The Primacy of the Classical? Saul Kripke Meets Niels Bohr

ABSTRACT

Kripke's theory of partial truth offers a natural solution of the Liar paradox and an appealing explanation of why the Liar sentence seems to lack definite content. It seems vulnerable, however, to the objection that it cannot state important facts about partial truth. I point out that the same vulnerability infects the quantum logic developed by Garrett Birkhoff and John von Neumann, among others. It is often claimed that the only way to record these facts is within a classical metalanguage, but Kripke showed that the same language can function both as the language of partial truth and also as a classically bivalent language. An explanation of why we need a classical explanation of a non-classical system was advanced in the context of quantum mechanics by Niels Bohr, and it applies also, I argue, to the partial truth situation.

Notes

1 Reinhardt (1986, 228): ‘If the formal language is to provide an adequate explication of the informal language that we use, it must contain its own metalanguage. I take it that this is in fact a desideratum for success in formulating a theory of truth'. Hence, he concluded, truth must be partial.

2 Feferman (1991, 40) also found ‘helpful’ the analogy with partial recursive functions.

3 Kripke (1975, 71): 'What hitherto has been, as far as I know, an intuitive concept with no formal definition, becomes a precisely defined concept in the present theory'.

4 t is a faithful definition in L⁺ of the diagonal function d(x) := sub(x,num(x)) at the argument which is the Gödel number of A(x) (a property or relation which needs only the vocabulary of L for its definition is said to be arithmetical; Tarski’s celebrated theorem on the undefinability of truth shows that truth for the language of arithmetic is not itself arithmetical).

5 Gödel’s Diagonal Lemma is not sufficient for self-reference. The lemma states that for any formula A(x) of L⁺ there is a sentence B (a so-called fixed point of A(x)) such that Σ |- B ↔ A(˹B˺), where Σ is strong enough for recursive arithmetic, but as Craig Smoryński points out, Σ |- 0=0 ↔ Pr(˹0=0˺), where Pr(x) numeral wise represents the provability predicate for Σ. But ‘0=0’ clearly does not express its own unprovability. (Smoryński 1991, 114).

6 In this scheme conjunction is defined as a minimum and disjunction a maximum operator with respect to the ordering in which 'true' dominates u dominates 'false'; in the Weak scheme any u entry takes the result to u. Kripke's preference for the Strong scheme is probably due to the fact that the (imagined) statements of Dean and Nixon in Kripke's well-known example (1975, 60) are grounded in the Strong but not in the Weak scheme.

7 Although groundedness is far from absolute: different monotonic valuation schemes will give different smallest fixed points.

8 In the valuation scheme using supervaluations the codes of all logical truths will be in the extension of T in the smallest fixed point (in Kripke's, many classical logical truths will not), but fixed point supervaluational truth is nor compositional, nor is there a corresponding phenomenon of groundedness (Feferman 1991, 42).

9 Gödel called assertions which lack a truth-value on pain of contradiction 'singularities' (quoted in Reinhardt 1986, 221–222).

10 Halbach and Horsten have given a sequent-based axiomatization of Kripke’s theory in the Strong Kleene logic (2006).

11 The measurement problem seems not to have existed as such for Bohr: there was no need to appeal to any collapse of the wave packet because for him there was no real wave packet: the wave functions of quantum mechanics, involving essentially as they do imaginary numbers, had only symbolic existence. Thus rather than solving the measurement problem, Bohr dissolved it, as Zinkernagel puts it (2016, 16).

12 A radically different version of quantum logic has been developed within topos theory, whose associated algebra is not a nondistributive lattice but a distributive Heyting algebra (see Döring 2011 for example). For an exhaustive list and discussion of the current variants of quantum logic see the entry ‘Quantum logic in historical and philosophical perspective’ in the Internet Encyclopedia of Philosophy.

13 A further non-classical feature of quantum probabilities is that there is no joint distribution over the values of incompatible propositions.

14 Under the subset-to-subspace mapping these are pretty much the only choices for the negation, meet and join: the set-theoretical intersection of two subspaces is a subspace but set-theoretical complements and unions are not: one has to go to orthocomplements and linear spans. The linear span of two subspaces is the smallest subspace including both, thereby accommodating the uniquely quantum-mechanical feature of closure of states under superpositions. 

15 In his own seminal work on the foundations of quantum mechanics George Mackey resorted to an axiom (axiom VII) to make the identification, an axiom which he conceded might seem ‘entirely ad hoc’ (1963, 71). For a spirited defence of the view that no further justification is needed see Dickson (1998).

16 This is of course not satisfied in the topos-based approach whose logic is intuitionistic.

17 Observables are compatible if they can be measured in the same experimental set-up. Spin up and spin down in any given direction clearly can be, but the apparatus has to be rotated to measure the spin value in any other direction. Observables are compatible just in case their operators commute, and propositions (subspaces) are compatible derivatively: two propositions A and B are compatible if they describe outcomes of compatible observables (this can be shown to be the case just when (A∧B)∨(A∧B^⊥) = A).

18 It might seem strange that two incompatible propositions can even be meaningfully conjoined, given that the respective observables cannot be measured by the same set-up. But the lattice L(H) is of course closed under both join and meet so the conjunction always exists even if it is the zero element. Some other developments of quantum logic, those based on what are called Partial Boolean Algebras, do not allow conjunctions and disjunctions of incompatible propositions.

19 Note also that Prob_v(O∈Δ) = <v|P_L(O,Δ)|v> = E _v(P_L(O,Δ)) where E is ‘expectation’; classically of course the expected value of the indicator of a proposition A is equal to A’s probability. 

20 Where projectors P and Q are compatible they obey the same additivity law as classical indicator functions: P∨Q = P + Q - P∧Q.

21 That is, As Beltrametti and Cassinelli (1981, 142) put it, v ‘causes with certainty the yes answer’ to the question whether the observation yields a value in Δ. Beltrametti and Cassinelli’s quantum logic is that of yes/no questions, i.e. a logic of projectors rather than subspaces, but given the one-one relationship between projectors and subspaces the two are of course equivalent.

22 There is no truth-functional conditional in this logic, but there is a conditional definable in terms of the basic quantum-logical operations: it is the Sasaki hook A→B defined by A^⊥∨(A∧B). Like the classical material conditional it supports a rule of modus ponens, in the form A∧(A →B) ≤ B (recall that the partial order ‘≤’ is lattice-entailment). Unlike the material conditional it is a type of counterfactual conditional, in that strengthening of the antecedent, transitivity and contraposition are not valid for it (Hardegree 1979). It also has an intimate connection with quantum-probabilistic conditionalisation as represented by the Lüders rule (Gibbins 1987, 164,165).

23 This can of course have remarkable consequences: the famous example is of course Schrödinger’s cat. The problem is not dissolved by claiming that the indeterminacy of value-ascriptions to observables arises only from a particular way of interpreting quantum mechanics: the Kochen-Specker Theorem shows that if the dimension of the Hilbert space is greater than two, and if the value-assigning function adds and multiplies over compatible observables then it is impossible to ascribe – at any rate consistently – values to all the observables independently of the quantum state.

24 Since a valuation on propositions whose values are ‘true’ and ‘exclusion-false’ is a total function, it might seem that adjoining the exclusion sense of ‘false’ is inconsistent with the Kochen-Specker theorem. In fact this is not so because the assumption of value-additivity for that theorem is violated (I owe this observation to Ben Eva): suppose that v is a nontrivial superposition of two orthogonal unit vectors, with projectors P₁ and P₂. Then such a valuation will assign both the value 0. However their disjunction is P₁+P₂ which has the value 1, violating the K-S additivity condition for values of compatible observables.

25 See also Gibbins (1987, 94).

26 If the theory is first-order formalised, capture in its consequences; if full second order, in its theorems.

27 ‘The liar paradox refutes the naïve theory of truth’, (McGee 1991, 103). There is an intimate relation between the T-schema and comprehension: intertranslatability. A well-known interpretation of second order predicative arithmetic into the Tarski hierarchy up to the ordinal ω exploits the following form of the (Class) comprehension principle:∀y (y ∈ {x: A(x)} ↔ A(y)). The translation proceeds by interpreting ‘y ∈ {x: A(x)}' as ‘sat(y, ˹A(x)˺)' for each formula A free in one variable, where ‘sat’ stands for ‘satisfies’. So translated, the comprehension principle becomes in effect the T-schema: ∀y (sat(y, ˹A(x)˺) ↔ A(y)).

28 It is not clear that Cantor’s theory of transfinite numbers fits such an account. Probably like many at the time Bohr would have thought that because it is so disconnected from experience it fails to be authentic mathematics.

29 There is a good deal of evidence from cognitive psychology that those operations are functionally represented in the brain’s neural structures, together with a range of inferential processes employing them. The system appears to be a multi-level one, with more sophisticated processing and inferential techniques appearing later in our evolutionary history, where the representation of hypothetical situations and possibility-ranges becomes adaptively important.

30 That is, Gibbins (1987, 159): ‘Quantum logic is the logic of the microphysical world, and to grasp it we must climb up the ladder of classical mathematics and classical logic’. This might have been uttered by Bohr himself.

31 The same, I claim, goes for suitable paradoxical ordered sets of statements, like Yablo’s (1985). Here a predicate Y(x) ↔ ∀y> x ¬ ­Y(y) is asserted for all natural numbers x. There is clearly a paradox, but it does not seem to involve any form of self-reference. Stephen Yablo himself denied that it did, and the denumerably many predicates Y(x) seem only to refer to those farther down the list. Indeed, Forster has described a simple infinitary propositional language admitting (denumerably) infinite conjunctions in which the set A_n = p_n ↔ ¬ p_k is inconsistent (1996; such a language, a propositional fragment of the Lω₁ω family, is described in Bell 2000). It is true that Priest (1997) has shown that there is a predicate fixed point of the form (∀k>x)(¬Sat(˹Y(z)˺,k)) = Y(x), from which the paradox can be derived, but the fact that there is a simple non-circular derivation of the paradox without even mentioning satisfaction or truth shows, I believe, that self-reference is not the cause (this is the burden of Roy Cook’s remarkable book (2018)).

32 According to Gödel, ‘It might even turn out that it is possible to assume every concept to be analogous to dividing by zero. Such a system would be most satisfactory in the following respect: our logical intuitions would then remain correct up to certain minor corrections, i. e., they could then be considered to give an essentially correct, only somewhat “blurred”, picture of the real state of affairs.’ (Quoted in Reinhardt 1986, 221). The rate of decoherence is extremely rapid for macroscopic bodies where there are enormous degrees of freedom.

33 Thus Feferman: ‘A second line of defense is that the assumption of classical logic accords with ordinary informal reasoning about statements such as the "Liar" which are judged to be indeterminate using proof by contradiction.’ (1991, 41).