Searching for appropriate fatalism candidate of Lazy argument

Further open questions

Why does this argument look to be Megarian? Let us briefly recapitulate the premises of LA as they are given above. The premises of the argument are these:

1. 
   P => [(Q v ~Q) => P]
   
2. 
   ~P => [(Q v ~Q) => ~P].
   
3. 
   P v ~P
   

Without its prefixes (i.e. ‘to be fated’ in the A-version or ‘to be true from eternity’ in the B-version), premises a) and b) are paradoxes of material implication while c) is LEM. Let us here recall that Chrysippus’ critical notes are against inserting a LEM particle in a) and b) ( i.e. Qv~Q or reputedly free will). Chrysippus’ request is to put the necessary condition (in his external or internal sense, i.e. of a simple fated or co-fated condition) instead of this particle. The polemics about LA now clearly grow into polemics about the problem of valid implication and the nature of conditionals. From one side, we have Chrysippus’ request that the antecedent condition has to be connected with the consequent. From the other side, the argument affirms the claim that variables in implication need not be connected and that valid implication is not necessarily tied to its antecedent content. Since all three premises are tautologies we will expect that in the background of the argument is the logical fatalism approach. Furthermore, we could expect that the conclusion has to be reached ‘solely on logical grounds’ in conformity with the line of the ideal of logical fatalism. Who is or are Chrysippus’ opponent(s)?

Some solutions of this form of LA (for example of Dummett, in line with a futility solution) are going toward a confutation against taking any precautions and toward the negation of a free will particle Qv~Q – “any precautions you take cannot be considered as being effective in bringing about your survival—that is, as effecting it” [1978, p. 340]. However, the negation of the inserted free will LEM particle ~(Qv~Q) simply cannot be validly inferred as a conclusion from the above three theorems. The idea of this procedure is very familiar to another historical argument probably originated in the same school and established on the same principles.

In only one place, as we know, LA is mentioned together with The ‘Reaper’ Argument (RA). Plutarch [Ps.-Plutarch, fat. 547e] mentions both as sophisms. Stephanus [in Int. 34,34-35,10], Ammonius [in de int. 131,20; 132,7] and an anonymous commentator of Aristotle [in Int. 54,8-55,5 Tarán] held RA to be ‘parabolic’ – i.e. the parallel argument. Both features are common to LA, too. Not all versions of RA given in the literature could be compared to LA, but one of Ammonius’ has some interesting features. He is introducing RA as the argument that destroys possibility and leaves true propositions about future events just to necessity. This is the argument:

“'If you will reap', it says, 'it is not the case that perhaps (takha) you will reap and perhaps you will not reap, but you will reap, whatever happens (pantos); and if you will not reap, in the same way it is not that perhaps you will reap and perhaps you will not reap, but, whatever happens, you will not reap. But in fact, of necessity, either you will reap or you will not reap'. Therefore the 'perhaps' has been destroyed (aneiretai), given that it has no place either in the opposition of reaping to not reaping, one of these occurring of necessity, or in what follows from either of the hypotheses” [ibid. 131,25-31].

We could read the premises in this manner: ‘if p then (whatever happens implies p)’; ‘if ~p then (whatever happens implies ~p); pv~p; the conclusion will be about ‘destroying possibility’, i.e. ~(<>P&<>~P). There are different readings of the argument [cf. Seel, 1993]. Also, different interpretations of the expressions, especially of the phrase ‘whatever happens,’ are possible. Let us suppose that the phrase instead of ‘whatever happens’ is something what is negated in the conclusion: i.e. <>P&<>~P. We omitted a temporal reading of the sentences and a prefix of the future as redundant here. We have:

a’) P=>[(<>P&<>~P)=> P]

b’) ~P=>[(<>P&<>~P)=> ~P]

c’) Pv~P

d’) ~(<>P & <>~P).

Even the three premises are theorems, the conclusion is not logically valid. Several things are of interest to us. One, the first two premises are paradoxes of material implication, the third is LEM. The same case is in LA. Second, the argument is, as it seems according to sources, probably Megarian. Third, here we have a truth-functional reading of implication – as not valid only in the case when the antecedent is valid and the consequent not valid, i.e. the material reading of Philo. Fourth, inference in the argument leads to the negation of the second antecedent (the stable one) of premises a) and b) by help of LEM. Fifth, by analogy with RA, the conclusion of LA could have also been similar to a negation of the second antecedent of a) and b). The sixth item is a little bit more complicated. Let us only say that the conclusion, derived from the conjunction of the complementary pair, is the strongest Megarian principle, the principle of plenitude – there are no unactualized possibilities (in the Megarian reading of temporal succession, which is considerably different from Aristotle’s non-temporal interpretation of the principle and that equals <>P and []P). There are several equivalent forms of the principle: one could be found in the RA’s conclusion, i.e. ~(<>P&<>~P). Other interesting forms of the statement are <>P=>[]P and []Pv[]~P. Neither of them are theorems. The last, []Pv[]~P, resembles a principle criticized by Aristotle in de interpretatione Ch. ix – i.e. it looks like (one precluded by him) an unrestricted distribution of the necessity operator in front of the bracketed LEM to the particular variables inside brackets. Aristotle does not deny this distribution completely and without restriction, but just for future cases. Distribution is not logically allowed and has nothing in common with LEM, whose legitimate modal version is []Pv~[]P (or Pv<>~P) but not []Pv[]~P (or Pv~<>P).⁶ However, we could imagine how this distribution is obtained (for example, in Aristotle or in the Stoics) by the application of LEM together with either the ‘from truth to necessity’ principle or the principles that ‘whatever is the case is true’ and ‘whatever is true is necessary’ [cf. Fitting & Mendelsohn, 1998:37; Kneale & Kneale, 1962:47-8; Haack, 1974:79-80]. All the past fatalists (like Aristotle and his commentators were, as well as most of the ancient philosophers, except Cleanthes [Cic. fat. vi, 13] and perhaps Epicurus), would agree with such a distribution of necessity for the past since it is in accordance with the principle of past-conservation. What they saw in this step to be problematic is the application of this distribution for the future sentences.

Let us now cast a glance at the so-called ‘proofs’ – one, (A), is obtained by application of the ‘case to necessity’ principle, the other (B) by application of the Tarskian correspondence step i.e. by the ‘case to truth’ principle accompanied by the ‘truth to necessity’ principle:

(A)	Pv~P	(B)	Pv~P
	P=>[]P		P=>TP
	~P=>[]~P		~P=>T~P
	[]Pv[]~P		TP=>[]P
			~TP=>[]~P
			[]Pv[]~P

The outcomes of both versions would be of help in the version of LA premises extended by prefixed modalities and it corresponds to the phrase ‘to be true from infinity’ in the B-version of Cicero’s source of argument.⁷ The modally equipped premises will be:

a”) []P => [(Qv~Q)=>P]

b”) []~P => [(Qv~Q)=>~P]

c”) []Pv[]~P

If Diodorus really accepts usual interdefinability between necessity and possibility (as Øhrstrøm & Hasle think [1995, p.25]), than the expression in c”) perhaps makes sense. Step c”) would be in some sense equivalent to the Diodorean principle of plenitude <>P=>[]P (where the implication has to be read in the sense of ‘follows after’, i.e. <>P_t1=>[]P_t2 and t₁t₂). However, a”) and b”) are no longer valid principles of material implication. They are neither genuine Philo versions nor are they theorems at all (at least not in the usual modern sense). On this basis, the conclusion with a negation of LEM – inserted as the common antecedent in both premises, that plays a role in the negated disjunctive conclusion, i.e. ~(Qv~Q) – will not be acceptable, at least not for Diodorus. Here we simply lost the thread of the analogy.

From here onward we can continue only on the basis of not very clearly grounded conjectures and extrapolations. One among many possible solutions of this kind could be to borrow the formulation of an inserted LEM in the second antecedent (‘whatever happens’) of a”) and b”) and to substitute it for its RA formulation from a’) and b’), i.e. <>Q&<>~Q, and then to transform it in such a way as to obtain the intended negated form in the conclusion, which implies the principle of plenitude, i.e. ~(<>A&<>~A) <=> (<>A=>[]A). This step gives us nothing more than we already know since the outcomes resemble RA – there is no possibility for free will either to call or not to call the doctor and everything that could be done is necessitated in advance, since possibilities cannot be unrealized. It is in accordance not just with RA but also with Diodorus’ Master Argument (MA). It is in conformity with his intended conclusions toward the logical fatalism position. However, there is nothing in common here with the futility conclusion of LA in the versions quoted in our historical sources. Also, it is hard to imagine some alternative reading of basic Megarian principles that would enable us, in this construction, to obtain a conclusion in a logically valid way. Like in the RA example, the conclusion here cannot be obtained in a logically valid procedure (and without some additional, here tacitly presupposed, assumptions).

In our opinion and in respect to these three similar forms of the argument (RA, LA, MA) – either in Philo’s or Diodorus’ way of reading implication – to prove futility was not the intention imbedded in the arguments. The more acceptable assumption would be to expect that the originally offered Megarian conclusions had something in common and are projected with approximately the same mission and with the same metaphysical background that corresponds to logical fatalism.