A Survey of Symbolic Logic

Description

My friend mentioned this when I asked him what modal logic is:

Modal logic (to give you a very condensed overview of what it is and how I happen to use it in my own renewed ontological interpretation of physics based on the idea of substantial relation and the tetra-categorization of the concept of existence) is basically designed to extend the field of propositional logic by introducing two new categories of existence-based analysis to express the states or modes under which things are necessarily (□) and possibly (◊) true or false.

In a paper I wrote in 2013 (submitted but not yet published), entitled The Scholastics’ Neglected Heritage: Thought and Denotation Before the Post-Aristotelian Development of Logic , I address the power and relevance of its use in relation to a renewed treatment of St. Anselm’s ontological argument through both the limits and strengths of Gödel’s quantified modal retranslation and proof of the same argument.

What is called (or used to be called) alethic logic (from the Greek aletheia) is concerned with modalities , or modal statuses relative to the truth of propositions. Modalities are used to logically treat of necessarily true/false, possibly true/false, or contingently true/false information. Modal logic goes back to Aristotle himself and the representatives of the Megarian school (e.g. Eubulides of Miletus).

Around 1918, Lewis (Clarence Irwing) introduced new logical calculi using modalities (see A Survey of Symbolic Logic). He came up with the operators commonly used today in both standard and second-order modal logic. He also introduced the strict conditional (or strict implication) operator (a slightly squeezed <). For example, the strict conditional allows you to account for strict deductibility (crucial in mathematical logic and physics):

(p < q) = DF ∼◊ p(p & ∼ q)

The strict deductibility (by way of strict implication) of q from p means that it is not possible that p be true and q false. If p is true, then q is necessarily true, or:

(p < q) = DF □(p ⊃ q)

Modal logic thus basically deals with propositions built upon modes of existence :

p : the proposition is true in actuality (contingently and/or necessarily).

◊ p : the proposition is possibly true.

□ p : the proposition is necessarily true.

My contention is that existence requires the inclusion of its own negation (namely the negation of ◊ p , of impossible existence) to be treated in a way which allows logic to more exhaustively (though not completely) be used as consistent footing of mathematical physics.

What I call the tetra-structural relation of existence (and denote E ∋ 0 → 3) can be expressed in standard modal logic (NB: here I mix my own notation, in bold , with standard modal notation, p , ◊, and □):

(0) Impossibility (ncb) ≡ ∼◊ p ; (1) Potentiality (cb) ≡ ◊ p ; (2) Contingency (cc∼b) ≡ ∼□ p ; (3) Necessity (cn∼b) ≡ □ p.

Thus instead of three values of existence (E) modally computed as follows:

E 0 ** => E 1 E 1 ** => E 0

E 1 => E2 E 2 ** => E 1

E 0 ** => E2 E 2 ** => E 0

The modal inclusion of impossibility will give us the ability to generate 4 permutations of modally accountable truth values of existence (E ≡ E ∋ 0 → 3**)

E 0 ** <=> E 1 ** <=> E2 ** <=> E 3

E 1 <=> E0 <=> E3 ** <=> E 2

E 2 ** <=> E3 ** <=> E0 ** <=> E 1

E 3 ** <=> E2 ** <=> E1 ** <=> E 0

Briefly about Ryan’s remarks on Hilbert: while his certainly was a committed and optimistic attempt to secure the self-contained premise of set-theoretic formalization in order to thoroughly severe mathematics from all extra-axiomatic reference (and thereby fulfilled Leibniz’s and Boole’s common dream of a perfect artificial language for deductive reasoning), I can appreciate that Hilbert’s program deserves more than is usually acknowledged (115 years after his address to the international mathematical community gathered in Paris).

When I say he was a minimal Platonist, I mean it quite respectfully, as Hilbert finally was and remained, amidst the growing community of lesser and positivistically-driven thinkers, an absolutist in his own right. Yet he also did say: “Mathematics is a game played according to certain simple rules with meaningless marks on the paper.” In asserting this (like the true color of his programmatic statement), he set the stage for the Bourbakist conception of an autarchic mathematics free of ontological intrusions from the pre-mathematical order. All one finally needs to do is be completely thorough as far as axioms, algorithmic rules of inference, and theorems to generate an absolutely complete and consistent formal system. There lies Hilbert’s absolutism: deflating all of mathematics by formalizing it in axiomatic form. An ambitious and optimistic program that Gödel did a little more than just put a quiet halt to by way of a clever version of the liar’s paradox. Hilbert’s absolutism was especially directed toward securing the consistency of the axioms of arithmetic. Without such an absolute demonstration, the axiomatic solution to the foundations of mathematics would simply be left to the formalists’ dream land. And, honestly, this is what Gödel’s devastatingly paradoxical proof reduced Hilbert’s program to (the stuff of a dream land). The crucial nature of Gödel’s mere adaptation of “the liar paradox via Russell’s paradox” (which is not quite accurate to say; it is more accurate to say that Gödel opens his incompleteness proof by reprising the liar’s paradox) lies in his reverse-like use of inconsistency to eventually (once and for all!) prove that any attempt to consistently formalize elementary arithmetic as a first order formal system recursively axiomatizable, will be incomplete—i.e. there will always be some statement about natural numbers that can neither be proved nor disproved in the system.

To end with (as I must), I would not trust Stephen Hawking’s take on Gödel (nor would I trust him on most of what he writes and has to say, but this is for another conversation).

Sorry for the lengthy note. And thank for the previous (yes indeed interesting!) remarks, information, and references.

Sub umbra Altissimi.

Sebastien