Essay:On Explication

Introduction
Russell was a pioneer: he effectively founded both analytic philosophy and scientific philosophy. However, this aspect of Russell is often neglected in favour of his political work. In particular, Principia Mathematica which he coauthored with A.N. Whitehead (PM hereafter) and Our Knowledge of the External World as Field for Scientific Method in Philosophy we’re incredibly influential on developing philosophical and scientific thought; scientific standards of rigour were bought to philosophy. Logic was now a central focus of philosophy I.e. instead of reasoning about intuitive entities, we instead focus on the logical form of sentences concerning those entities. Russell believed that Logic is the essence of philosophy. Russell’s method appealed greatly to some: “Philosophy from early times, has made greater claims, and achieved fewer results, than any other branch of learning. Ever since Thales said that ‘all is water’, philosophers have been ready with glib assertions about the sum-total of things; and equally glib denials have come from other philosophers ever since Thales was contradicted by Anaximander. I believe that the time has now arrived when this unsatisfactory state of things can be bought to an end. (Russell 1914a, p. 13).

Within this essay, I intend discuss these topic: rational reconstruction, explication/conceptual engineering, and the use of formal logic for dealing with problems that arise from the vagueness of natural language.

I want to discuss these topics, primarily, through the works of: Bertrand Russell, Rudolf Carnap, and W.V.O. We shouldn’t just focus on refuting someone’s argument; we should also focus on diagnosing where the argument went wrong.

Ordinary language distorts both our perception of the world and our reasoning about it. The formal methods of science, in contrast, present a new way of thinking, that has yet to realise its full potential: algorithmic methods for computation, Bayesian stats for abductive reasoning, and crucially the use of a logically perfect language (viz. first-order logic)—all help circumvene the distorting problems of ordinary language.

Conventions
== §1. Explicating the Empty Class ==
 * ‘:=’ is the definition symbol.
 * 1) ‘FS’ shall signify the formal system I am working in.
 * 2) ‘L’ shall signify the object-language of FS
 * 3) ‘ML’ shall signify the metalanguage of FS
 * 4) ‘…’ single quotation marks shall signify mentioning of object language expressions in the metalanguage of L.
 * 5) ‘para’ signifies my paraphrase of text from a source.
 * 6) ‘qte’ signifies quoting verbatim from the source, which shall be signified with standards quotation marks viz. “…”.
 * 7) ‘*’ signifies  my own notes.
 * 8) ‘(…)’ signifies parentheses, shall be used for references, parenthetical comments, and as, and when, they appear verbatim in quoted material.
 * 9) ‘[…]’ signifies square brackets, which indicate my modifications of quoted material.
 * 10) ‘{…}’ braces, shall signify mathematical classes.
 * 11) ‘&#9001;…,…&#9002;’ = angle brackets shall signify ordered pairs.
 * 12) ‘Λ’ in O(L) signifies the empty class.
 * 13) I may use shorthand where convenient (viz. where spellcheck isn’t convenient) e.g. ‘wrk’ for work or ‘cspdng’ for corresponding.

Background details and History of the Null Class
para: Viewed in the context of historical intellectual traditions, the empty class (null class, hereafter) can be viewed as serving as an extensional surrogate for age-old issues about the concept of ‘Nothing’, and the logical operation of Negation (The empty set, the singleton, and the ordered pair [ESO, hereafter], 274).

para: George Boole’s wrk in the algebraic trdtion of 19th century rprsntd peak extensionalism (ESO, 274). Boole introduced ‘0’ (The mathematical analysis of logic [1847, hereafter], 21), without explanation, and used it as an “elective symbol” that complimented his use of the numeral ‘1’ which, unlike ‘0’, was intended by Boole to denote the “Universe” (1847, ibid; ESO loc. cit. ). *However, Boole gave his binary numerals sundry directives*—para: ‘0’ and ‘1’ were paraphrased variously as predicates, classes, states of affairs, and as numerical quantities (ESO, loc. cit.).

para: Significantly, Boole asserted (1847, 3; ESO, loc. cit.) that in “Symbolic Algebra … the validity of the process of analysis doesn’t depend on the interpretation of the symbols employed, but solely on their laws of combination” (*emphasis is mine*); *in quote Boole appears to be claiming that validity can be demonstrated purely by syntactic means viz. without any recourse to semantics.*

para: In Boole’s better known “An Investigation into the Laws of Thought” [1854, hereafter] (1854) Boole had “signs” representing “classes” and he had signs accommodating the arithmetical property of 0 viz. that ‘0 × y = 0’ for every ‘y’ assigned (1854, 47) to “0” the interpretation “Nothing” viz. the class consisting of no individuals (1854, ibid; ESO, 274). Moreover, Boole makes clear in this work (1854, 28) that the “meaning of class shall be extended so as to include […] the cases denoted by the terms ‘nothing’ and ‘universe’ which as classes shall be understood to comprise respectively [the class of] ‘no beings’ [viz. what ‘nothing’ denotes] and [the class of] ‘all beings’ [viz. what ‘everything’ denotes]. According to Kanamori it can “justifiably be argued that Boole invented the null class” (ESO, loc. cit.). Boole was part of the “algebraic tradition” members of which tried to mathematically analyse logic; both C.S. Peirce and Schröder, both of whom also belonged to the algebraic tradition, adopted and adapted Boole’s use of “0” (ESO, loc. cit.).

para: Unlike the Algebraic tradition who tried to mathematically analyse Logic, Frege tried to provide a Logical analysis of mathematics; and the null class played a key role in Frege’s analysis of number (ESO, 274). In his “Grundlagen” Frege eschewed the terms ‘set’ (“Menge”) and ‘class’ (“Klasse”) but regardless the extension of the concept ‘not identical with itself’ was key to Frege’s definition of zero as a logical object (ESO, loc. cit.). Frege held that Logic gives unity to a class as the extension of a concept (*i.e. in this case, the concept of ‘zero’, or equivalently, the concept of ‘nothing’*) thus makes the null class viable (ESO, loc. cit.).

para: Peano (cf. 1889) used upper-case lambda (‘Λ’) to denote both the falsity of propositions, and to represent the null class (ESO, 275). Peano later (1897) provided a definition of the null class:= the intersection of all classes; making it explicit that there is exactly one null class (ESO, loc. cit.). Moreover, in the same year (cf. 1897), Peano produced the first occurrence of the—now cannonical—symbol ‘&#8707;’ for signifying the existential quantifier; Peano used ‘&#8707;’ to indicate that a class is not identical to the null class (ESO, loc. cit.). Whilst Frege took the existential quantifier to be derivative from the universal quantifier viz. from ‘~(x)~’, for Peano the existential quantifier is intimately related to the null class (ESO loc. cit.) qte: “Thus, in the logical tradition the null class played a focal and pregnant role” (275, Kanamori, sic.).

para: It was during the emergence of class theory, that the null class came to the fore as a primitive notion (ESO, 275). The grandfather of class theory himself—Georg Cantor—didn’t assign much importance to the null class (ESO, loc. cit.). For example, early on during Cantor’s study of “point sets” (“Punktmengen”) of Real numbers, Cantor proposed (1880, 355) that “for the absence of points … we choose the letter ‘O’ such that ‘P←→O’ indicates that the set P contains no single point'”; ‘←→O’  is better viewed as being a predication'' for ‘being empty’ (ESO, loc. cit.).

qte: “Richard Dedekind in his groundbreaking essay on arithmetic Was sind und was sollen die Zahlen? deliberately excluded the null class “for certain reasons”, though he saw it’s possible usefulness in other contexts” (ESO, 275). para: Whereas Frege, the logician, tried to get at what numbers are via definitions based on equinumerosity; Dedekind, the mathematician, worked to define the number sequence structurally, up to isomorphism or “inscrutability of reference” (ESO, loc. cit.). *Frege was focussed on finding one correct analysis; Dedekind was perhaps aware, that there isn’t one correct/ultimate “analysis”.* para: Moreover, whereas zero was crucial to Frege’s logical investigations, the particular “base element” was immaterial for Dedekind’s work, and he simply denoted it by the symbol ‘1’ before proceeding to define the numbers by “abstraction” (ESO, loc. cit.).

para: Zermelo (1908a) wrote in his Axiom II: “There exists an improper set, the null set 0, that contains no elements at all”, and due to Zermelo’s Axiom I of extensionality, there exists only one null set (ESO, 275). Felix Hausdorff the qte: “first developer of the transfinite after Cantor […] unequivocally opted for the null set (Nullmenge)” (ESO, loc. cit.) in his Grundzüge der Mengenlehre. pIn addition, Hausdorff rejected the notion that the null class doesn’t exist, insisting that it did exist but with no elements in it (ESO, 276). In particular, Hausdorff employed the null class as the extension of the conjunction of mutually exclusive properties.

para: Just as ‘0’ became a basic placeholder for numeral systems, so too did ‘Λ’ for the mathematics of classes to indicate the empty extension (ESO, 276).

Nothing, nonreferring names, and nonexistence
The Importance of Logic for Philosophy (TILP, hereafter) was delivered by W.V.O. Quine as a guest speaker at the Harvard Philosophy Club in 1947.

para: The word ‘nothing’ behaves in ordinary language like a noun, which misleads people into treating ‘nothing’ like a name, which in-turn engenders fallacious reasoning about ‘nothing’. In general the use of ‘nothing’ engenders confusion because the noun form that ‘nothing’ has, induces in us the belief that it is a thing, and that ‘nothing causes an event’ is to say that something, called “nothing”, has caused the event.

Something similar seems to have been going on in Heidegger’s mind when he asked “What about Nothing? What does Nothing do?”, and to resolve these issues Heidegger decided that “Nothing nothings”, and that we have a special way of sensing nothing, this sense being anguish. Lewis Carroll has demonstrated that such handling of the word ‘nothing’—a false substantive—can lead to ridiculous results.

For example, take the following sentence: ‘Nothing causes x’, this sentence can be translated into symbolic logic, thus, which translates back into ordinary language as With this translation there is no term corresponding to ‘nothing’, and hence no opportunity to apply false inferences that depend on treating ‘nothing’ like a name. In FOL false susbstantives such as ‘nothing’, ‘something’, ‘everything’, simply vanish. Translation into first-order Logic enables us to translate a troublesome segment of ordinary language into another segment which is well-behaved. Idiomatically, (2) may be construed as Significantly, the translation into FOL operates only on whole sentences; the word ‘nothing’ has no translation by itself, and due to this, this part well-behaved part of language includes no misleading substantive such as ‘nothing’. Thus, the perplexities and fallacies conjoined to the word ‘nothing’ are mere artefacts of an eliminable part of our language i.e. by this process ‘nothing’ may be viewed as either having been, There is, however, no significant difference between (4) and (5)—the difference is merely between different ways of phrasing one and the same thing.
 * 1) (y) ~(y causes x),
 * 1) Whatever y may be, y does not cause x.
 * 1) Whatever you may select, it does not cause x.
 * 1) dispensed with in favour of (2), or having been
 * 2)  identified with, and explained in terms of, (2).