Those of us who believe in the truth typically accept contingent truths or, to use possible worlds semantics, truths that are true in some worlds and false in others. For example, the sentence ‘Dwight Goodyear was born in 1970’ expresses a true proposition that is contingently true since we can easily imagine it being false in possible worlds where I was never born. But necessary truths, or truths that are true in all possible worlds, are a bit more controversial. To be sure, it is intuitively clear to many people, including me, that certain propositions like ‘2+2=4’, ‘A or not A’, and ‘A=A’ are necessarily true propositions that are true in all possible situations: they are true in all possible worlds. But what if someone lacks this intuition? What if someone argues that even so-called necessary truths are contingent? What if someone embraces universal possibilism or the view that there are no necessary truths? Here are few ways one might respond in defense of necessary truths:
(1) Axiom 5 of modal logic (logic that deals with contingency, necessity, possibility, or impossibility), an axiom included in the widely accepted system of modal logic S5, states that, as Raymond Bradley and Norman Swartz nicely put it in prose, “if a proposition P is possibly true, that is, if P is true in at least one possible world then the proposition that it is true in at least one possible world will be true in all possible worlds and thus necessary” (see their book Possible Worlds, pp. 223-224). For example, the sentence ‘Dwight Goodyear was born in 1970’ expresses, as we have seen, a proposition that is contingently true (true in some worlds and false in others) and thus possible since something is possible if it is true in at least one world. But once we have this possibility in place then, according to S5, it is logically equivalent to the following necessary truth that holds in all possible worlds:
‘The proposition ‘Dwight Goodyear was born in 1970′ is true in at least one world’.
So according to S5 all possibility claims, not just necessity claims, turn out to be necessary. If this is the case then we would have to accept necessary truths if we accept possibly true and contingently true propositions. Indeed, universal possibilism, the view that there are no necessary truths, would entail that it is not possible that a proposition, p, is necessarily true. But then from the interdefinability of possibility and necessity (that if p is not possible then p is necessary) it is necessary that p not be necessarily true. And this would, of course, establish a necessary truth which contradicts universal possibilism.
(2) We can also present a transcendental argument (an argument that gains its strength by revealing the conditions for the possibility of something we want to understand) and maintain that without the concept of absolute necessity we can’t adequately explain our ability to grasp forms of relative necessity. Examples of relative necessity: (a) It is not physically possible to go faster than the speed of light given the laws of nature as we currently understand them. But these laws, we can easily imagine, could be different in a different possible world or indeed our own. (b) It is not biologically possible for me, given my genetic makeup, to naturally sprout some wings and fly. But we can easily imagine a different genetic makeup for me in a different possible world. (c) It is not possible for someone living in New York to legally have many wives given the laws of the state. But we can easily imagine the laws being changed. In each of these cases there is a form of necessity in place which, upon consideration, is revealed to be relative or contingent: relative to certain conditions such as laws of nature, biology, and the state. Now, most of us also want to distinguish forms of necessity that are not contingent: the laws of arithmetic and logic for example. These laws just seem categorically different from the aforementioned forms of contingent necessity since they seem to hold in all possible worlds. As we think through these various examples, we see that the condition for the possibility of thinking about their diversity is the notion of absolute necessity: without this notion we wouldn’t be able to distinguish the types of necessity that make sense of our experience. Joseph Melia, in his book Modality, explains: “There are indeed many forms of necessity. But many can be defined in terms of what is (absolutely) compatible with a certain set of facts, which we have arbitrarily decided to keep fixed. Throughout, a particular kind of necessity – absolute necessity – is needed to define these various relative modalities. Moreover, the mere notion of relative modality is not enough to capture a distinction we want to capture: a distinction in kind between the laws of logic and the laws of biology; a distinction in kind between the laws of mathematics and the laws of physics – a distinction between what is absolutely necessary and what is absolutely contingent” (pp. 17-18).
(3) A third reason for believing in necessarily true propositions is this: if we don’t postulate them then, as Alvin Plantinga points out, certain counterfactual claims (a counterfactual is a subjunctive conditional containing an if-clause which is contrary to fact) cannot be coherently described such as “if there had been no human beings, it would have been true that there are no human beings” (see Plantinga, Warrant and Proper Function, pp. 117-120). After all, if propositions aren’t necessary – if they are just contingently existing things that arose when humans began to entertain and express them – then there would be no propositions if there were no humans. And if there were no humans then the proposition ‘there are no human beings’ would not be true (since there would be no propositions at all to be true or false). But isn’t that absurd? Don’t we want to avoid being committed to, as Plantinga puts it, “possibly, (there are no people and it is not true that there are no people)”? (118)
(4) Finally, we could argue that the astonishing advances in the applied sciences (engineering, aeronautics, and so on), advances that would not be possible without the application of the seemingly necessary truths of mathematics, become impossible to explain without necessary truth. Bradley and Swartz put it nicely: “[I]f these necessary truths are merely the result of arbitrary human conventions for the use of mathematical symbols, all this becomes a seeming miracle. Why should the world conform so felicitiously to the consequences of our linguistic stipulations?” They argue that it is precisely by embracing necessary truths as propositions which are true in all possible worlds, including worlds without humans, that we can provide an adequate explanation of the puzzling application of these truths in our world: “Necessary truths, such as those of mathematics, apply to the world because they are true in all possible worlds; and since the actual world is a possible world it follows that they are true in (i.e., apply to) the actual world” (Possible Worlds, p. 61).
For a related post on why truth cannot be contingent on humans go here.