I have just begun studying Stoic Philosophy in a philosophy class of mine, and it is interesting to me that the Stoics had different doctrines of logic than modern logicians have. I thought I would take a brief look at two of these, both coming from Diogenes Laertius’ Logic and the Theory of Knowledge.
Diogenes Laertius says: “A disjunctive is that which is disjoined by the disjunctive conjunction, for example, ‘either it is day or it is night'” (7.72). It is clear by the way it is being used that what is here meant by a conjunction is what we today call a connective, and thus could be ‘and’, or any of the other connectives such as ‘or’ ‘if and only if’, ‘for’ et cetera. What is more interesting is that Diogenes goes on immediately afterwards to say that “this conjunction indicates that one of the two propositions is false.”
In this respect there is a notable difference between the Stoic system of logic and modern logic, since in modern logic a disjunctive connective (or ‘conjunction’) means only that it is not the case that both disjuncts are false, but allows that both may be true. For instance, if I say “A or B” and I establish that A is true, I cannot infer that B is not true. A and B may both be true, and the statement “A or B” would be true in virtue of either disjunct being true. The only way the whole disjunction is false is if both A and B are false. However, for the Stoics, according to Diogenes, the disjunction indicates that at least one of the disjuncts is false. I presume the Stoics would have also said that the whole disjunction is false if neither of the disjuncts are true, but it is interesting that they maintained that it was also false in the case that both disjuncts were true.
Counterfactual Conditionals and Conditionals
Diogenes says: “A conditional is true if the opposite of the conclusion conflicts with the antecedent, for example, ‘if it is day, it is light’. This is true; for ‘it is not light’, being the opposite of the conclusion, conflicts with ‘it is day’.” (7.73)
A good friend of mine, Yousuf Hasan, pointed out that this is very different from the modern conception of conditionals in logic (that is to say, in modern logic; though he did note that some ‘relevant-logics’ may adopt a more sympathetic view of the Stoic philosophy on this point). Consider the conditional “if the sky is blue, then I am currently in Montreal” (his example). In modern logic, suppose the antecedent were true, and the consequent true; the whole conditional would be true. However, for the Stoics, it looks like even if both the antecedent and the consequent were true, the conditional would not be true precisely because the negation of the consequent does not imply the negation of the antecedent.
This made me think about the theory of conditional statements of the form (A⊃B). We all learn in logic today that if the antecedent is false, the whole conditional must be true. However, though I recognize that this is a truth-functional commitment, I wonder if it accurately tracks the truth value of conditionals. Let us think of some counterfactual conditionals as examples. Suppose I think of “if Billy catches the bus, then Billy gets to school on time.” Suppose, then, that the antecedent is actually false (Billy does not catch the bus in the actual world). Does it follow from Billy’s not catching the bus that the conditional “if Billy catches the bus, then Billy gets to school on time” is true? No, clearly not, since there are logically possible worlds in which Billy catches the bus and still does not get to school on time.
Moreover, consider not counterfactual conditionals, but factual conditionals: eg. “if the sky is blue, then I am in Montreal”. Does it really follow from the antecedent and consequent’s being true that the conditional statement is true? Truth-functionally we want to say yes, but the point is that we only say yes because we have a vested interest in adopting a logic which is truth-functional. If we instead wanted to adopt a logic which was accurate at the expense of being truth-functional then our logic might look quite different (for instance, perhaps some statements would have no truth values at all, such as subjunctive counterfactual conditionals of creaturely freedom).
Coming back, then, to counterfactual conditionals, one might ask ‘in a logically possible world where P is false, and P⊃Q is true, what is the truth-maker for P⊃Q?’ Mere truth-functionality is not a truth-maker. Moreover, counterfactuals by their very nature seem to be propositions about other logically possible worlds, rather than propositions about the actual world (as indicated by being counter-factual). Then what would the truth-maker be? Perhaps it could be true if in all the nearest logically possible worlds the antecedent implies the consequent, but that isn’t strong enough to make it true in the actual world. Alternatively, perhaps one can stipulate that a conditional is true ceteris paribus, such that we can say “if you had planted that acorn, [ceteris paribus] then it would have grown into a tree.”
There may be some counterfactual conditionals which are true at our world not ceteris paribus, but rather in virtue of being necessary truths, such as P⊃P. If somebody objects that P⊃P has no propositional content (makes no assertion, or does not propose anything), then we can think of other better examples. Consider the conditional, which might be a counterfactual for some relativist with respect to preferred logics, that “if I accept the law of non-contradiction, then I do not reject the law of non-contradiction.” One who rejects the law of non-contradiction (conditionally) might accept it conditionally, but one who accepts it unconditionally cannot reject it (conditionally or unconditionally).
Here, there is an obvious truth maker because the truth of the antecedent causes the truth of the consequent. So, perhaps some conditionals can be straightforwardly true in this respect: that they are true when the antecedent’s truth is causally related to the consequent’s truth, or when the consequent’s truth is causally related to the antecedent’s truth (as, for instance, ‘if I am in Montreal then I am geographically located’ since being geographically located is part of the causal story or explanation of my being in Montreal, but my being in Montreal is not part of the causal story of my being geographically located). After that, one can have conditionals which are true ceteris paribus. After that, perhaps a conditional can be true in virtue of describing all the nearest logically possible worlds in which the counter-factual antecedent is true. Finally, a conditional can be true truth-functionally.