps or gluts, or both. Intuitionistic logic denies the law of excluded middle. Paraconsistent logic tolerates contradiction.
'Intuitionistic logic', sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination.
'Paraconsistent logic' refers to so-called contradiction-tolerant logical systems in which a contradiction does not necessarily result in trivialism. In other words, the principle of explosion is not valid in such logics.
Some (namely the dialetheists) argue that the law of non-contradiction is false. They are motivated by certain paradoxes which seem to imply a limit of the law of non-contradiction, such as the liar paradox. In order to avoid a trivial logical system and still allow certain contradictions to be true, dialetheists will employ a paraconsistent logic of some kind.