Some Remarks on Logical Form

1 Some Remarks on Logical Form Hartley Slater Wittgenstein’s discussion of the colour exclusion problem in ‘Some Remarks on Logical Form’ led to a series of papers by Lewy, von Wright and others on Entailment. Central to this stream of thought was concern with C.I. Lewis’ discussion of the so-called ‘Paradoxes of Strict Implication’. Lewy and the rest were concerned to avoid Lewis’ entailments to necessary consequents, and from impossible antecedents, for instance. Later, Anderson and Belnap took up the issue of Entailment in a different way. Indeed, it was not until Anderson and Belnap’s work that a thorough study of the distinctive grammar of remarks like ‘That something is red all over entails that it is not green’ was undertaken. Nevertheless followers of this second train of thought were happy to take ‘entails’ as a sort of propositional connective, although naturally everyone realised that the plain hook of standard propositional logic would not suffice to express the relation, because of the so-called ‘Paradoxes of Material Implication’. The problem for both of these traditions, however, was that they had no way of formalising ‘that’-clauses as referential phrases, allowing the relation to be formalised in predicate logic terms. Indeed that would have brought up for them difficult philosophical questions to de with Realism. But now that we are moving out of the Empiricism of those times it is easier for us to see just what it was that Wittgenstein had in mind. When Wittgenstein returned to philosophy in the late 1920s he realised that the propositional logic in his Tractatus was incomplete. In his paper ‘Some Remarks on Logical Form’ he pointed out, amongst other things, that contrarieties between colour descriptions could not form any part of his earlier propositional logic. That something is red at some place entails that it is not green at that place, so not all elementary propositions could be independent, as he once thought. The point he was making in his paper he put in terms of a seemingly needed modification of the Tractatus’ Truth Tables. These allowed for four possibilities with two elementary propositions: TT, TF, FT, FF. But in the case of something’s being red and that thing’s being blue the first of these possibilities is ruled out, he said, leaving merely: TF, FT FF. Few later thinkers have taken up the matter in these terms, but Len Goddard was an exception. For Goddard tried to introduce a ‘contrariety operator’, ‘c’, akin to the negation operator ‘it is not the case that’, ‘¬’. The above abbreviated truth possibilities ‘TF, FT, FF’ were then supposedly appropriate for the pair ‘φ, cφ’, just as the even more abbreviated possibilities ‘TF, FT’ were appropriate for the pair ‘φ, ¬φ’. With ‘φ’ as ‘it is red’ Goddard made out that ‘it is blue’ (with the same referent for ‘it’) was ‘cφ’, displaying the inherent contrariety of the two expressions. There is one main problem with this proposal of Goddard’s however. For contrariety is a relation, and so needs nominal expressions on either side of it. And the required nominal expressions are not mentioned sentences, since Wittgenstein’s novel truth was a well-known synthetic a priori truth. So what are related are not sentences, but what is expressed by those sentences, which shows that the verb ‘is contrary to’ is to be completed using ‘that’-clauses and similar referential phrases. The facts are that that something is red is contrary to its being blue, and to its being green. But the following generations of philosophers, starting with the Logical Empiricists, not only failed to have ‘that’-clauses and the like in their preferred 2 languages, they expressly disowned synthetic a priori truths, being only interested in the analytic a priori (see, e.g. Ayer 1946, 77f). It is the historical and philosophical depth of the issue that must be hard to grasp, by anyone immersed in the mainline formal tradition. Wasn’t modern logic’s language supposed to be able to express all that was required for science? So how can anything have been missed out? Most puzzling, given the increased use of mathematics in the logic of the period, is the lack of attention to certain identities found in mathematics itself, such as ‘Goldbach’s Conjecture is that every even integer greater than 2 can be expressed as the sum of two primes’ and ‘Pythagoras’ Theorem is that the square on the hypotenuse of a right angled, Euclidean triangle is equal to the sum of the squares on the other two sides’. These have the form ‘x = λφ’, and so the introduction of some symbol like ‘λ’ to formalise the nominaliser ‘that’ is essential to gain a formal expression of them. This lambda symbolism is an extension of that in Church’s Lambda Calculus, with a more general lambda expression like ‘λxFx’ being standardly read as ‘the property of being F’. But Quine, for instance, would have as little to do with properties as with propositions. The consequences of ignoring ‘that’-clauses have multiplied in the formal logic tradition since the 1930s. Anderson and Belnap, for instance, could themselves, in the end, only follow the Logical Positivist tradition in their otherwise very thorough grammatical appendix to volume 1 of Entailment, although they knew that they were treading on shaky ground. For they wanted to make it philosophically respectable, they said, to ‘confuse’ two things. Their attempt was to ‘make it philosophically respectable to “confuse” implication or entailment with the conditional’ (Anderson and Belnap 1975, 470). But there are other cases they needed to consider besides the ones they did which show that they definitely created confusion, thereby failing to make the matter ‘philosophically respectable.’ Anderson and Belnap were exceptionably good on the grammar of ‘that’-clauses. These are referential phrases, Anderson and Belnap admitted, and they refer to propositions. They are also the standard complements on both sides of verbs like ‘entails’ and ‘implies’. So Anderson and Belnap were quite clear that ‘entails’ expresses a relation, and at one point they are also quite clear that the conditional ‘if … then …’ does not. Nevertheless, in their conclusion, they think that it is excusable to ‘confuse’ the two forms as above, and write an entailment in the form ‘φ → ψ’ with ‘→’ some supposedly improved form of ‘if … then …’. The improvement intended was on the standard formulation of the conditional by means of hook, which was therefore deemed to be inappropriate. The first problem for Anderson and Belnap’s grammar arises because not only ‘that’clauses can flank verbs like ‘entails’. There are also propositional names and definite descriptions, as in the cases of propositional identity before. Anderson and Belnap were not unaware of this possibility, and gave one instance – ‘Euclid’s First Proposition’ (Anderson and Belnap 1975, 478). However, in their conclusion this insight gets lost and there they only consider ‘that’-clause complements. But evidently there are not just forms like ‘that φ entails that ψ’ to consider, there are also ones like ‘x entails that ψ’ (e.g. ‘Pythagoras’ Theorem entails that a 3-4-5 triangle on a plane is right angled’) and even ‘x entails y’ (e.g. ‘The Axiom of Determinacy entails the Continuum Hypothesis’). The problem is that there is no way that these latter forms are of the ‘if φ then ψ’ form, since there is nothing in them corresponding to the ‘φ’ (or the ‘ψ’ in the second case). So the implication and conditional forms are distinct, and the implication form must be symbolised in a relational way, i.e. not as a conditional but as ‘xEy’. That will then allow, as special cases, the ‘x’ or ‘y’ to be ‘that’- 3 clauses, permitting the general relational form ‘xEy’ to be the one Anderson and Belnap were pre-occupied with, namely ‘λφEλψ’. Of course a formal connection can then be made between such entailments and conditionals using the propositional truth scheme: Tλφ ≡ φ. For if it is true that φ, and that φ entails that ψ, then it is true that ψ, and this can be formalised (Tλφ & λφEλψ) ⊃ Tλψ, Hence λφEλψ ⊃ (Tλφ ⊃ Tλψ), and so λφEλψ ⊃ (φ ⊃ ψ). More generally xEy ⊃ (Tx ⊃ Ty). But one cannot improve upon this conditional to make the two sides equivalent, since the socalled ‘Paradoxes of Material Implication’ are against it. For it is well known that we can say such things as ‘There is jam in the cupboard, if you want some’, and also ‘If there is jam in the cupboard, then I am a Dutchman’, and in neither case is there any claim that the consequent is implicated in the antecedent. Examples like these have been thought to be anomalies because ‘If … then …’ was expected to express an implication. But instead they simply and directly show, against Anderson and Belnap, and indeed the whole ‘Relevance Logic’ tradition that followed them (see, e.g. Read 1988), that ‘If … then …’ does not express an implication. But now a second, and even more severe problem arises for this latter tradition. For, as Anderson and Belnap so clearly point out, not only are ‘that’-clauses referring phrases, what they refer to are propositions. It follows, therefore, that entailments are relations between propositions, i.e. that they are not relations between sentences, but between what sentences express. So there is no way that some relation between sentences, like the popular ‘sharing a variable’, can be involved. The idea has been that such standard propositional truths as ‘(φ & ¬φ) ⊃ ψ’ and ‘φ ⊃ (ψ v ¬ψ)’ cannot formalise entailments or implications because the antecedent and consequent in each case have nothing in common. So they have no relevance to one another. In the classic case that Wittgenstein considered in 1929 neither ‘it is red‘ nor ‘it is green’ is a molecular sentence; they are both elementary sentences, as Wittgenstein was quick to point out. So it is not a verbal connection, but a connection of meaning that has to be involved. The move throws a quite different light on the so-called ‘Paradoxes of Strict Implication’ that C. I. Lewis faced when he tried to defend his account of entailments as expressed by means of fishhook, i.e. by what has been called ‘strict implication’. Lewis wanted to say that if not the material conditional then at least the necessary truth of such a conditional expressed an entailment, and gave a couple of independent proofs that he said showed that this was the case. The first moved from ‘φ & ¬φ’ to ‘φ’ and ‘¬φ’; then from ‘φ’ to ‘φ v ψ’, and finally from ‘¬φ’ with ‘φ v ψ’ to ‘ψ’, establishing that it is necessary that if φ & ¬φ then ψ. The second moved from ‘φ’ to ‘φ & (ψ v ¬ψ)’ via ‘(φ & ψ) v (φ &¬ψ)’, and then finally to ‘ψ v ¬ψ’, showing that necessarily if φ then ψ v ¬ψ. Relevance theorists have objected, as above, that both these theses involve irrelevance, and so have tried to break the chains of steps in Lewis’ arguments, in an attempt to ensure that the given conditionals do not follow. For instance in the first case it has come most widely to be believed that there is some problem with Disjunctive Syllogism, i.e. the move that gets one from ‘¬φ’ with ‘φ v ψ’ to ‘ψ’. But if entailments are not conditionals then there is no problem with Lewis’ proofs 4 other than the fact that they do not establish entailments. All they establish are the given conditionals, and conditionals do not necessarily correspond to entailments, as we saw before. Why doesn’t its being necessary that φ ⊃ (ψ v ¬ψ) reflect an entailment, i.e. why is it not the case that that φ (for arbitrary ‘φ’) entails that ψ v ¬ψ (with classical ‘¬’)? That is because it is no part of the meaning of a proposition, in general, that the Law of the Excluded Middle holds. If propositional identity were a matter of strict equivalence, or equally, if propositions were functions from sentences to possible worlds, then this would follow, since then any proposition at all would be identical to the conjunction of itself with any necessary truth. But Fermat’s Last Theorem is clearly different from Pythagoras’ Theorem, while each of them is true in all possible worlds, so propositional identities cannot be a matter of strict equivalence. They are a matter of the truth of such remarks as ‘Goldbach’s Conjecture is that every even integer greater than 2 can be expressed as the sum of two primes’ and ‘Pythagoras’ Theorem is that the square on the hypotenuse of a right angled, Euclidean triangle is equal to the sum of the squares on the other two sides’, as we saw at the start. So again the introduction of some symbol like ‘λ’ is essential to gain an understanding of them Why doesn’t its being necessary that (φ & ¬φ) ⊃ ψ reflect an entailment, i.e. why is it not generally the case that that φ & ¬φ (with classical ‘¬’) entails that ψ (for arbitrary ‘ψ’). This time the supposition in question, namely that it is true that φ & ¬φ (with classical ‘¬’) is impossible. It is against the Law of Non-Contradiction. So there is nothing that could entail anything. As the earlier tradition on Entailment insisted, there is nothing to start with which might have entailments (c.f. Lewy 1976, von Wright 1957). Contradictions are meaningless they held. The irony is that Anderson and Belnap provided the clearest, and most sustained analysis of the grammar of ‘that’-clauses up to their time. In particular this was what was crucially missing in the immediately preceding period, in the work on Entailment by Lewy and von Wright. The so-called ‘paradoxes of material and strict implication’ had bothered logicians since at least the time of the Stoics. But it is only on the basis of Anderson and Belnap’s grammatical appendix that we can now see at all clearly that the mistake has simply lain in calling hook and fishhook forms of ‘implication’. Take that description away and there are no paradoxes. The description confuses verbs with connectives, specifically ‘entails’ and its cognates with ‘only if’ and is equivalents; and there is no paradox once these are kept separate. Literature Anderson, Alan and Belnap, Nuel Jr 1975 Entailment Vol 1, Princeton: Princeton University Press. Ayer, Alfred 1946 Language, Truth and Logic, London: Gollancz.. Goddard, Len 1960 “The Exclusive ‘Or’”, Analysis 20.5, 95-105. Lewy, Casimar 1976 Meaning and Modality, Cambridge: C.U.P.. Read, Stephen 1988 Relevance Logic, Oxford: Blackwell. Von Wright, George 1957 Logical Studies, London: Routledge and Kegan Paul.. Wittgenstein, Ludwig 1929 “Some Remarks on Logical Form”, Proceedings of the Aristotelian Society Supplementary Volume 9, 162-171.