Dan Kervick, in his review of my Bertrand Russell on Modality and Logical Relevance, finds the book “confusing and difficult.” For example, Kervick says, “At times, Dejnoka seems to suggest only that Russell has an implicit modal logic. In other passages it is asserted that the modal logic is explicit” (Kervick 2003, p. 31). However, I indicate many times in the book that the logics are implicit (Dejnoka 1999, pp. 16, 17, 61 twice, 62, 66, 96), and there is an entire chapter devoted to paraphrasing Russell’s modal texts into implicit logics. I found it otiose to add “implicit” every time I wrote “logic.” Besides, it is obvious that Russell never expressly states any modal logics. I never thought anyone would think otherwise. Take it from me, I am talking about implicit modal logics.
Kervick does not understand what I mean when I say Russell rejects modal entities and modal notions, yet “functionally” has a modal logic a logic which “behaves as if it were” based on modal entities or modal notions, which “simulates” a modal logic which is based on modal entities or modal notions (Kervick 2003, p. 30).
The idea is simple, and it is Russell’s. I am finding implicit in Russell logical analyses of the same sort that Russell is always doing. Namely, Russell finds that often, “supposed entities can be replaced by purely logical structures [which] substitute [for the supposed entities] without altering [the truth-value] of the ... propositions in question” (Russell 1971, p. 326, my emphasis). The two most famous examples in Russell are his definition of numbers as classes of classes in Principia Mathematica, and his logical analysis eliminating definite descriptions in “On Denoting”.
Russell’s greatest achievement was to develop a logic which “functions” as, “substitutes” for, or “replaces” mathematics. He analyzes all arithmetical expressions away, and uses logical expressions in their place. No arithmetical entities are assumed, and no arithmetical notions are involved. Arithmetical entities and notions are eliminated across the board. Yet Russell can say and do everything in his logic that arithmeticians can say and do in arithmetic. This is just how I describe Russell as analyzing all modal expressions away and using quantificational expressions in their place (Dejnoka 1999, p. 2). Russell does not alter truth-values in modal logic when he rejects modal entities any more than he alters truth-values in arithmetic when he rejects numbers. He expressly preserves arithmetic, and if my formalization is right, he implicitly preserves modal logic, though I believe this is “surely unintentional on Russell’s part” (Dejnoka 1999, p. 97).
Kervick says that I seem “to be aware” that the concept of logical necessity and the concept of analyticity are “quite definitely” different for Russell (Kervick 2003, p. 36-37). “Yet,” he proclaims, “there is a surprise in store when Dejnoka turns [to define the implicit necessity operator of implicit FG-MDL]. For FG-MDL, it turns out, is based on reading ‘it is necessary that P’ as it is analytically true that A!” (Kervick 2003, p. 37, Kervick’s emphasis).
The eliminative analysis of necessity as analyticity is Russell’s (Russell 1994, p. 519), not mine. And once again, his idea is simple. Far from being a problem, such a difference is a necessary requirement of a successful logical analysis. For a logical analysis to be significant (informative), the analysans and analysandum must differ in connotative meaning; otherwise the analysis would be circular. In Russellian analysis, the sense in which they must be the same is extensional salva veritate, and the sense in which they must differ (prior to defining) is intensional. This is known as the paradox of analysis.
Imagine noting that the concept of a number and the concept of a class are “quite definitely” different for Russell, and then proclaiming “Yet there is a surprise in store when Russell comes to define number. For Principia, it turns out, is based on reading ‘number’ as class of classes!”
I state the formalization three times in Chapter Six:
S1. P → ◊P
I state it three times so as to cover seven implicit modal logics three alethic, one causal, one epistemic, and two deontic. I state that all seven implicit logics have the same S5 formalization, and differ only as to interpretation of the modal operators (Dejnoka 1999, p. 80). And I carefully discuss the paraphrase of Russell into each implicit logic one formal axiom at a time.
Kervick also says, “There also appears to be some confusion between modal logics and modal languages.... [T]he appropriate medium for paraphrases of Russell’s thinking would presumably be some sort of fully interpreted language, rather than a logic” (Kervick 2003, p. 31). Not at all. I say “logic” more times than I would care to count-thirty-six times in Chapter Six alone. Nor are the logics lacking an interpretation. Strictly speaking, describing Russell’s implicit interpretation is not necessary to my task of showing an S5 logic implicit in Russell. But I also describe Russell’s intended model for his quantified logic twice (Dejnoka 1999, pp. 72, 101).
Kervick goes on to observe that everyone uses casual modal language, even, say, the Marx Brothers, and that it would be otiose to delineate whatever modal logic might be implicit in the casual modal talk of the Marx Brothers (Kervick 2003, p. 32). This is disingenuous. I am paraphrasing a great logician’s technical theories concerning philosophical topics of modality, including several expressly stated semi-formal logical analyses, not the comedy routines of a vaudeville act.
Kervick says that modalities in MDL are not “relative to” specific variables (my term is “with respect to”), but apply non-relatively or simpliciter to entire propositional functions (Kervick 2003, p. 34). He says our interpretations of MDL are “significantly different” (Kervick 2003, p. 34), but does not explain why.
In fact Kervick’s interpretation of MDL is definable in terms of mine. For a propositional function is MDL-necessary simpliciter just in case it is MDL-necessary with respect to every specific variable it contains. But still my interpretation of MDL is the correct one. Russell describes MDL possibility as follows:
When you take any propositional function and assert of it that it is possible, that it is sometimes true, that gives you the fundamental meaning of ‘existence’. You may express it by saying that there is at least one value of x for which that propositional function is true. Take ‘x is a man’, there is at least one value of x for which this is true. That is what one means by saying that ‘There are men’, or that ‘Men exist’. (Russell 1971, p. 232, my emphasis)Note that where I say “with respect to x”, Russell twice says “of x for which”. This is a “smoking gun” text showing that MDL-possibility always binds a specific variable. The text is semi-formal and is thus more perspicuous than the casual talk of “any propositional function” in the same passage. Kervick’s repeated reliance on casual language is not a good idea for reading Russell. Russell is not an ordinary language philosopher, and what counts is how he formalizes things. Obviously, Russell would formalize this text as existential quantification, and as we know, the existential quantifier binds (“is relative to”) specific variables.
Let us think about the implications of this famous text. The text states that existence and MDL-possibility are defined as being the very same notion not always false. Thus existence and MDL-possibility are interchangeable salva veritate, even salva analycitate. Thus Kervick’s account implies that existence is predicated of propositional functions as simpliciter as MDL-possibility is. And that is absurd. The heart of the Frege-Russell logical revolution, multiple nested quantifiers, would be destroyed. And all the subtlety of MDL as I interpret it would be correspondingly lost, since the corresponding multiply nested modal operators would be destroyed.
Many modal statements are not even expressible on Kervick’s interpretation of MDL. For example, “Logical analysis is endless,” i.e., “Everything is a logical constituent of something”, or “(∀x)(∃y)Cxy” (compare Russell 1971, p. 202). On my account of MDL, this is synonymous with “(x)(◊y)Cxy”, but in Kervick-MDL, it is unwritable. Due to Russell’s repeated identification of existence and MDL-possibility as the same “fundamental logical idea” (Russell 1971, pp. 232, 254), Kervick cannot even write “(∀x)(∃y)Cxy”!
We may now distinguish four logical stages. Stage 1 is my version of MDL, on which a propositional function is necessary with respect to a variable it contains if it is always true with respect to that variable. This stage is faithful to Russell’s equation of possibility with existence, and of necessity with universality, since his existential and universal quantifiers are applied with respect to, i.e., bind, specific variables.
Stage 2 is Kervick’s version of MDL, on which a propositional function is necessary simpliciter if and only if it is always true with respect to every variable it contains. The previous sentence defines stage 2 in terms of stage 1, thus showing how to get along without stage 2, the only stage Russell never expressly defines.
Stage 3 is Russell’s definition in ‘Necessity and Possibility’ and ‘On the Notion of a Cause’ of a proposition as necessary with respect to a determinate constituent if, when we replace that determinate constituent with a variable, the resulting propositional function is always true. Stage 3 is definable in terms of either stage 1 or stage 2. In fact, the previous sentence states the definition, which may be taken either in Kervick’s way or mine.
Stage 4 is Russell’s analysis of a necessary proposition as analytic, where “Analytic propositions have the property that they are necessary with respect to all of their constituents except such as are what I call logical constants” (Russell 1994, p. 519, my emphasis). Clearly, stage 4 is definable in terms of any of the preceding stages. Thus all four stages are distinct only in reason.
We may also speak of a mix-and-match matrix. Stages 1 and 2 apply to propositional functions, while stages 3 and 4 apply to propositions. Stages 1 and 3 make modalities “relative to” specific variables or determinate constituents, while stages 2 and 4 do not.
Kervick calls MDL, or associates MDL with, my “second account of Russell’s modal logic,” my “modality as quantification account” (Kervick 2003, p. 37). He then criticizes MDL because it applies, and is intended by Russell to apply, modal notions to propositional functions, not propositions, and thus does not study logical relationships among propositions prefixed by modal operators (Kervick 2003, p. 38). Folks, MDL is not a modal logic! I indicate that eight times (Dejnoka 1999, pp. ix, 3, 16, 62, 80, 96, 194, 196). MDL is never on the list of seven modal logics (Dejnoka 1999, p. 16, 80). “MDL is not the modal logic” (Dejnoka 1999, p. 196), but the “basic element” (Dejnoka 1999, p. 16), the “building block” (Dejnoka 1999, pp. 96, 194), the “stepping-stone” (Dejnoka 1999, p. 3). MDL is stage 1. Only stage 4 is a modal logic, the early alethic FG-MDL. Kervick claims I give two accounts of FG-MDL, one analytic and one MDL-quantificational. But FG-MDL-analyticity is just what is definable (eliminable) in terms of MDL quantificational notions. This is just how FG-MDL functions as a modal logic without using modal notions. There is no second account.
Russell describes and rejects stages 1 (MDL), 3, and 4 (FG-MDL) in his landmark early paper, ‘Necessity and Possibility’, ca. 1903-1905. In that paper, Russell finds that no one theory captures all our modal intuitions, and concludes that the topic of modality ought to be banished from logic (Dejnoka 1999, p. 112; see 6). But if we stop there, we miss the big picture. Russell basically banishes modal entities and notions from then on. But the banishment of the topic ends the very next year when Russell accepts eliminative logical analysis MDL as his own theory of modality. Russell accepts MDL in eight published works from 1906 to 1940, a period of thirty-six years. Russell evidently accepts MDL from 1906 to the end of his life. And FG-MDL is definable in terms of MDL according to Russell’s own definition of “analytic” in ‘Necessity and Possibility’. Thus Russell implicitly accepts FG-MDL from 1906 until 1914, when he implicitly modifies FG-MDL into FG-MDL* by adding the requirement of truth in virtue of logical form (Dejnoka 1999, p. 8). Thus Russell implicitly holds mature logic FG-MDL* from 1914 to the end of his life, a period of fifty-six years. That is the big story of the tenth of the book Kervick reviewed. The FG-MDL* necessity operator is implicitly the Principia thesis assertion sign, construed as iterable; and so is the relevance logic entailment operator, implied by Russell’s repeatedly stated whole-part “containment” theory of logical deduction, which Anderson-Belnap overlook.
Dejnoka, Jan. 1999. Bertrand Russell on Modality and Logical Relevance. Aldershot, England: Ashgate.
Kervick, Dan. 2003. Review of Dejnoka 1999. The Bertrand Russell Society Quarterly, number 117 (February 2003), 29-38.
Russell, Bertrand. 1994. “Necessity and Possibility.” In Foundations of Logic 1903-05. The Collected Papers of Bertrand Russell, vol. 4. Alasdair Urquhart, ed. London: Routledge.
1971. Logic and Knowledge. Robert C. Marsh, ed. New York: Capricorn.