Abstract
According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard (Heyting) semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the dispute between classicists and intuitionists. I argue for this by presenting an account of consequence, and a cognate semantic theory for the language of the propositional calculus, which (a) respects the meanings of the connectives as embodied in the familiar classical truth-tables, (b) does not presuppose Bivalence, and with respect to which (c) the rules of the intuitionist propositional calculus are sound and complete. Thus the disagreement between classicists and intuitionists, at least, need not stem from their attaching different senses to the connectives; one may deny the doctrine without changing the subject. The basic notion of my semantic theory is truth at a possibility, where a possibility is a way that (some) things might be, but which differs from a possible world in that the way in question need not be fully specific or determinate. I compare my approach with a previous theory of truth at a possibility due to Lloyd Humberstone, and with a previous attempt to refute Quine’s thesis due to John McDowell.
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Notes
Intuitionists do not deny excluded middle in the sense of holding it to be false; they repudiate the law only in being unwilling to assert arbitrary instances of it.
Note that the sequents directly recoverable from the truth-tables do not by themselves constitute a complete axiomatization of intuitionist propositional logic. Both classicists and intuitionists accept the following elimination rule for ∧: from ⌈ A ∧ B ⌉ infer A; and from ⌈ A ∧ B ⌉ infer B. Without the assumption of Bivalence, the sequents (1) to (4) in the text do not ensure the soundness of this rule.
Thus, in a recent paper, Stalnaker writes that a possible world ‘would be more accurately labelled a possible state of the world, or a way that a world might be. It is something like a property that a total universe might have, and it is a maximal property in the sense that saying that a world has a particular property of this kind is enough to determine the truth or falsity of every proposition’ (Stalnaker 2010, 21; last emphasis added).
There are deductive arguments—notably, in set theory itself—for which this assumption does not hold. But I suppress the complications that a more general theory would bring in.
The relation of determination must not be confused with that of relative possibility, familiar from Kripkean possible-worlds semantics. A possibility y determines a possibility x if it is logically necessary that x does obtain if y does. Per contra, a world or possibility v is possible relative to a world or possibility w if, given that w obtains, v (logically) could have obtained. Since I accept the widely held thesis that S5 is the logic of logical necessity, I have suppressed the relation of relative possibility in my analysis.
Proof: Suppose that x ∈ U. Then, for any statement A that is true throughout U, A will be true with respect to x. So x ∈ Cl(U).
Proof: To show that Cl Cl(U) ⊆ Cl(U), suppose that x ∈ Cl Cl(U). Then x is a truth-ground of any statement that is true throughout Cl(U). Suppose now that statement A is true throughout U. Then A will also be true with respect to any y ∈ Cl(U), i.e., A will be true throughout Cl(U). Hence A will be true with respect to x. Since this holds for any statement that is true throughout U, we have that x ∈ Cl(U), showing that Cl Cl(U) ⊆ Cl(U). The converse inclusion follows directly from INCREASING.
Proof: Suppose that U ⊆ V. Then any statement that is true throughout V will be true throughout U. Now suppose further that x ∈ Cl(U). Then any statement that is true throughout U will be true with respect to x. So any statement that is true throughout V will be true with respect to x. So x ∈ Cl(V). Thus if U ⊆ V then Cl(U) ⊆ Cl(V), as required.
Proof: We need to show that when U = Cl(U) and V = Cl(V), (U ∩ V) = Cl (U ∩ V). By INCREASING, we have (U ∩ V) ⊆ Cl (U ∩ V), so it suffices to show that Cl (U ∩ V) ⊆ (U ∩ V). Now (U ∩ V) ⊆ U, so by MONOTONE, we have Cl (U ∩ V) ⊆ Cl (U). Since U = Cl(U), this yields Cl (U ∩ V) ⊆ U. Similarly, since V = Cl(V), we have Cl (U ∩ V) ⊆ V. Thus Cl (U ∩ V) ⊆ (U ∩ V), as required.
Because the truth-grounds of any statement form a closed set, we still have ∧-introduction in the form: if A is a logical consequence of some premisses X, and B is a logical consequence of X, then ⌈ A ∧ B ⌉ is a logical consequence of X. See the proof of soundness for the rule &R in Theorem 4 of Sambin (ibid., 868).
But how are we to interpret ‘x • y’ when x and y are not compossible? See the beginning of the next section.
For proof, see Sambin (1995), especially the remarks on distribution (864), lemma 2 (865) and theorem 4 (868).
Proof: We need to show that Cl (U → V) ⊆ U → V when U and V are closed. Suppose then that x ∈ Cl (U → V) and take an arbitrary u ∈ U. By Stability, x • u ∈ Cl (U → V • u). By definition of → , x • u ∈ Cl (V). Since V is closed, this means that x • u ∈ V. That is, for any u ∈ U, x • u ∈ V, i.e. x ∈ U → V, as required.
If x • x = ⊥ then ∀w (x ≤ w → w = ⊥). But x ≤ x, so x = ⊥.
Note that if f is a statement that is true only at the impossible state of affairs ⊥, then ⌈¬A ⌉ is equivalent to ⌈ A →f ⌉ and the closure of |¬A| follows from the result about conditionals proved in note 16.
Relevant and dialetheist logicians, who reject explosion, will reject this way of ensuring that negated statements conform to (R). There are other ways of ensuring this, and thereby of developing the nascent logico-semantic theory of truth at a possibility in directions that they too could accept, but I cannot explore these alternatives here.
As Sambin points out: ‘the result now is a simple and fully constructive completeness proof for first-order BL [basic linear logic] and virtually all its extensions, including the usual, or structured, intuitionistic and classical logic. Such a proof clearly illustrates the fact that stronger set-theoretic principles and classical metalogic are necessary only when completeness is sought with respect to a special class of models, such as the usual two-valued models’ (ibid., 861).
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I am grateful to David Charles and Timothy Williamson for comments that improved a draft.
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Rumfitt, I. On A Neglected Path to Intuitionism. Topoi 31, 101–109 (2012). https://doi.org/10.1007/s11245-011-9108-5
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DOI: https://doi.org/10.1007/s11245-011-9108-5