- The Megaric logician Diodorus defined the possible as that which either is or at some time will be true, the impossible as that which neither is nor ever will be true, and the necessary as that which both is and always will be true. These definitions assume – as ancient and medieval logic generally assumes – that the same proposition may be true at one time and false at another …
- Against those who rejected his definition on the ground that some propositions are possible though they neither are nor ever will be true, Diodorus had an argument which came to be known as the 'Master Argument'. Its main premisses were that
Other ancient logicians rejected the Diodoran conclusion (that what neither is nor will be true is not possible), but agreed with him that they could only do so by rejecting at least one of the propositions (a) and (b); that is, they admitted that his reasoning was valid. Modern scholars have wondered why they did so; but before attending to this problem, we may glance at a somewhat simpler one, namely that of the consistency of these propositions with the Diodoran definitions of modal terms.
- Every true proposition about the past is necessary;
- An impossible proposition never follows from a possible one.
- The necessary, according to Diodorus, both is and always will be true, and it must be admitted that some true propositions about the past do not always remain so. Thus 'I had soup for tea last night' may be true now, but unless I have soup tonight also, it will be false tomorrow. This makes (a) as it stands inconsistent with the Diodoran account of necessity. It is true, however, of all propositions of the form 'It has been the case that p' that once they are true they are true for ever, and we shall see that if we understand (a) as referring to propositions of this form, it will suffice for the purpose to which Diodorus puts it. (b) holds in the Diodoran system, as it holds in S4.
- In dealing with the main problem, our starting-point must be a little different. We cannot assume the Diodoran definitions, but must take necessity and possibility as undefined, and use (a) and (b), which we might restate as
to prove the conclusion
- When anything has been the case, it cannot not have been the case,
- If anything is impossible, then anything that necessarily implies it is impossible,
Stated more precisely, our problem is to discover what broad assumptions about time, likely to have been taken for granted both by Diodorus and by his main opponents, would make (z) demonstrable from (a) and (b).
- What neither is nor will be true, is not possible.
- In answer to this question, it can be shown that the following two will suffice :-
- When anything is the case, it has always been the case that it will be the case;
- When anything neither is nor will be the case, it has been the case that it will not be the case.
- The way the proof proceeds may be best understood by considering an example. One of the opponents of Diodorus is said to have contended that we may rightly say of a shell at the bottom of the sea that it can be seen there, even if in fact it is not being seen arid never will be. But by (d), if the shell neither is nor will be seen, it has been the case that it will not be seen. Hence, by (a), it cannot (now) not have been the case that it has been the case that it will not be seen. That is, the proposition that it has not been the case that it will not be seen, i.e. that it has always been the case that it will be seen, is impossible. But by (c), the proposition that the shell is now being seen entails this impossible proposition that it has always been the case that the shell will be seen. Hence, by (b), the proposition that the shell is being seen is itself impossible.
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