6 The Strength of Russell’s Modal Logic I discussed Russell’s theory of modality in chapters 2–4. It is time to discuss Russell’s modal logics and their strength. First, I sketch my methodology for imputing a modal logic to Russell. Then I distinguish Russell’s modal theory from his modal logic in general. Then I define seven modal logics which may be implicitly attributed to Russell, including three alethic logics, a causal logic, an epistemic logic, and two deontic logics, and assess their strength. I save the deontic logics for last because they are the most difficult. Next, I discuss grades of modal involvement and problems of paraphrase of ordinary de re modal talk, especially problems of quantification into modal contexts. Last, I refute Thomas Magnell’s critique of my 1990 Erkenntnis paper. My methodology is this. I shall impute a modal logic to Russell if either of two tests is met: (i) it is more reasonable than not to paraphrase Russell’s thinking into the modal logic, or (ii) it is more reasonable than not to suppose that Russell would have substantially assented to the modal logic as a paraphrase of his thought. Test (i) is Russell’s own philosophy of paraphrase applied to whole theories. Paraphrase is regimentation of informal discourse into a formal notation. Some replacement of vague thoughts by a more determinate notation is permissible, if that replacement is more reasonable than not. This involves our judgment in balancing the letter and the spirit of the discourse, in assessing what Russell meant or intended. It is ultimately a matter of philosophical interpretation. Russell states test (i) in his famous “Mr Strawson on Referring” (MPD 178–79). Test (i) is met to the extent that a certain modal logic is logically implicit in Russell’s thinking. Test (ii) boils down to test (i), if we suppose that Russell would have substantially assented to formal paraphrases of his thinking which are more reasonable than not. I think that supposition is fairly safe. But no one is perfectly rational all the time, so it may be safer to say only that tests (i) and (ii) are not wholly distinct in the case of thinkers such as Russell. I think there is a sense in which test (ii) is more speculative. Namely, test (ii) calls for express speculation about what Russell would or might have said, if asked. This is good. Such speculation is involved in applying test (i), and it 61 62 Bertrand Russell on Modality and Logical Relevance should be brought out expressly. But test (i) is the more basic, because more purely rational, test. It is not always the case that philosophers would assent to what is logically implicit in their thinking; it might, for example, lead to a reductio ad absurdum of their beliefs. To that extent there is tension between tests (i) and (ii). But the tension is resolved by the very fact that test (i) is more basic. My disjunctive two-prong test is somewhat weaker than a test of finding it more reasonable than not that Russell actually did admit the modal logics I impute to him. On the other hand, my test is far stronger than a test of merely finding these modal logics to be logically consistent with everything or nearly everything he says. On either prong, my test requires finding a substantial positive basis in Russell for imputing the logics to him. In fact, it may be called a substantial positive basis test. This concludes my discussion of methodology. We are after fair philosophical game, if not game to a mere historian’s taste. I distinguish Russell’s modal theory from his modal logic in general. Russell’s full modal theory is MDL {1,2,3}, in which MDL appears as level (3). MDL defines a propositional function F(x) as possible with respect to x just in case F(x) is not always false. From the ontological perspective, MDL {1} is basic to understanding Russell’s theory of modality. But from the logical point of view, MDL {3} is only the stepping-stone to the modal logics. I proceed to the task of imputing modal logics to Russell and assessing their strength. I do not impute a modal logic to Principles of Mathematics. I do impute the full range of Leibniz’s possible worlds as logically implicit in that work. In fact, the 1903 Russell takes such worlds far more seriously than Leibniz does. For Leibniz, they are mere ideas in God’s mind. For Russell, they would be mind-independent complex entities. Such worlds could very easily be the semantic basis for a modal logic. But in Principles, Russell expressly adopts G. E. Moore’s theory of logical necessity as being degrees of implication among propositions. I conclude that there is no possible worlds modal logic in Principles. There are only the possible worlds themselves. I suppose you could call the totality of degrees of implication among propositions a modal logic. But that seems too remote from what is ordinarily meant by the term “modal logic.” Russell’s first modal logic is FG–MDL, meaning fully generalized MDL. Russell presents FG–MDL in his unpublished paper, “Necessity and Possibility” (Russell 1994a; c. 1903–5). FG–MDL defines logical truths as truths which are fully general, i.e., truths containing nothing but logical The Strength of Russell’s Modal Logic 63 constants and bound individual and predicate variables. I follow Gregory Landini in proposing these definitions for FG–MDL: ~A =Df (F1...Fm, x1...xn)AF1...Fm, x1...xn and A =Df (›F1...›Fm, ›x1...›xn)AF1...Fm, x1...xn (Landini 1993) It is easy to see how FG–MDL is based on MDL. In FG–MDL, truths are necessary (“~”) or possible (“”) with respect to all the variables they contain. Russell’s second and mature modal logic is FG–MDL* (“*” is pro- nounced “star”). FG–MDL* defines logical truths as truths which are both (i) fully general in accordance with Landini’s definitions and (ii) true in virtue of their form, i.e.. tautologous. I explained why Russell abandoned FG–MDL and adopted FG–MDL* in chapter 1. I proceed to define the modal operators, using the usual boxes and diamonds to represent “Necessarily” by ~ and “Possibly” by . The modal operators, ~ and , must be defined differently in the two modal logics. In FG–MDL, “~” means “true when fully generalized using only universal quantifiers.” In FG–MDL*, “~” means “true when fully generalized using only universal quantifiers and true in virtue of its form.” Both interpreta- tions are modally innocent. They contain no modal notions. In FG–MDL*, the operator for necessity might be informally called “Analytically,” or even “Logically.” That will not cover the whole field of necessary truths, but Russell aims to cover only logically necessary truths. Iteration of modal operators is admissible in both FG–MDL and FG–MDL*, since we are prefixing whole propositions with the operators. We can iterate operators all we want. Theory of types is not a problem, since an iterated operator is never predicated of itself, but always of a proposition. I shall now discuss the strength of Russell’s two modal logics for logical necessity, beginning with FG–MDL*.1 The distinguishing axiom of alethic S1 is that if P, then it is logically possible that P. That is, P 6 P. That is trivially true in FG–MDL*, due to considerations of logical form. The distinguishing axiom of alethic S2 is that if it is logically possible that P and Q, then it is logically possible that P. That is, (P & Q) 6 P. That is trivially true in FG–MDL* as well. 64 Bertrand Russell on Modality and Logical Relevance The distinguishing axiom of alethic S3 is that if P deductively implies Q, then the logical possibility of P deductively implies the logical possibility of Q. That is, (P 6 Q) 6 (P 6 Q). That is clearly true in FG–MDL*. Iteration of logical modal operators is syntactically possible in FG–MDL*, since they are predicated of statements. Thus we may proceed to consider S4 and S5. Iterated modal operators entirely of the same kind will be trivially collapsible in FG–MDL*, since they will all have the same conditions of application, namely appropriate kind of logical form. Thus ~~P 6 ~P, and P 6 P. The only way it could be logically necessary for P to be logically necessary is if P is logically necessary in virtue of its logical form. And the only way it could be logically possible for P to be logically possible is if P is not logically impossible in virtue of its logical form. The distinguishing axiom of alethic S4 is that if it is logically possible that P is logically possible, then P is logically possible. That is, P 6 P. That is true in FG–MDL*, as we just saw. The distinguishing axiom of alethic S5 is that if P is logically possible, then it is logically necessary that P is logically possible. That is, P 6 ~P. In FG–MDL*, if a statement is possible, then it is necessarily possible, since its possibility is in virtue of its logical form, which is an essential part of the statement’s being the statement it is, and which is timeless and unchanging. And if a statement is necessary, then it is both necessarily possible and necessarily necessary, in virtue of its tautological character. Thus FG–MDL* is closest to S5; note that Arthur Prior and Kit Fine compare standard quantificational logic to S5 (Prior 1977: chapter 1).