‘Necessary truths’ are (or are internally related to) rules of representation and reasoning, which form the network of concepts and transitions between concepts and propositions in terms of which we describe how things are. Although we present them to ourselves as truths, and although we conceive of them as necessarily true and think of them as describing objectively necessary facts, they are not descriptions at all, but expressions of rules for forming descriptions. They are forms of representation.
G.P Baker and P.M.S Hacker, ‘Grammar and Necessity’, in Wittgenstein: Rules, Grammar and Necessity, p. 320
Here we have a nice encapsulation of Wittgenstein’s important insight into the nature of necessity, an insight that goes to the heart of what philosophy is. Already in the Tractatus, Wittgenstein was claiming the only kind of necessity that existed is logical necessity:
A necessity for one thing to happen because another has happened does not exist. There is only logical necessity.
That is, whenever we describe something as necessary, though we may not be aware of it, we are describing a logical connection. For example, say we contend that a fruit cake necessarily contains raisins, that without raisins, it cannot count as a fruitcake. This is asserting a logical or conceptual relationship, not an empirical one. We are defining what a fruitcake is. Necessity and logic go hand in hand in this way. What about if we say that in order to get to the job interview on time, you must (necessarily) catch the 9.20 from Paddington station? Is this a conceptual/logical or an empirical connection? Are there really no circumstances at all under which you could miss this train but still make your appointment? Might you not, for example, call them up and let them know you will be late? Might there not after all be a bus that gets you there on time? Whenever the ‘must’ is not iron-clad, it is an empirical, not a logical, proposition.
Saying necessary connections are logical is, on Wittgenstein’s view, a tautology, for these two terms mean the same. The necessary/contingent distinction is exactly the same as the logical/empirical distinction. He does not eradicate the distinction, as Quine tried to do. Contingent truths are true, but might have been false. Necessary truths could not have been otherwise. Their falsity makes no sense. This means it is excluded from empirical discourse. It has no use.
But there are nuances among necessary/logical truths. There are necessary propositions/necessary truths which belong to the domain of logic, such as the law of the excluded middle (P v ~P), the law of non-contradiction (~(P & ~P)), and other laws (e.g. if all F’s are G, and x is F, then x is G). Their common element is that they are all tautologies. It is the recognition of these as tautologies that consigns them to the domain of logic. They are not descriptions of any state of affairs; they say nothing about the world–they are unconditionally true. They are not meaningless however: they are tautologies. They can be employed in reasoning and the establishment of proofs or contradictions. That is their purpose. That is their use, which is, for Wittgenstein, the chief determinant of meaning.
Arithmetical propositions are similar in that they are not descriptions of the world, nor indeed of a mathematical reality, and say nothing. They are not useless however. They license the transformation of empirical propositions. For example, because it is (necessarily) true that 2 + 2 =4, we are entitled to go from ‘I had two apples and I bought two more’ to ‘I have four apples’. Empirical claims such as this are, of course, contingent; I might have eaten one of the apples. Indeed in any applied example that involves counting or arithmetic, the arrival at the correct answer (e.g. 4 apples) is never inevitable. It is not ‘in the nature of things’, as Frege thought, that if we start with two things then add two more things we will inevitably have four things. There is no such inevitability unless we explicitly say so by our knowledge that nothing else was added, or we apply a ceteris paribus clause. Arithmetical claims are different; they are necessary. This is not because arithmetic is in the nature of things, but because we reserve arithmetic from falsification. Nothing could show 2 + 2 is not 4. This is a social practice, but that does not mean arithmetic is true by convention, or true because we say it is. We do not make arithmetic true. Nothing makes it true. It is unconditionally true. As before, their meaning is closely related to their use–which is only in connection to empirical propositions. If they have no use in this regard, they are like the rules of a theoretical game nobody ever plays.
Then there are what Wittgenstein called ‘grammatical’ propositions which specify other rules for the use of words. For example, ‘red is a colour’ is a grammatical proposition that specifies a rule for the use of the words ‘red’ and ‘colour’. These cause trouble in philosophy because they can look similar to empirical claims; in fact, out of context, they can look indistinguishable from empirical claims. For example, ‘This (pointing to an object) is red’. In one case, this might be an ostensive definition of red: teaching the meaning of ‘red’ to someone by pointing at a sample object. On another occasion, it might be an empirical claim: ‘This car is red. It was supposed to be green.’ One is definitional, conceptual, unconditional, i.e. necessary. To deny it is to say nothing about the world; it is only to dispute the validity of employing a sign for a certain concept. The latter usage is factual, empirical, falsifiable, i.e. contingent.
Logic, mathematics, grammar: these are rules for the use of language and constitute forms of representation. We sometimes call them true, and this is one way in which philosophers sometimes get into trouble and fail to clearly understand their difference from empirical propositions. Empirical propositions are true or false. Logic, mathematics and grammar are not the same. For them to be ‘true’ is just to be a rule, to be accepted as something one has to be in accord with in order to make sense, to be understood. To be false is to be nothing, not even a rule. In chess, there is are rules for the movement of each piece. Moves that do not accord with these are simply not part of chess. So although we might say 2 + 2 = 4 is true, and 2 + 2 = 5 is false, these should not be understood as descriptions of any kind of reality, but just as statements of rules that ought to be followed, like ‘(it is true that) the bishop moves diagonally’.
Outside of these three kinds of rules or necessary truths, there are some propositions which are somewhere between the empirical and the necessary, and these are dealt with in Wittgenstein’s late paper, On Certainty. ‘The world is billions of years old’, ‘water is H2O’. These are examples of proposed rules which have come to be accepted as rules. ‘Water is H2O’ is a rule that partially determine the meanings of the constituent terms. Such examples are cases of empirical propositions that have hardened into rules. Our forms of representation can in this sense change over time. Rules can be thrown out, and new ones adopted.
Philosophers have certainly spent ages pondering the metaphysics of the domain of logic and mathematics. Do numbers exist? Are there objects corresponding to the logical constants? The metaphysics of grammatical truths have also been rife in philosophy. Does ‘redness’ exist? Is the fact that red is a colour a ‘law’ of some special kind? Wittgenstein’s correlation of necessary, logical and conceptual truths sweeps away metaphysics as a mythology resulting from treating some necessary truths as a special kind of super-hard contingent truth. In a quite ordinary or common sense, red things exist, and the concept of red exists, but apart from this, asking whether ‘red’ or ‘redness’ exists is meaningless. It is like asking whether Mozart’s music smells. You can say ‘no’, but it is clearer to say there is no meaning to the question; that nothing counts as music with an odour.
On this account, the medieval debate in Tibetan philosophy concerning the ontological status of universals–which corresponds closely to the medieval Western debate on nominalism and conceptualism–is confused. The only kind of necessity is logical or conceptual necessity. The only point of logical and conceptual truths is their use in providing rules for the use of empirical propositions. They are not descriptions of anything. This is a sweeping away of metaphysics.
Everything empirical is contingent. It is is logical feature of the empirical domain that everything might have been otherwise. This insight itself is logical; it is a rule to clarify the proper employment of ’empirical’ and ‘contingent’. The casualty of this conclusion is metaphysics: nothing counts as a description of a non-empirical or ultimate reality. This is the insight that has important ramifications for traditional madhyamaka philosophy. Our understanding of emptiness, buddha nature, the kayas and wisdom, and all the ontology of Buddhist philosophy, will be misunderstood if thought of as metaphysical in nature, as a super-reality.