Megarian Necessity in Forward-Branching, Backward-Linear Time

Megarian Necessity in Forward-Branching, Backward-Linear TimeAuthor(s): Michael ByrdSource: Noûs, Vol. 12, No. 4 (Nov., 1978), pp. 463-469Published by: WileyStable URL: http://www.jstor.org/stable/2214500 .

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Megarian Necessity in Forward- Branching, Backward-Linear Time

MICHAEL BYRD

UNIVERSEIY OF WISCONSIN - MADISON

I

Studies by Mates [4], Hintikka [3], Prior [5], and Rescher [7] have drawn attention to the importance of temporally- defined modalities in the writings of Aristotle, Aquinas, and other ancient and medieval philosophers. The most widely- known concepts of this sort are the Diodorean and the Megarian-Aristotelian treatments of necessity. According to the former, a statement is necessary now just in case it is true now and will be true at all future times. According to the latter, a statement is necessary now in case it is true now and at all earlier and later times.

A major concern of the initiators of tense logic was the question of what modal structure these two conceptions of necessity possessed. Arthur Prior's interest centered on the Diodorean view. And, in [6], he reports how his own dis- coveries and those of Geach, Dummett, Kripke, and Bull revealed the logical structure of Diodorean necessity. They found, of course, that the logic of Diodorean necessity de- pended on the underlying conception of time. If time is linear and continuous, the right logic is the logic S4.3, obtained by -adding the axiom:

4.3. L(Lp D Lq) v L(Lq D Lp),

to S4. If time is linear and discrete, the correct system is the system D, formed by adding:

D. L(L(p D Lp) D p) D L(MLp Dp),

NOOS 12 (1978) 463 01978 by Indiana University



464 NOCS

to S4.3. If, however, time is conceived as branching toward the future, but linear toward the past, then S4 is the logic of Diodrean necessity.

Investigation of Megarian necessity has been much less thorough; this is surprising in view of its centrality in the works of figures like Aristotle and Aquinas. Now, it is clear that if time is conceived as linear, then Megarian necessity is the system S5. However, it is not at all evident what the correct logic is if time is treated as branching toward the future but linear toward the past. To my knowledge, no serious attempt has yet been made to solve this problem.' Moreover, this fail- ure is serious: important proponents of Megarian necessity-AristQtle, for instance-can be plausibly repre- sented as having held the view that time is forward-branching and backward-linear.

The only recent conjecture about the logic of Megarian necessity in this sort of temporal structure was advanced by Rescher and Urquhart in their book Temporal Logic ([8]: 129). They suggest that the correct system is the system T+ (for more on this sort of logic, see [9] and [1]). This logic is axiomatized, as follows:

ASO. All truth-functional tautologies

Al. L(p D q ) D. Lp D Lq

A2. I- D p

A3. p D LMp A4. LLp D LLLp

Rules: Modus Ponens

Substitution

Necessitation

The presence of A3 is required, since if time t is earlier or later than t', then t' is earlier or later than t. To understand the import of A4, note that if LLp is true at t, then Lp is true at all earlier or later moments. But, in forward-branching, backward-linear time, all moments are earlier or later than some moment earlier or later than t. Thus, if Lp is true at all moments earlier and later than t, p is true at all moments. So, Lp is true at all moments, and LLp is true at all moments



MEGARIAN NECESSITY 465

earlier and later than t. So, LLLp is, true at t. On the other hand, the S4 axiom Lp D LLp is not valid. The sentenced may be true at all moments before and after t, though Lp is false at some moment prior to t.

Although plausible, the Rescher-Urquhart conjecture is false. The formula

Meg. (Lp &-Lq) D L(L D Lp),

is not a theorem of T . Consider the structure (W. R), where W = {O, 1, 2, 3, 4} and where aRb if and only if the absolute difference between a and b is 0, 1, or 4. The reader may verify that all theorems of T+ are valid on this structure. However, let V be a valuation which makes p true at 0, 1, 2, and 4 and which makes q true at 1, 2, 3, and 4. Meg is false on V at point 1: Lp is true at 1, sincep holds at 0, 1, and 2; Lq is false there, since q fails at 0. However, Lq is true at 2, since q holds at 1, 2, and 3, but Lp is false, sincep fails at 3. Since 1R2, M(Lq & -Lp) is true at 1. But this entails the falsity of L(Lq D LP) there.

On the other hand, the formula Meg is valid for Megarian necessity in forward-branching, backward-linear time. Sup- pose Lp & -Lq is true at t, but that L(Lq D Lp) is false there. This entails that there is a time t', t' > t or t' S t, where Lq is true and Lp is false. Clearly t # t'. Furthermore t -/ t'; other- wise Lp is true at t'. So, t' < t. But, this would make Lq true at t, contrary to hypothesis.

II

What then is the logic of Megarian necessity in forward- branching, backward-linear time? It is the logic T+ (Meg) obtained by adding Meg as an axiom to T . This claim is proved by examining the Scott-Lemmon canonical models for T+ - (Meg). I show how to impose a forward-branching, backward-linear ordering on the elements of this model. This ordering < is defined in such a way that LA is true at x iff A is true atx and at ally such thatx <y ory < x That is, LA is treated as Megarian necessity. Since every non-theorem of T2 (Meg) fails somewhere in the canonical model, it follows that every non-theorem of T+ (Meg) is invalid for Megarian necessity in forward-branching, backward-linear time. Since the converse clearly holds, equivalence follows.



466 NOOS

The canonical model MMeg for T+ (Meg) is the structure (WMeg, RMeg, VMeg), where:

(1) WMeg is the set of all maximal T2 (Meg)-consistent sets.

(2) For all A, B E WMeg, A RMeg B iff { S: LS E A} C B.

(3) For all A E WMeg and all propositional variablesp, VMeg(p, A) = t iff p E A.

Familiar arguments suffice to establish that:

(1) For all sentences S and all A E WMeg, VMeg(S, A) = t iff S E A.

(2) All theorems of T2(Meg) are true throughout WMeg on VMeg.

(3) Every non-theorem of T2(Meg) is falsified somewhere in WMeg on VMeg.

(4) RMeg is reflexive and symmetric.

I now describe how to impose a forward-branching, backward-linear ordering on MMeg. First, two definitions:

DI. For any C, D E WMeg, C-t D iff for all sentences S LS E C iffLS E D.

D2. For any C, D E WMeg, C <t D iff for all sentences S. if LS E C, then LS E D, and there is a sentence T for which LT E D and-LT E C.

Note that if either C -t D or C <t D, it follows that C RMeg D, and conversely. Furthermore, observe that et is an equivalence relation. For any equivalence class [C] generated by this relation, let <[C] be an arbitrarily chosen linear ordering of the elements of [C].

The required ordering < is defined by cases:

(1) if A <t B. then A < B.

(2) if, for some C, A <[c] B. then A < B.

It must be proved that < satisfies the ordering postulates of forward-branching, backward-linear time. Specifically, <




must be irreflexive, asymmetric, transitive, and backward linear. (See [8]: 68) That < is irreflexive and asymmetric is apparent. For transitivity, assume A < B and B < C. If A t B and B =t C, then A < C by construction. If A <t B and B <t C, there is an S such that LS E C and[-LS E B. Since A <t B, -LS E A. Further, by transitivity of implication, if LT E A, then LT E C. So, A <t C. The other two cases are similar.

That leaves backward linearity; < must be such that if A < B, C < B. and A f C, then A < C or C < A. Now if A-t C, the result holds by construction. So, I assume:

(1) not A-t C.

For the first case, let us also assume that:

(2) A <t B;

(3) C <t B;

(4) not A< C;

(5) not C < A.

By definition, (4) and (5) entail:

(6) not A <t C;

(7) not C <t A.

Assumption (1) implies that one of the following hold:

(8) (3U)(LU E A and -LU E C) and (3V)(LV E C and -LV E A) or;

(9) (3U)(LU E A and -LU E C) and (V)(if LV E C, then LV E A) or;

(10) (3V)(LV E C and -LV E A) and (U)(if LU E A, then LU E C).

Claims (9) and (10) contradict (6) and (7); so (8) must hold. Thus, LU & -LV E A. Now, it is true that A RMeg C. For let LS E A; then, by (2), LS E B. By (3), B RMg C; so S E C. Since LU & -LV E A, L(LV D LU) E A by Meg. Given A RMeg C, LV D LU E C. But this cannot be, since LV E C and -LU E C. So, given (2) and (3), the result holds. If we replace (2) or (3) by either



468 NOOS

(2') A <[A] B or,

(3') C <[C] B,

the argument follows in much the same way. Finally, it must be proved that necessity is Megarian; that

is, LS E A iff S E A and S E B, for all B such that A < B or B < A. The 'only if part is straightforward. For the other half, assume that S E A and that S E B, for all B such that A < B or B < A, and that LS # A. Then there is a C such that A RMeg C and -S E C. Since C /t A, A it C, and not A-t C, there are V, U with LU E A, LV E C, -LV E A, and -LU E C. Then the argument of the previous paragraph suffices to exhibit the required contradiction.

III

Let me conclude by mentioning a few significant facts and problems regarding T+ and T (Meg).

(a) It can be shown in a standard way that T+ (Meg) is sound and complete relative to the class of frames (W. R), where R meets the conditions: C1. (x)x Rx

C2. (x)(y)(if x R y, then y R x)

C3. (x)(y)(z)(u)(ifx Ry,y Rz, andx Ru, thenx Rz ory Ru).

(b) Lemmon-Scott filtration methods suffice to show that T (Meg) possesses the finite model property relative to the class of frames just defined. Consequently T+ (Meg) is decid- able.

(c) The argument of [1] and [2] entails that T2 (Meg) has finitely many modalities. In fact, just 8 positive ones: LLp Lp-* MLp-* MMLp-> LLMp-* LMp-* Mp-* MMp.

(d) Axiom A4 is not independent in T+ (Meg).

(e) Two intriguing open questions are: (i) A sentence S is a modal function of its propositional

variables. Two sentences S and T, containing the same variables, express the same modal function in




T+(Meg) if S is provably equivalent to T in T,(Meg). Given a finite set of variables, are infinitely many distinct modal functions expressible in T+(Meg)? (Compare with [1], where I show that, for any set containing at least two variables, infinitely many distinct modal functions are expressible in T+.)

(ii) What logical system charaterizes forward- convergent, backward-linear time?

REFERENCES

[1] Michael Byrd, "On the Addition of Weakened L-Reduction Axioms to the Brouwer System," forthcoming.

[2] and David Ullrich, "The Extensions of BAlt3," to appear in theJournal of Philosophical Logic.

[3] Jaakko Hintikka, "Aristotle and the 'Master Argument' of Diodorus," American Philosophical Quarterly 1(1964): 101-14.

[4] Benson Mates, Stoic Logic (Berkeley: University of California Press, 1953). [5] Arthur Prior, "Diodorean Modalities," The Philosophical Quarterly 5(1955): 202-

13. [6] ' Past, Present, and Future (London/New York: Oxford University Press,

1967). [7] Nicholas Rescher, Temporal Modalities inArabic Logic (Dordrecht/Boston: Reidel,

1966). [8] and Alasdair Urquhart, Temporal Logic (Vienna: Springer-Verlag, 197 1). [9] Ivo Thomas, "Modal Systems in the Neighborhood ofT." Notre Dame Journal of

Formal Logic 5(1964): 59-61.

NOTE

'After this paper was accepted for publication, I discovered that its main results are antedated in papers by G. Kessler and G. E. Hughes in Theoria 41, 1975. My work is independent of theirs and of work by Segerberg to which they refer.



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Megarian Necessity in Forward-Branching, Backward-Linear Time