Zolin and Pizzi: Defining Necessity from Noncontingency

Abstract

The point of the present paper is to draw attention to some interesting similarities, as well as differences, between the approaches to the logic of noncontingency of Evgeni Zolin and of Claudio Pizzi. Though neither of them refers to the work of the other, each is concerned with the definability of a (normally behaving, though not in general truth-implying) notion of necessity in terms of noncontingency, standard boolean connectives and additional but non-modal expressive resources. The notion of definability involved is different in the two cases (‘external’ for Zolin, ‘internal’ for Pizzi), as are the additional resources: infinitary conjunction in the case of Zolin, and for Pizzi, first, propositional quantification, and then, later, most ingeniously, the use of a propositional constant. As well as surveying and comparing of the work of these authors, the discussion includes some some novelties, such as the confirmation of a conjecture of Zolin’s (Theorem 2.7).

Appendices

Appendix 1: The Distinguishable Model Property

Toward the end of Sect. 2 the question of whether, for every model for the language with $\Updelta$ rather than $\square$ (or $\diamondsuit$) for every model there is an equivalent distinguishable model. (Since we are considering all models, this asks about the distinguishable model property for the minimal or basic logic, called ${\mathsf {K}}\Updelta$ in Sect. 4, and otherwise known as ${\mathsf {K}}^{\Updelta}$.) This would imply the distinguishable model property for (the minimal logic in) Zolin’s $\boxtimes$-language, by Lemma 2.8. As was remarked, the most straightforward method for proving this for the $\square$-based language, the method of filtrations, fails for the $\Updelta$-based language, as we shall now show. However, a variation on this method will turn out to establish this property for the $\Updelta$-based language after all.

Although the main interest in filtrations has been in turning an infinite model into a finite model agreeing with the original on all the subformulas of a given formula ([3, 4, 24]) it is useful in modal logic in the usual $\square$-based language for proving that a logic has the distinguishable model property (see Bellissima and Bucalo 1995), whose treatment is followed below) with respect to a given semantics, which in the current setting is understood to be the usual Kripke model theory: this means that given any any model for the logic in question, an equivalent distinguishable model can be found. In saying that the models are equivalent what is meant is that any formula is true throughout the one is true throughout the other: any formula at all, not just those which are subformulas of some initially given formula.

Given a model ${\mathcal{M}} = \langle W, R, V\rangle$ one produces a model, the finest filtration of ${\mathcal{M}}, {\mathcal{M}}_0 = \langle W_0, R_0, V_0\rangle$ by taking as the elements of W ₀ the equivalence classes [u] of elements u  ∈ W by the relation “verify (in ${\mathcal{M}}$) exactly the same formulas”, and putting $[x]R_0[y]$ iff for some $x^{\prime} \in [x]$, {\text{some}}\,$y^{\prime} \in [y], xRy$, {\text{and}}\,$V_0(p_i) = \{[w] \vert w \in V(p_i)\}$, then verifying by induction on the complexity of A that for all $u \in W: {\mathcal{M}} \models_u A$ if and only if ${\mathcal{M}}_0 \models_{[u]} A$, from which we conclude that ${\mathcal{M}}_{0}$ is a distinguishable model equivalent to ${\mathcal{M}}$. If every model for ${\mathsf {S}}$ has an equivalent distinguishable model, ${\mathsf {S}}$ is said to have the distinguishable model property, and the above filtration method shows that this is so for every $\square$-based normal modal logic ${\mathsf {S}}$.³¹ But the method fails for logics in the $\Updelta$-based language because the induction step in the proof alluded to fails for $\Updelta$-formulas. Specifically, suppose we have a situation like that depicted in Fig. 2, in which ${{\mathcal{M}} = \langle W, R, V\rangle}$ is a model on the frame on the left of the diagram with $\{x, x^{\prime}, x^{\prime\prime}\} \subseteq V(p_i)$ for all i and $y \notin V(p_i)$ for all i, and V making some but not all propositional variables true at w. The element of the filtrated model ${{\mathcal{M}}_0 = \langle W_0, R_0, V_0\rangle}$ are then $[w] = \{w\}, [x] = \{x, x^{\prime}, x^{\prime\prime}\}, [y] = \{y\}$ and R ₀ as indicated on the right-hand side of the diagram. In ${{\mathcal{M}}}$, the points $x, x^{\prime}, x^{\prime\prime}$, all verify the same formulas because we stipulated this for the propositional variables (all true at each of these points) and, each having at most one R-successor, they verify all $\Updelta$-formulas. (This would be so even if we had given x and $x^{\prime}$ several R-successors each, as long as the elements of R(x) in turns verified the same formulas as each other, and likewise for the elements of $R(x^{\prime})$.) Here we see the failure of the desired equivalence ${{\mathcal{M}} \models_u A \Leftrightarrow {\mathcal{M}}_0 \models_{[u]} A}$ taking u as x, say, and A as $\Updelta p$: although ${{\mathcal{M}} \models_x \Updelta p, {\mathcal{M}}_0 \not\models_{[x]} \Updelta p}$, since [x] has a p-verifying R ₀-successor ([x] itself) and a p-falsifying R ₀-successor (namely [y]).

The failure of the finest filtration method in the present case suggests that if the distinguishable model property is to be established for the present language, some variation is called for; the following variation was devised, in response to this need, by Toby Meadows. Again we work with a kind of quotient model, and will describe the construction along the lines of the finest filtration above but replacing the ‘0’ subscripts by ‘1’. Given a model ${{\mathcal{M}} = \langle W, R, V\rangle}$ we pass to the model ${{\mathcal{M}}_1 = \langle W_1, R_1, V_1\rangle}$, which will actually depend on an additional function f described below—though Prop. 6.1 will not itself depend on how f is chosen—by taking as the elements of W ₁ the equivalence classes [u] of elements $u \in W$ by the relation “verify (in ${{\mathcal{M}}}$) exactly the same formulas” (exactly as for W ₀, that is. We now come to the f just mentioned. This is to be a function from $\wp(W)$ to W satisfying $f(U) \in U$ for all nonempty $U \subseteq W$. Now we are in a position to define R ₁: put [x]R ₁[y] iff for some $y' \in [y],\,f([x])Ry$. As with V ₀, we set $V_1(p_i) = \{ [w] \vert w \in V(p_i) \}$. With all this in place, one shows by induction on formula complexity:

Proposition 6.1

(Meadows) For any models ${{\mathcal{M}}}$ and ${{\mathcal{M}}_{1}}$ as above, we have for all formulas A and all ${u \in W: {\mathcal{M}} \models_u A}$ if and only if ${{\mathcal{M}}_1 \models_{[u]} A}$.

Corollary 6.2

$K\Updelta$ has the distinguishable model property.

Proof

For any ${{\mathcal{M}}, {\mathcal{M}}_1}$ is a distinguishable model, equivalent to ${{\mathcal{M}}}$ by Prop. 6.1. $\square$

Appendix 2: Pizzi’s Cryptic Comment Elaborated

In Sect. 4, Pizzi was quoted as saying that the “equivalence of the two systems which will be proved does not mean that the box operator defined in ${\mathsf {K}}\Updelta\tau{\mathsf {w}} + {\rm Def} \square$ has the same meaning as the box operator axiomatized in ${\mathsf {K}}\square\tau{\mathsf {w}} + {\rm Def} \Updelta$,” a, remark which, it was observed, is somewhat puzzling. So here we look at his elaboration of the idea.

What follows is the opening portion of §5 of Pizzi [18], in which a disanalogy is alleged to obtain between the cases of ${\mathsf {K}}\square\tau + \hbox{Def} \Updelta$ and ${\mathsf {K}}\Updelta\tau$ + Def $\square$ on the one hand, and the “${\mathsf {w}}$” versions of these logics on the other.³²

As a final comment, we have to stress that the relation between the modal system ${\mathsf {K}}\square\tau$ + Def $\Updelta$ and the equivalent contingential system ${\mathsf {K}}\Updelta\tau$ + Def $\square$ is not the same which obtains between ${\mathsf {K}}\square\tau{\mathsf {w}}$ + Def $\Updelta$ and the equivalent contingential system ${\mathsf {K}}\Updelta\tau{\mathsf {w}}$ + Def $\square$. The characteristic axiom $\diamondsuit \tau \land \diamondsuit \neg \tau$ of ${\mathsf {K}}\square\tau$ is the $\square$-variant of the characteristic axiom $\nabla \tau$ of ${\mathsf {K}}\Updelta\tau$; and, conversely, $\nabla\tau$ is equivalent to the $\Updelta$-variant of $\diamondsuit \tau \land \diamondsuit \neg \tau$. But this one-one translation among the characteristic axioms does not hold when we consider the couple of systems ${\mathsf {K}}\square\tau {\mathsf {w}} + \hbox{Def} \Updelta$ and ${\mathsf {K}}\Updelta\tau{\mathsf {w}}$ + Def $\square$. In fact the axiom $(\square\tau \lor \square\neg\tau) \to \square p$ of ${\mathsf {K}}\square\tau{\mathsf {w}}$ is not the $\square$-variant of the axiom $\Updelta \tau \to \Updelta p$ the $\square$-variant of $\Updelta \tau \to \Updelta p$ is in fact $(\square \tau \lor \square\neg\tau) \to (\square p \lor \square \neg p)$, which is weaker than $(\square \tau \lor \square\neg\tau) \to \square p$. In the other direction, the $\Updelta$-variant of the axiom $(\square \tau \lor \square \neg \tau) \to \square p$ of ${\mathsf {K}}\square\tau{\mathsf {w}}$ is equivalent to $\Updelta \tau \to (\Updelta p \land \Updelta(\tau\to p))$, which is stronger than $\Updelta \tau \to \Updelta p$. So it appears that the necessity operator axiomatized in ${\mathsf {K}}\square\tau{\mathsf {w}}$ + Def $\Updelta$ has not exactly the same properties of the operator introduced in ${\mathsf {K}}\Updelta\tau{\mathsf {w}}$ + Def $\square$ by Def $\square$. To illustrate this point let us remark that if we consider the simple ${\mathsf {K}}\Updelta$ + Def $\square$, thanks to Def $\square$ we obtain the equivalence $\square\tau \leftrightarrow \Updelta\tau$; so that, given the equivalence $\Updelta \tau \leftrightarrow \Updelta\neg\tau$ granted by axiom ${\mathsf {K}}\Updelta$1, by replacement we obtain the equivalence $\square\tau \leftrightarrow \square \neg \tau$. But the equivalence $\square\tau \leftrightarrow \square \neg \tau$ is introduced in ${\mathsf {K}}\Updelta\tau{\mathsf {w}}$ + Def $\Updelta$ as an axiom, Nec$\tau{\mathsf {w}}$ (equivalent in ${\mathsf {K}}\square\tau{\mathsf {w}}$ to $(\square \tau \lor \square \neg \tau) \to \square p$) which, when subjoined to ${\mathsf {K}}$ + Def $\Updelta$, yields as a theorem $\Updelta \tau \to \Updelta p$, while this wff is independent from ${\mathsf {K}}\Updelta$ + Def$\square$.

Fig. 2

A Model (with $x, x^{\prime}, x^{\prime\prime}$ equivalent) and Its Finest Filtration

The problem is that if $(\square \tau \lor \square \neg \tau) \to (\square p \lor \square \neg p)$ is taken as an axiom for the necessity operator in place of $(\square \tau \lor \square \neg \tau) \to \square p$ or $\square \tau \leftrightarrow \square \neg \tau$ the resulting system—let us call it ${\mathsf {K}}\tau{\mathsf {w}}$ – is a system weaker than ${\mathsf {K}}\square\tau{\mathsf {w}}$, and it may be seen$\ldots$

I have included this opening part of the second paragraph because of its opening phrase: “the problem is$\ldots$”, though reading on, one sees that it proceeds to consider a weaker logic than ${\mathsf {K}}\square\tau{\mathsf {w}}$, as do some parts of the first paragraph, making it irrelevant to the question of the relation between ${\mathsf {K}}\square\tau{\mathsf {w}}$ itself and ${\mathsf {K}}\Updelta\tau{\mathsf {w}}$, which are definitionally equivalent, or that between ${\mathsf {K}}\square\tau{\mathsf {w}}$ + Def $\Updelta$ and ${\mathsf {K}}\Updelta\tau{\mathsf {w}}$ + Def $\square$, which is the relation of identity, this logic being, as noted in Sect. 4 above, a definitional extension both of ${\mathsf {K}}\square\tau{\mathsf {w}}$ and of ${\mathsf {K}}\Updelta\tau{\mathsf {w}}$. (The problematic sentiment quoted from Pizzi there re-appears in the above passage as: “the necessity operator axiomatized in ${\mathsf {K}}\square\tau{\mathsf {w}}$ + Def $\Updelta$ has not exactly the same properties of the operator introduced in ${\mathsf {K}}\Updelta\tau{\mathsf {w}}$ + Def $\square$ by Def $\square$.”) So we know a priori that any remarks in the first paragraph from this passage of Pizzi’s suggesting otherwise must be mistaken. Unless, that is, he is simply making a comment about particular axiomatizations of these logics—a comment somewhat reminiscent (as Rohan French has reminded the author) of the point made in Hiż (1958). In particular, it is not clear what Pizzi means by “weaker” and “stronger” when he writes “$\ldots(\square \tau \lor \square\neg\tau) \to (\square p \lor \square \neg p)$, which is weaker than $(\square \tau \lor \square\neg\tau) \to \square p$,” and “$\ldots\Updelta \tau \to (\Updelta p \land \Updelta(\tau\to p))$, which is stronger than $\Updelta \tau \to \Updelta p$,” since the formulas alluded to are all provable in the logics actually under discussion in (what appears here as) Theorem 4.3(ii).