# Diodorus Cronus

Diodorus Cronus (fourth century, B.C.E.) was a Greek philosopher of the Megarian school who made important contributions to the development of modal logic and theories of conditionals. His teacher was Apollonius Cronus, a student of Eubulides; he studied with Philo of Megara; and his most famous student was Zeno of Citium, founder of the Stoa. His five daughters, Menexene, Argeia, Theognis, Artemisia, and Pantacleia, are all said to have been logicians.

Diodorus examined the conditions under which conditional, “if…then” propositions might be true. Diodorus stated that a conditional proposition was true if there was never a time when the antecedent statement was true and the consequent statement was false. If there was any time when the consequent statement was false, the proposition was false.

Diodrous devised the Maser Argument, widely discussed during antiquity but now lost, to prove that, “Everything that is possible either is or will be true.” He also raised the paradox of future contingency with his question, “Will there be a sea battle tomorrow?” The logic of Diodorus influenced the logic of the Stoics, which was later taken up by twentieth century logicians. Diodorus Cronus’ modal theory and his Master Argument served as a major philosophical inspiration for Arthur Prior, who founded tense logic.

## Life

The most important philosopher of the Megarian school was Diodorus Cronus, who taught in Athens and Alexandria around 315-284 B.C.E. He left no writings, and his most famous pupil is Zeno of Citium, founder of the Stoa. Little is known about the life of Diodorus Cronus. Diogenes Laertius recounts two possible sources for the surname “Cronus.” One story is that Diodorus of Iasos, the son of Aminias, took the name of his teacher Apollonius Cronus, who was a student of Eubulides. Another story is that, while staying at the court of Ptolemy Soter, Diodorus was asked to solve a dialectical subtlety by Stilpo. When he was not able to answer on the spur of the moment, he was given the nickname “Cronus” by Ptolemy, referring to the God of time and mocking his slowness. He left the banquet, wrote an essay on Stilpo’s question, and died of despondency. Strabo, however, says that he took the name from Apollonius, his master. Laertius also credits him with being “the first person who invented the Concealed argument, and the Horned one” (Diogenes Laertius, Lives, "Life of Euclides," VII).

Like the rest of the Megarian school, he reveled in verbal arguments, proving that motion and existence are impossible. The impossible cannot result from the possible; a past event cannot become other than it is; but if an event, at a given moment, had been possible, from this possibility would result something impossible; therefore the original event was impossible. This problem was taken up by Chrysippus, who admitted that he could not solve it.

Diodorus Cronus’ five daughters, Menexene, Argeia, Theognis, Artemisia, and Pantacleia are all said to have been logicians. Philo of Megara studied with Diodorus; the logic of Philo and Diodorus influenced the logic of the Stoics, which was later taken up by twentieth century logicians. Diodorus Cronus’ modal theory and his Master Argument served as a major philosophical inspiration for Arthur Prior.

## Thought

Diodorus Cronus made important contributions in logic to the development of theories of conditionals and modal logic. Diodorus devised a forerunner of strict implication, and developed a system of modal concepts that satisfies the basic logical requirements of modern modal theory. In antiquity, Diodorus Cronus was famous for his so-called Master Argument, which aimed to prove that only the actual is possible.

### Conditional propositions

Historical evidence confirms that Diodorus conceived of logic as a logic of propositions. The Stoics later systematically developed propositional logic and created subtle classifications of predicates. At the time of Diodorus and Philo, philosophers distinguished between “simple propositions” and “complex propositions.” Simple propositions were either positive or negative statements of a single fact. Complex propositions were thought to be composed of two or more simple propositions, and could be disjunctions, conjunctions or conditional statements. Diodorus probably examined the conditions under which all three types of complex propositions might be true, but most of his work was with conditional, “if…then” propositions.

Diodorus stated that a conditional proposition was true if there was never a time when the antecedent statement was true and the consequent statement was false. If there was any time when the consequent statement was false, the proposition was false. It was not a requirement that the antecedent and consequent statements be relevant to each other in any way. The antecedent statement could be something impossible, even nonsensical, such as, “If the earth flies,” but if the consequent statement was always true, or necessary, the whole proposition was true. This created a “paradox of strict implication;” one example found in Greek texts of Diodorean-true conditional, "If it is not the case that there are indivisible elements of things, then there are indivisible elements of things" (SE, Outlines of Pyrrhonism 2.111), suggests that there was some awareness of these paradoxes in antiquity.

### Modalities

Diodorus considered the modalities “possible,” “impossible,” “necessary,” and “non-necessary” as properties of propositions or states of affairs, rather than as components of a proposition. One text reports all four definitions of Diodorus' modal notions: Possible is that which either is or will be true; impossible that which is false and will not be true; necessary that which is true and will not be false; non-necessary that which either is false already or will be false (Boethius, On Aristotle's On Interpretation 2.II.234-235). The modality of a particular proposition depended on its range of truth-values, in the present or in the future. A proposition that is always true, such as, ”If I walk, I move,” is both possible and necessary. If a proposition is sometimes, but not always, true, it is possible, but not necessary. The proposition, "It is day," is such a case, because it is true if spoken during the day, and false if spoken at night. In defining truth-values in terms of time, Diodorus and and other ancient Greek philosophers considered was true for their own time and place, but probably were not aware of time changes, or the fact that when it was night in Athens, it was day on the other side of the world. They did not include a specific date or time in their propositions. Since Diodorus based the modality of a proposition on what was true at a specific time, certain time-based propositions could change their modality from possible to impossible and from non-necessary to necessary as time passed. If the proposition, "Artemisia is five years old" was now true, then that proposition was now possible; but after she reached her sixth birthday, the proposition would become impossible, because it would never be true again. Diodorus also distinguished between propositions in the present tense like "Helen has three husbands" and "These men are marrying" and propositions in a tense of completion, "Helen had three husbands" and "These men married," and observed that it is possible for propositions like the latter two to be true, without there ever having been a time at which a corresponding one of the former type was true.[1]

### Master Argument

Diodorus used distinct claims to define what is “possible:” Everything that either is or will be true is possible, and, “Everything that is possible either is or will be true.” The first statement was not questioned by Hellenistic philosophers, but the second statement was considered counterintuitive required justification. Diodorus attempted to support the second claim with his Master Argument. (Epictetus, Dissertations 2.19). The Master Argument was widely discussed in antiquity, but the complete thread of the argument has been lost. One brief passage in the Dissertations of Epictetus makes reference to it:

There is a general conflict between the following three statements: (I) Every past true proposition is necessary; and (II) the impossible does not follow from the possible; and (III) something is possible which neither is true nor will be true. Being aware of this conflict, Diodorus used the plausibility of the first two statements in order to show that (IV) nothing is possible that neither is nor will be true (Epictetus, Dissertations 2.19.1).

Hellenistic philosophers generally regarded Diodorus' modal notions as jeopardizing freedom because they characterize as “impossible” anything that never happens or is never true. This amounted to a sort of logical determinism, since it limited the scope of contingency.

### Atomism

Diodorus Cronus is reported to have offered new arguments that there must be partless bodies or magnitudes, using logical arguments that depended on mutually exhaustive alternatives. Diodorus apparently used the idea that there is a smallest size at which an object at a given distance is visible, as the basis for an argument that there are indivisible magnitudes. His argument began with the idea that there is a difference in size between the smallest size at which a given object is visible, and the largest size at which it is invisible. Unless one concedes that there is a magnitude at which a body is both invisible and visible (or neither), there cannot be any other magnitude intermediate between these two magnitudes. Therefore, magnitudes must increase by discrete units. Sextus Empiricus (AM 10.48ff) also reported an argument of Diodorus' concluding that magnitudes have discrete intervals. The argument denied the existence of moving bodies, insisting that bodies move neither when they are in the place where they are, nor when they are in the place where they are not. These alternatives were presented as exhaustive, and the conclusion was that bodies are never moving. However, rather than assert that everything is static, Diodorus took the view that bodies must have moved without ever being in motion: They were simply at one place at one moment, and at another place at another moment.

### Contingency of a future event

The "problem of the future's contingents" is a logical paradox concerning the contingency of a future event, first posed by Diodorus Cronus under the name of the "dominator," and then reactualized by Aristotle in Chapter Nine of De Interpretatione. It was later taken on by Leibniz. Deleuze used it to oppose a "logic of the event" to a "logic of signification."

Diodorus' problem concerned the question: "Will there be a sea battle tomorrow?" According to this question, two propositions are possible: "Yes, there will be a sea battle tomorrow" or "no, there will not be a sea battle tomorrow." This was a paradox in Diodorus' eyes, since either there would be a battle tomorrow or there wouldn't be one. According to the basic principle of bivalence (A is either true or false), one of the two propositions had to be correct and therefore excluded the other. This posed a problem, since the judgment on the proposition (whether it was right or wrong) could only be made after the event had happened. In Deleuze's words, "time is the crisis of truth."[2] The problem thus concerns the ontological status of the future, and therefore of human action: Is the future determined or not?

#### Aristotle's solution

According to the principle of bivalence, something concerning reality is either true or false (A is B, or A is not B). Logic is thus based on disjunctive syllogism. This poses a problem when logic is applied to future possibilities instead of present reality. Diodorus' famous propositions are: "Will there be a sea battle tomorrow?" and/or "Will there not be a sea battle tomorrow?" Are future events determined or not? Logical necessity seems to be defeated by real necessity.

It can be said that the proposition is neither true nor false: Some possible futures make it true and others false; this may be called "indeterminacy intuition." It could also be said that the truth-value of the proposition will be only given in the future, that is, when the future unfolds. Thus, the truth value will always be given but never given in the present.

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: The observer 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 (Aristotle, On Interpretation §9).[3]

Diodorus concluded that the future battle was either impossible or necessary, meaning that the chain of causal events which would determine tomorrow’s action was already in place today. Aristotle added a third term, contingency, which preserves logic while at the same time leaving room for indetermination in reality. What is necessary is not that there will or that there won't be a battle tomorrow, but the alternative itself is necessary:

A sea-fight must either take place to-morrow or not, but it is not necessary that it should take place to-morrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place to-morrow (Aristotle. De Interpretatione'' 9, 19 a 30).[4]

Thus, an event always comes in the form of a future, undetermined event; logic always come afterwards. Hegel conveyed the same meaning by claiming that wisdom came at dusk. Aristotle also viewed this as a practical, ethical question: Pretending that the future is already determined would have unacceptable consequences for humanity.

## Notes

1. Sextus Empiricus, Against the Physicists 2.97-8.
2. Gilles Deleuze, Cinema 2: The Time-Image (1989), chapter VI, section 1.
3. E.M. Edghill, trans., On Interpretation by Aristotle §9. Retrieved May 21, 2007.
4. Ibid.

## References

• Ammonius, David L. Blank, Norman Kretzmann, and Boethius. 1998. On Aristotle's On interpretation 9. Ithaca, N.Y.: Cornell University Press. ISBN 0801433355
• Diogenes Laertius and Robert Drew Hicks. 1942. Lives of Eminent Philosophers. Harvard University Press.
• Epictetus and T. W. Rolleston. 1888. The Teaching of Epictetus: Being the Encheiridion of Epictetus; with Selections from the 'Dissertations' and 'Fragments.' London: W. Scott.
• MacFarlane, John. "Sea Battles, Futures Contingents, and Relative Truth" and "Future Contingent and Relative Truth," The Philosophical Quarterly 53 (2003), 321-36.
• Mates, Benson. 1949. Diodorean Implication.
• Mikel, Anton F. 1992. The Master Argument of Diodorus Cronus. Thesis (Ph. D.)—Florida State University, 1992.
• Sextus. 1949. Outlines of Pyrrhonism, Book I. Great Books Foundation series. Chicago: Regnery.
• Sextus and Robert Gregg Bury. 1987. Sextus Empiricus. The Loeb classical library, 311. Cambridge, Mass: Harvard University Press. ISBN 0674993446
• Spranzi, Marta. 1992. Diodorus Cronus and Aristotle: Motion, Partless Quantities and the Possible. Ferrara: Università degli Studi di Ferrara.