Laws of Thought

During the 18th, 19th, and early 20th Century, scholars who saw themselves as carrying on the Aristotelian and Medieval tradition in logic, often pointed to the “laws of thought” as the basis of all logic.  One still encounters this approach in textbook accounts of informal logic.  The usual list of logical laws (or logical first principles) includes three axioms: the law of identity, the law of non-contradiction, and the law of excluded middle.  (Some authors include a law of sufficient reason, that every event or claim must have a sufficient reason or explanation, and so forth.)  It would be a gross simplification to argue that these ideas derive exclusively from Aristotle or to suggest (as some authors seem to imply) that he self-consciously presented a theory uniquely derived from these three laws.  The idea is rather that Aristotle’s theory presupposes these principles and/or that he discusses or alludes to them somewhere in his work.  Traditional logicians did not regard them as abstruse or esoteric doctrines but as manifestly obvious principles that require assent for logical discourse to be possible.

The law of identity could be summarized as the patently unremarkable but seemingly inescapable notion that things must be, of course, identical with themselves.  Expressed symbolically: “A is A,” where A is an individual, a species, or a genus.  Although Aristotle never explicitly enunciates this law, he does observe, in the Metaphysics, that “the fact that a thing is itself is [the only] answer to all such questions as why the man is man, or the musician musical.” This suggests that he does accept, unsurprisingly, the perfectly obvious idea that things are themselves.  If, however, identical things must possess identical attributes, this opens the door to various logical maneuvers.  One can, for example, substitute equivalent terms for one another and, even more portentously, one can arrive at some conception of analogy and induction.  Aristotle writes, “all water is said to be . . .  the same as all water  . . .  because of a certain likeness.” If water is water, then by the law of identity, anything we discover to be water must possess the same water-properties.

Aristotle provides several formulations of the law of non-contradiction, the idea that logically correct propositions cannot affirm and deny the same thing:

“It is impossible for anyone to believe the same thing to be and not be.” 

“The same attribute cannot at the same time belong and not belong to the same subject in the same respect.”

“The most indisputable of all beliefs is that contradictory statements are not at the same time true.”

Symbolically, the law of non-contradiction is sometimes represented as “not (A and not A).”

The law of excluded middle can be summarized as the idea that every proposition must be either true or false, not both and not neither.  In Aristotle’s words, “It is necessary for the affirmation or the negation to be true or false.”  Symbolically, we can represent the law of excluded middle as an exclusive disjunction: “A is true or A is false,” where only one alternative holds.  Because every proposition must be true or false, it does not follow, of course, that we can know if a particular proposition is true or false.

Despite perennial challenges to these so-called laws (by intuitionists, dialetheists, and others), Aristotelians inevitably claim that such counterarguments hinge on some unresolved ambiguity (equivocation), on a conflation of what we know with what is actually the case, on a false or static account of identity, or on some other failure to fully grasp the implications of what one is saying.