List: There are numerous formal systems of modal logic, which are differentiated by which axioms and inference rules they add to classical logic. The attached diagram shows the most discussed ones, arranged from weakest (left) to strongest (right) and showing how they are contained within each other. As indicated, those in the bottom row treat only classical tautologies and (for S1-S3) their added modal axioms as necessarily true, while the others extend this to all their theorems; in fact, that is the sole difference between S0.5 and T (sometimes called M). C. I. Lewis introduced S1-S5, and the two strongest of these are now generally considered to be the "best" for most purposes, as John P. Burgess observes.
JPB: The question Which is the right validity logic? has been answered at the sentential level, which is the only level that will be considered here: it is the system known as *S5*. This result is essentially established already in Carnap (1946). The question Which is the right demonstrability logic? goes back to the earliest days of modern modal logic. ... To the extent that there is any consensus or plurality view among logicians today, I take the view to be that the right demonstrability logic is *S4*. ( https://doi.org/10.1305/ndjfl/1039096306, 1999, p. 82) On the other hand, John L. Pollock proposes a "Basic Modal Logic" and calls it B, although it is very different from the system named B after L. E. J. Brouwer and shown in the attachment as containing T and contained in S5 but independent of S4. Pollock's approach is to eliminate all "iterated" modalities, which correspond to compounded expressions like "possibly possible" and "necessarily possibly necessary." He describes his motivation and results as follows (https://www.jstor.org/stable/2270778, 1967). JLP: [W]hen philosophers and logicians *apply* modal logic to concrete problems, they rarely need principles which involve iterated modalities. For most practical purposes, principles involving only one layer of modalities are all that are needed. This suggests that if we try to construct a theory of modal logic in which there are no iterated modalities, we can avoid most of the controversy and still have a theory that is strong enough for all of the normal uses to which modal logic is put. (p. 355) JLP: [A]ll of the controversy over which theory of propositional modal logic is correct stems from disagreement about principles involving iterated modalities. S1-S5 and M[=T] are the most common theories of propositional modal logic, and ... we see that they all give exactly the same theorems not involving iterated modalities. Thus there is a very real sense in which B is *basic* modal logic. B gives us the core theorems that everyone accepts. (p. 363) Jean Porte summarizes, "Indeed, *S1-S5* and Feys' *T* all have the same theses of modal degree at most one and, among the systems which have this property, *S5* is the strongest, and *S0.5* is the weakest" ( https://doi.org/10.1305/ndjfl/1093883250, 1980, p. 672). In other words, all theorems of S5 in which no propositional variable is within the scope of more than one modal operator, which together comprise Pollock's B, are also theorems of S0.5-S4 and T, as well as the Brouwer-inspired B. Moreover, "A formula of modal order greater than one is a thesis of S0.5 if and only if it is a substitution instance of a first-order thesis" (p. 675). Accordingly, Pollock's B matches E. J. Lemmon's original specification for S0.5 (https://www.jstor.org/stable/2964179, 1957, pp. 180-181), except that such substitution is precluded. Lemmon even anticipates this in a footnote, referencing William T. Parry's brief suggestion of the same idea, which he calls S.1 (https://www.jstor.org/stable/2266559, 1953, p. 328). As indicated in the attachment, without such a limitation, infinitely many higher-degree modalities are irreducible in all the systems shown except S3-S5. Even in S3 and S4, there are 42 and 14 non-equivalent modalities, respectively. By contrast, as G. E. Hughes and M. J. Cresswell point out, "Every formula of higher than first degree is reducible in S5 to a first-degree formula." They go on to present "an effective procedure for reducing any wff of higher than first degree to one of first degree by equivalence transformations" ( http://www.stoqatpul.org/lat/materials/huges_cresswell_modal_logic.pdf, 1996, pp. 98-101). Jean-Louis Gardies adds, "Due to this, every thesis of S5 may be related to a thesis being its translation in [Pollock's] B," although "different theses of S5 may have the same corresponding thesis of the first modal degree" ( https://filozof.uni.lodz.pl/~filozof/wp-content/uploads/logicatr/Gardies2.pdf, 1998, p. 32). Personally, I find this very appealing. After all, what does it mean to assert, "There is some possible state of things where there would be some possible state of things where the proposition A would be true"? Why and how is it different from merely asserting, "There is some possible state of things where A would be true"? As I noted previously, a fundamental principle of pragmaticism is that there are *real* possibilities, namely, those accessible from the *actual* state of things. With that in mind, since B is already taken, perhaps Pollock's "basic" system should instead be named P for him along with Parry, Peirce, and pragmaticism Regards, Jon S. On Thu, Apr 21, 2022 at 7:49 PM Jon Alan Schmidt <jonalanschm...@gmail.com> wrote: > List: > > In a recent post ( > https://list.iupui.edu/sympa/arc/peirce-l/2022-04/msg00003.html), I > discussed two alternatives for deviating from classical logic to reason > about a universe of discourse that is general rather than > individual--intuitionistic logic and multi-valued logics. I probably should > have included a third candidate, intensional modal logic, which retains > true and false as the only truth values but adds operators for possibility > and necessity that are commonly interpreted as analogous to the existential > and universal quantifiers, respectively. The idea is that rather than > quantifying predicates over subjects, they quantify propositions over > "possible worlds." > > C. I. Lewis is widely credited with launching this now-standard approach > in 1912 by drawing a distinction between (classical) material implication > vs. (modal) strict implication and insisting that the latter is the > "correct" formalization of a conditional proposition. However, as B. Jack > Copeland acknowledges at the very beginning of his historical overview ( > https://cdn.preterhuman.net/texts/science_and_technology/the%20genesis%20of%20possible%20world%20semantics.pdf > , 2002), Peirce anticipates this position decades earlier. > > CSP: A hypothetical proposition, generally, is not confined to stating > what actually happens, but states what is invariably true throughout a > universe of possibility. (CP 3.366, 1885) > > CSP: Now, the peculiarity of the hypothetical proposition is that it goes > out beyond the actual state of things and declares what would happen were > things other than they are or may be. The utility of this is that it puts > us in possession of a rule, say that "if *A* is true, *B* is true," such > that ... throughout the whole range of possibility, in every state of > things in which *A* is true, *B* is true too. ... If, then, *B* is a > proposition true in every case throughout the whole range of possibility, > the hypothetical proposition, taken in its logical sense, ought to be > regarded as true, whatever may be the usage of ordinary speech. If, on the > other hand, *A* is in no case true, throughout the range of possibility, > it is a matter of indifference whether the hypothetical be understood to be > true or not, since it is useless. (CP 3.374, 1885) > > CSP: The quantified subject of a hypothetical proposition is a > *possibility*, or *possible case*, or *possible state of things*. (CP > 2.347, c. 1895) > > CSP: The consequence *de inesse*, "if *A* is true, then *B* is true," is > expressed by letting *i* denote the actual state of things, *Ai* mean > that in the actual state of things *A* is true, and *Bi* mean that in the > actual state of things *B* is true, and then saying "If *Ai* is true then > *Bi* is true," or, what is the same thing, "Either *Ai* is not true or > *Bi* is true." But an *ordinary* Philonian conditional is expressed by > saying, "In *any* possible state of things, *i*, either *Ai* is not true, > or *Bi* is true." (CP 3.444, 1896) > > > He reiterates near the end of his life that some propositions purportedly > describing a general universe--namely, conditionals with antecedents that > would not be true in any possible state of things--are neither true nor > false, but vacuous. > > CSP: A conditional proposition,--say "if A, then B" is equivalent to > saying that "*Any *state of things in which A should be true, *would *(within > limits) *be *a state of things in which B is true." It is therefore > essentially an assertion of a *general *nature, the statement of a " > *would-be*." But when the antecedent supposes an *existential *fact to be > different from what it actually is or was, the conditional proposition does > not accurately state anything ... . A historian simply tells nonsense > when he says "If Napoleon had not done as he did before the battle of > Leipzig (specifying in what respect his behaviour is supposed different > from what it was,) he would have won that battle." Such historian may have > meant something; but he utterly fails to express any meaning. (RL 477:9-10, > 1913 Oct 14) > > > Note that Peirce here also dismisses conditionals whose antecedents are > "existential" and "counter to fact." As I explore at length in my paper, > "Temporal Synechism: A Peircean Philosophy of Time"--now finally > published officially in the journal *Axiomathes* ( > https://doi.org/10.1007/s10516-020-09523-6), nearly 18 months after first > appearing online (https://rdcu.be/b9xVm)--I interpret his writings to be > most consistent with the "growing block" theory. Accordingly, since > classical logic is for reasoning about actualities, it applies to the past; > and since modal logic is for reasoning about real possibilities and > conditional necessities, it applies to the future to the extent that it is > accessible from the present. > > CSP: That a possibility which *should* never be actualized, (in the sense > of having a bearing upon conduct that might conceivably be contemplated,) > would be a nullity is a form of stating the principle of pragmaticism. One > obvious consequence is that the potential, or really possible, must always > *refer* to the actual. The possible is what *can become actual*. A > possibility which could not be actualized would be absurd, of course. (R > 288:69[134-135], 1905) > > > This philosophical framework has bearing on which of the many different > formal systems of modal logic is most appropriate for such reasoning. > Perhaps that will be a subject for another post. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Structural Engineer, Synechist Philosopher, Lutheran Christian > www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt >
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