John, List: JFS: For every version of first-order logic, there is a fixed domain D1 of entities in the domain of quantification. Those entities could be anything of any kind--that includes abstractions, fictions, imaginary beasts, and even hypothetical or possible worlds.
When using Beta EGs to implement first-order predicate logic (FOPL), the fixed domain of quantification consists of *indefinite individuals *within the universe of discourse, such that heavy lines of identity (LoIs) correspond to variables in the standard notation. General concepts are attributed to those individuals by attaching names to LoIs, thereby making those individuals more definite and those concepts more determinate. Your position seems to be that modal logic is not just *analogous *to FOPL, but *formally equivalent* to FOPL--the fixed domain of quantification now consists of *possible worlds*, and the "predicates" being attributed to them are propositions that would be true in them. Accordingly, what Peirce scribes on R 339:[340r] could be interpreted as a new application of Beta EGs instead of a plausible candidate for the new Delta EGs. I continue to be skeptical of this suggestion because there is an obvious and fundamental *semiotic *difference between describing *things *with names (rhemes/semes) and describing *states of things* with propositions (dicisigns/phemes). As a simple test, we can examine whether the usual modal axioms can be translated into theorems of FOPL as implemented using Beta EGs. Distributive axiom K = □(*p* → *q*) → (□*p* → □*q*) becomes ∀*x*(P*x* → Q*x*) → (∀*x*P*x* → ∀*x*Q*x*), which is valid. However, serial axiom D = □*p* → ◇ *p* becomes ∀*x*(P*x*) → ∃*x*(P*x*), which is invalid--as Peirce himself emphasizes repeatedly, an oddly enclosed (shaded) LoI does not assert the *existence *of anything. Moreover, reflexive axiom T = □*p* → *p* cannot be translated into a well-formed formula (WFF) at all because there is no counterpart for asserting a proposition to be true in the *actual *world--a name must *always *be attached to at least one LoI. Symmetric axiom B = ◇□*p* → *p*, transitive axiom 4 = □*p* → □□*p*, and euclidean axiom 5 = ◇□*p* → □*p* likewise cannot be translated into WFFs because there is no counterpart for iterated modalities. The first two limitations might be overcome by implementing a different version of FOPL that includes existential import (serial) and singular terms (reflexive), with corresponding modifications to Beta EGs, but the others would require stipulated axioms just like the corresponding modal systems. JFS: Second order logic is the only kind of higher order logic that anybody uses for any practical applications in any version of science, engineering, or computer systems. ... Logicians (usually graduate students who need to find a thesis topic) publish papers about such things in the *Journal of Symbolic Logic*. And the only people who read them are graduate students who need to find a thesis topic. I would caution against making such sweeping and dismissive pronouncements. After all, there might very well be applications of logics beyond second-order in science, engineering, or computer systems that have not yet come to your attention or that get discovered in the future, perhaps by one of those graduate students. In any case, I remind you again that according to Peirce, "True science is distinctively the study of useless things. For the useful things will get studied without the aid of scientific men" (CP 1.76, c. 1896). JFS: In the passage below by Jay Zeman, "a different kind of line of identity, one which expresses the identity of spots rather than of individuals. This is an intriguing move, since it strongly suggests at least the second order predicate calculus, with spots now acquiring quantifications. Peirce did very little with this idea, so far as I am able to determine," Jay mistakenly used the term "second order PC." There is no quantified variable for some kind of logic. On the contrary, Zeman's statement is *not *mistaken. In the Gamma EG that he is discussing (CP 4.470, LF 2/1:165, 1903), a special LoI with dotted lines on either side of it facilitates quantification over general concepts (predicates) in addition to the usual quantification over indefinite individuals corresponding to ordinary LoIs in Beta EGs. In your own words, "For second order logic, the domain D2 consists of all possible functions and/or predicates that range over entities in D1." JFS: But these examples are a small fraction of the many instances of metalanguage throughout Peirce's publications and MSS. Once you start looking for them, you'll find them throughout his writings. I am not disputing this. I just see no evidence in R 514, R L376, or elsewhere to support the *specific *claim that what Peirce has in mind for Delta EGs is adding metalanguage to Beta EGs, since his only stated reason for needing "a *Delta *part" at all is "in order to deal with modals." It seems much more plausible that he was considering a new notation for representing and reasoning about modal propositions to replace his unsatisfactory broken cuts (1903) and tinctures (1906), such as the one that he introduces on R 339:[340r]. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Sat, Mar 9, 2024 at 12:02 PM John F Sowa <[email protected]> wrote: > Jeff, Jon, List, > > In his 1885 Algebra of Logic, Peirce presented the modern versions of both > first-order and second-order predicate logic. The only difference between > his notation and the modern versions is the choice of symbols. Since > Peano wanted to make his logic publishable by ordinary type setters, he had > to avoid Peirce's Greek letters and subscripts. Therefore, he invented the > practice of turning letters upside-down or backwards, which type setters > could do very easily. > > For every version of first-order logic, there is a fixed domain D1 of > entities in the domain of quantification. Those entities could be anything > of any kind -- that includes abstractions, fictions, imaginary beasts, and > even hypothetical or possible worlds. For second order logic, the domain > D2 consists of all possible functions and/or predicates that range over > entities in D1. > > Second order logic is the only kind of higher order logic that anybody > uses for any practical applications in any version of science, engineering, > or computer systems. When they use the term HOL, they actually mean some > kind of second order logic, which may be the one described above or > something with a different way of specifying D2. > > The first (and most widely cited or defined) version of higher order logic > that goes beyond second was developed by Whitehead and Russell (1910). It > goes beyond second order logic by introducing domains D3, D4,..., which are > so huge that nobody has ever found a use for them in any practical > application. > > Given D1 and D2 as above, W & R specified D3 as the set of all possible > functions or predicates that may be defined over the union of D1 and D2. > Then D4 is defined over the union of D1, D2, D3. And so on. Logicians > (usually graduate students who need to find a thesis topic) publish papers > about such things in the Journal of Symbolic Logic. And the only people > who read them are graduate students who need to find a thesis topic. > > Peirce never went beyond second order logic. But any statement in any > language or logic about any language or logic is metalanguage. Since that > word was coined over 20 years after Peirce, he never used it. But there > are many uses of metalanguage in Peirce's publications and MSS. But he > never chose or coined a word that would relate all the instances. > > In the example that Jon copied below, "the line of identity denoting the *ens > rationis",* Peirce used the term 'ens rationis' for that example of > metalanguage. But he described other examples with other words. > > In the passage below by Jay Zeman, "a different kind of line of > identity, one which expresses the identity of spots rather than of > individuals. This is an intriguing move, since it strongly suggests at > least the second order predicate calculus, with spots now acquiring > quantifications. Peirce did very little with this idea, so far as I am able > to determine", Jay mistakenly used the term "second order PC". There is > no quantified variable for some kind of logic. It is just another example > of metalanguage that makes an assertion about the EG. > > There is much more to say about metalanguage, which I'll discuss in a > separate reply to Jon. But these examples are a small fraction of the many > instances of metalanguage throughout Peirce's publications and MSS. Once > you start looking for them, you'll find them throughout his writings. > Unfortunately, Peirce had no standard terminology for talking about them. > > I hate to say it, but this is one time when I wish Peirce had found a > Greek word for it. > > John >
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