The Laws of Thought can be set out as follows:
(1) the Law of IdentityIt is also generally believed that laws (1) and (3) are tautologies.
This can be stated simply as the idea that any entity x is identical with itself (or x = x). With respect to propositions, it means that, if a proposition p is true, then it is true (or p → p).
A violation of the Law of Identity is the informal logical fallacy of equivocation: changing the meaning of terms in an argument or using the same term in different senses.
(2) Law of Noncontradiction
This can be stated as the idea that one cannot assert as true a proposition p and its negation at the same time (or p ∧ ¬p is false). Such a statement is self-contradictory. That is to say, the proposition p and its negation (¬p) asserted as true at the same time are mutually exclusive.
The source of the Law of Noncontradiction is Aristotle’s Metaphysics, although he gives three versions of the law, in (1) an ontological, (2) psychological and (3) logical sense.
(3) the Law of Excluded Middle
This states that every meaningful proposition is either true or false. There is no third option. Put another way, either the proposition p is true or its negation (¬p) is true. Symbolically, this means that p ∨ ¬p must be true.
There are of course many other tautologies important for deductive logic, such as De Morgan’s theorems, but the laws of thought remain fundamental.
But, as noted above, what is their epistemological status? Are they simply analytic a priori? And even if we can interpret them in that sense when considered only as pure tautologies, could they also be given an empirical synthetic a posteriori status when asserted as true of the real world? (just as pure geometry and applied geometry each have a different epistemological status.)
For it seems that the law of identity also expresses, or is bound up with, a deep metaphysical truth about existence: existence and identity are closely related concepts, as was stressed by Gottlob Frege and Quine (Scruton 1994: 143).
Indeed, Bertrand Russell gave the laws of logic an ontological interpretation:
(1) law of identity: ‘whatever is, is.’Russell thought that these were “self evident logical principles,” but the possibility that the law of excluded middle may not apply to certain quantum mechanical phenomena (Quine 1986: 86–87) should alert us to the idea that the laws of thought are ultimately empirical statements derived from human experience of the macroscopic world. (And this does not falsify the law of excluded middle with respect to the macroscopic world at all, but confines or limits it as a truth to that domain.)
(2) the law of contradiction: ‘nothing can both be and not be.’
(3) the law of excluded middle: ‘everything must either be or not be.’
That is to say, surely it is not unreasonable to assert that “every thing that exists is identical with itself” is, in the end, an empirical observation of the fundamental nature of the world of human experience. Of course, we cannot even conceive of how it could be false, but then human beings most probably could not conceive of the counter-intuitive truths of quantum mechanics until the 20th century.
Even in the realm of the laws of thought, when these laws are asserted as truths of the real world, they arguably must be considered empirical.
“Law of Thought,” Wikipedia
“Law of Noncontradiction,” Wikipedia
“Law of Identity,” Wikipedia
“Law of Excluded Middle,” Wikipedia
Scruton, R. 1994. Modern Philosophy: An Introduction and Survey. Penguin Books, London.
Quine, W. V. 1986. Philosophy of Logic (2nd edn.). Harvard University Press, Cambridge, Mass.