Διόδωρος Κρόνος の master argument
Diodorus Cronus
Aristotle, in his work On Interpretation, had wrestled with the problem of future contingents. In particular whether one can meaningfully regard future contingents as true or false now, if the future is open, and if so, how?
In response, Diodorus maintained that possible was identical with necessary (e.g., not contingent); so that the future is as certain and defined as the past. Alexander of Aphrodisias tells us that Diodorus believed that that alone is possible which either is happening now, or will happen at some future time. When speaking about facts of an unrecorded past, we know well that a given fact either occurred or did not occur, yet without knowing which of the two is true—and therefore we affirm only that the fact may have occurred: so also about the future, either the assertion that a given fact will at some time occur, is positively true, or the assertion that it will never occur, is positively true: the assertion that it may or may not occur some time or other, represents only our ignorance, which of the two is true. That which will never at any time occur, is impossible.
Diodorus went on to formulate an argument that became known as the master argument (or ruling argument: ho kurieuôn logos /ὁ κυριεύων λόγος/). The most succinct description of it is provided by Epictetus:
The argument called the master argument appears to have been proposed from such principles as these: there is in fact a common contradiction between one another in these three propositions, each two being in contradiction to the third. The propositions are: (1) every past truth must be necessary; (2) that an impossibility does not follow a possibility; (3) something is possible which neither is nor will be true. Diodorus observing this contradiction employed the probative force of the first two for the demonstration of this proposition: That nothing is possible which is not true and never will be.
Epictetus' description of the master argument is not in the form as it would have been presented by Diodorus, which makes it difficult to know the precise nature of his argument. To modern logicians, it is not obvious why these three premises are inconsistent, or why the first two should lead to the rejection of the third. Modern interpretations therefore assume that there must have been extra premises in the argument tacitly assumed by Diodorus and his contemporaries.
One possible reconstruction is as follows: For Diodorus, if a future event is not going to happen, then it was true in the past that it would not happen. Since every past truth is necessary (proposition 1), it was necessary that in the past it would not happen. Since the impossible cannot follow from the possible (proposition 2), it must have always been impossible for the event to occur. Therefore if something will not be true, it will never be possible for it to be true, and thus proposition 3 is shown to be false.
Epictetus goes on to point out that Panthoides, Cleanthes, and Antipater of Tarsus made use of the second and third proposition to demonstrate that the first proposition was false. Chrysippus, on the other hand, agreed with Diodorus that everything true as an event in the past is necessary, but attacked Diodorus' view that the possible must be either what is true or what will be true. He thus made use of the first and third proposition to demonstrate that the second proposition was false.
Aristotle solved the problem by asserting that the principle of bivalence found its exception in this paradox of the sea battles: in this specific case, what is impossible is that both alternatives can be possible at the same time: either there will be a battle, or there won't. Both options can't be simultaneously taken. Today, they are neither true nor false; but if one is true, then the other becomes false. According to Aristotle, it is impossible to say today if the proposition is correct: we must wait for the contingent realization (or not) of the battle, logic realizes itself afterwards:
One of the two propositions in such instances must be true and the other false, but we cannot say determinately that this or that is false, but must leave the alternative undecided. One may indeed be more likely to be true than the other, but it cannot be either actually true or actually false. It is therefore plain that it is not necessary that of an affirmation and a denial, one should be true and the other false. For in the case of that which exists potentially, but not actually, the rule which applies to that which exists actually does not hold good. (§9)
For Diodorus, the future battle was either impossible or necessary. Aristotle added a third term, contingency, which saves logic while in the same time leaving place for indetermination in reality. What is necessary is not that there will or that there won't be a battle tomorrow, but the dichotomy itself is necessary:
A sea-fight must either take place tomorrow or not, but it is not necessary that it should take place tomorrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place tomorrow. (De Interpretatione, 9, 19 a 30.)
樣相。アリストテレスによると、當時のメガラ學派は可能態と現實態の區別など存在しないと主張した。ディオドロス・クロノスは可能なものとは現在存在するもの又は未來に存在するだらうものだと定義し、不可能なものとは未來に眞でないだらうものだと定義し、不定のものとはすでにさうであるものか又は未來に僞であるだらうものだと定義した。ディオドロスは、「過去に存在したものの全ては眞かつ必然である」、「不可能なものは可能なものから生まれえない」、「現在存在しないし未來も存在しないだらうものは可能である」の三命題は定立しえない、といふいわゆるマスター・アーギュメントでも有名である。ディオドロスはこの三命題のうち前二者に尤もらしさを用いて、現在存在しないし未來も存在しないだらうものは不可能であると證明した。對照的にクリュシッポスは第二の前提を否定して、不可能なものは可能なものから生まれうると述べた。
A. N. Prior の時相論理に基附き、時相演算子$ F,$ Pを以下で定義する 時點の集合$ Wが得られ、現時點を$ c\in Wとする
時點$ t\in Wから時點$ s\in Wへ到達可能である事を$ t\le s或いは$ s\ge tと書く (S4 樣相論理 (KT4)) 時點$ tで命題$ Aが眞である事を$ @_tAと書く
命題$ @_cFAは、現時點$ cから到達可能な未來の少なくとも一つの時點$ tに命題$ Aが眞と成る事$ \exist t_{\ge c}@_tAを言ふ
命題$ @_cPAは、現時點$ cへ到達可能な過去の少なくとも一つの時點$ tに命題$ Aが眞と成る事$ \exist t_{\le c}@_tAを言ふ
Διόδωρος Κρόνος と A. N. Prior に據れば、可能$ \lozengeと必然$ \squareの樣相演算子は、任意の時點$ @_tからの未來に就いて定義される
可能なものとは現在存在するもの又は未來に存在するだらうものだ$ @_t\Diamond x\iff @_tFx
不可能なものとは未來に眞でないだらうものだ$ @_t\neg\lozenge x\iff @_t\neg Fx
不定のものとはすでにさうであるものか又は未來に僞であるだらうものだ$ @_t\neg\square x\iff @_t(Fx\land F\neg x)
必然$ @_t\square x\iff @_t\neg F\neg x
樣相
必然$ @_c\square p:=\forall t_{\ge c}@_tp
不可能$ @_c\square\neg p:=\forall t_{\ge c}@_t\neg p
偶有的 (不定)$ @_c\neg\square p:=\exist t_{\ge c}@_tp\land\exist t_{\ge c}@_t\neg p
命題
1. 過去についての眞なる命題はすべて必然である$ Px\to\square Px
2. 不可能なことが、可能なことから歸結することはない$ ((x\to y)\land\square\neg y)\to\square\neg x
3. 實現しない可能性がある$ \neg\square Fx
支配する者の議論 (master argument。hokyrieuōn logos)
太郎が死去する時點を$ dとし、太郎が支配する反實假想の時點を$ vとして、命題「太郎が支配してゐる」を$ mと書く
前提
太郎は結局生涯支配することのない者だとする$ @_d\neg Pm
主張
i. 太郎は今支配してゐる$ @_vm
ii.「太郎は支配するだらう」はずっとその通りであった$ @_v\neg P\neg\Diamond m
iii.「太郎は支配するだらう」はずっと僞であった$ @_dP\neg\Diamond m
議論
iii. は命題 1 に依り必然$ @_d\square P\neg\Diamond m
ii. は iii. と矛盾するから不可能$ @_v\square\neg(\neg P\neg\Diamond m)
前提が iii. を歸結する$ @_d\neg Pm\to @_dP\neg\Diamond mやうに、i. は ii. を歸結する$ @_vm\to @_v\neg P\neg\Diamond mから、命題 2 に依り i. も不可能$ @_v\square\neg m
依って命題 3 を否定できる$ \neg\exist v@_vm
あるものが存在してゐる閒、それは存在しないといふことはできない$ x\to\neg\neg x
以下の三命題は全立しない
$ \neg Fx\land \lozenge x.
Something is possible that neither is true nor will be
現在存在しないし未來も存在しないだらうものは可能である
その實現が、現在でも未來でも決して起こらなかった可能態といふものがある
實現することが決してない可能態はある
實現しない可能性がある$ \neg\square Fx
反實假想未來
A. N. Prior に依る樣相演算子の定義を認めれば$ \neg Fx\land Fxと、古典論理では矛盾 (ⅱ) 行動 A は、それを行なふか行はないかが私の意のままになる
投企の時閒はこれを否定する (統一を失ふ = 虛構を許す) $ Fx\lor\neg\lozenge x.
https://gyazo.com/bd425b7deb0bb3fb136a8b3a3bd0b8ca
現在存在しないし未來も存在しないだろうものは不可能である
あらゆる可能態は、現在においても、未來においても實現する
抑止
$ (\lozenge x\land\neg\lozenge y)\to\neg(\lozenge x\to\lozenge y).
The impossible does not follow from the possible
不可能なものは可能なものから生まれえない
可能態から不可能態となる、そのやうな因果關係はよくない (ありえない)
不可能なことが、可能なことから歸結することはない$ ((x\to y)\land\square\neg y)\to\square\neg x
A. N. Prior に依る樣相演算子の定義を認めれば$ ((x\lor Fx)\land\neg y\land\neg Fy)\to\neg((x\lor Fx)\to(y\lor Fy))
(ⅰ) 行動 A は、その後に起きる事件 B と積極的な相關をもつ
歷史の時閒はこれを否定する (全體を失ふ = 假象を許す) $ (\lozenge x\land\neg\lozenge y)\land(\lozenge x\to\lozenge y).
https://gyazo.com/3584c73fbfb1d4f45935ff4a147fce94
不可能なものは可能なものから生まれうる
豫防
$ Px\to\square Px.
Every past truth is necessary
過去に存在したものの全ては眞かつ必然である
過去は撤囘できないものである
過去についての眞なる命題はすべて必然である$ Px\to\square Px
A. N. Prior に依る樣相演算子の定義を認めれば$ Px\to(Px\land\neg F\neg Px),$ @_c(\exist i_{\in\{i|i<c\}}@_ix\to\forall j_{\in\{j|c\leq j\}}\exist k_{\in\{k|k<j\}}@_j@_kx)
A. N. Prior の時相論理の公理$ x\to\neg F\neg Px,$ @_cx\to\forall i_{\in\{i|c<i\}}\exist j_{\in\{j|j<i\}}@_i@_jxに對應する (ⅲ) 私は、行動 A をとるかとらないかといふ私の意圖とは獨立に、B が起きるか否かを知り得る
說話の時閒はこれを否定する (明晰を失ふ = 乖離を許す) $ Px\land\neg\square Px.
過去は別樣で有り得た
敎訓