Abstract
In this article I am going to argue for the possibility of a transcendental source of logic based on a phenomenologically motivated approach. My aim will be essentially carried out in two succeeding steps of reduction: the first one will be the indication of existence of an inherent temporal factor conditioning formal predicative discourse and the second one, based on a supplementary reduction of objective temporality, will be a recourse to a time-constituting origin which has to be assumed as a non-temporal, transcendental subjectivity and for that reason as possibly the ultimate transcendental root of pure logic. In the development of the argumentation and taking into account W.V. Quine’s views in his well-known Word and Object, a special emphasis will be given to the fundamentally temporal character of universal and existential predicative forms, to their status in logical theories in general, and to their underlying role in generating an inherently non-finitistic character reflected, for instance, in the undecidability of certain infinity statements in formal mathematical theories. This is shown also to concern metatheorems of such vital importance as Gödel’s incompleteness theorems in mathematical foundations. Moreover in the course of the discussion the quest for the ultimate limits of predication will lead to the notions of separation and intentional correlation between an ‘observing’ subject and the object of ‘observation’ as well as to the notion of syntactical individuals taken as the irreducible non-analytic nuclei-forms within analytical discourse.
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Notes
Husserl’s Späte Texte über Zeitkonstitution (Late Texts on Time Constitution), Husserl (2001b), edited by D. Lohmar and published in 2006, offer to the interested reader many insights to Husserl’s tantalizing efforts to reach a proper foundation for a timeless ego while doing justice all the same to its ‘self-reflexion’ as an absolute temporal flux; see, for instance, pp. 198–199.
See Husserl (1976b, pp. 33–35).
The meaning of the phenomenological terms noetic and noematic are mainly described in E. Husserl’s Ideen I [see: Husserl (1976b, pp. 230–231)]. A noematic object is constituted by certain modes as a well-defined object immanent to the temporal flux of a subject’s consciousness and it is possibly abstracted, in the sense of a formal-ontological object, as a syntactical object of a formal theory. A noematic object can be said to be given apodictically in experience inasmuch as: (1) it can be recognized by a perceiver directly as a manifested essence in any perceptual judgement (2) it can be predicated as existing according to the descriptive norms of a language and (3) it can be verified as such (as a reidentifying object) in multiple acts more or less at will. In contrast, a noetic object can be only characterized in terms of the multiplicities of real moments of a hyletic–noetic perception intentionally directed to it by its sole virtue of being given as an absolute evidence.
The terms immanent and intentional, among others, which are used very often in this text are very common in phenomenological analysis and familiar to any reader with a minimum of knowledge of phenomenology. Yet for the sake of self-sufficiency of the content of this article I enter hereby a very brief description.
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Intentionality is a phenomenological notion which is not to be understood as a relation of a psychological character towards the objects of experience. To a non-expert in phenomenology it can be roughly described as grounding the a priori necessity of orientation of a subject’s consciousness towards the object of its orientation.
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An immanent object and generally immanence is thought of as a correlate of intentional consciousness in contrast with a transcendent to the consciousness common physical object whose objectivity is anyway put by phenomenology into brackets.
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Concerning the meaning of identity in logic and the underlying ambivalence relative to this concept one should take into account a persisting confusion in positing under the same terms the identity between the signs of a language and the identity of corresponding physical objects. In any case, Quine’s reference to the Heraclitean flux, Hume’s claim of the non-identity of physical objects and also Whitehead’s and Wittgenstein’s essential refutation of the notion of the identity of objects can be taken as pointing, in a proper interpretation, to a notion of temporal-kinetic foundation of the concept of identity (Quine 1960, pp. 116–117).
The hyletic–noetic perception described in terms of corresponding moments of intentionality based on sensory (hyletic) data is not by necessity conditioned on a temporal-noematic constitution, this latter considered by Husserl as ‘posterior’ to noetic perception. For instance, the figure of a tree trunk, as a noematic object, is the invariably one and the same immanent object of a temporal consciousness constituted through a multiplicity of real hyletic–noetic moments of the concrete experience intentionally directed to the particular tree trunk in its various angles, adumbrations, etc. (Husserl 1976b, p. 226).
As well-known the universal–existential forms of quantification are logically interconnected since \(\forall x\; P(x)\) is equivalent to \(\lnot \exists x \;\lnot P(x)\).
The term metatheoretical as applied here and elsewhere in the text should be taken in a broad sense as referring to a theory pertaining to and yet transcending the bounds of formal theory in the first place, and further as ultimately associated with a subjective constitution within the world of phenomena.
By Weierstrassian sense I basically mean the application of some form of the well-known \((\epsilon , \delta )\)-Weierstrass formula which is a universal–existential quantification formula initially intended to dispense with the need of introducing infinitesimal numbers in coping with the subtleties of differential calculus.
Let it be noted here that predication in the Aristotelian Categories is associated with the definition of the primary and secondary substances where the former ones as substances in the strictest and primary sense of the term are defined to be those which are neither asserted of nor are present in the subject of a predication form. For instance, such primary substances can be regarded a particular man or a particular horse, which in their most abstract sense of a certain ‘something’ [τóδε τl], can be regarded as grounding the irreducible character of syntactical individuality (Aristotle 1983, pp. 18 and 28).
We take each \(I_{i}\) to be a countable set having as elements parametrized temporal instants corresponding to original impressions with the exclusion in each phase of instants corresponding to previous objectifications.
The statement (2) makes an assertion about the cardinality of prime numbers up to a certain bound, something that is dependent on the asymptotic character of the Prime Number Theorem and the inherent vagueness of the particular distribution law which is moreover associated with the intricacies of the still unsolved Riemann hypothesis. The Prime Number Theorem states that the limit of the quotient of the following two functions: \(\pi (x)\) (the function that gives the number of primes less than or equal to a real number x) and \(\frac{x}{{\text{ ln}}(x)}\), as x approaches infinity, is 1; this is expressed by the formula \(\lim \limits _{x\rightarrow +\infty } \frac{\pi (x)}{\frac{x}{{\text{ ln }}(x)}}=1\).
The equivalence between the two descriptions can be easily seen by associating with each function F from N to {0, 1}, the subset X of all naturals n such that F(n) = 1; inversely, each member X of S(N) determines a function F given by the equivalence \(F(n)=1\;\leftrightarrow \;n\in X\).
Nonetheless one can say that even in intuitionistic analysis one has to resort in rem to some implicit sense of completed infinite totality. One may cite, for instance, the continuity of functions in intuitionism founded on the notion of choice sequences taken as complete objects (Heyting 1966, pp. 42–46).
In this assertion the predicate \({\mathcal {X}}(u,w)\) formally represents a recursive map of finite sequences. The application of this assertion is a key part in the proof of the general form of Tarski’s undefinability of truth lemma (Kunen 1982, pp. 40–41).
Cantor’s diagonal argument (method) is a simple but ingenious technique introduced by G. Cantor to prove the existence of sets that cannot be put onto one-to-one correspondence with the set of natural numbers; in other words, it proves the existence of uncountable sets. The method essentially consists in constructing a set whose elements do not belong to any conceivable list of sequences of 0s and 1s, yet this is implicitly conditioned on the concept of an infinite set as a completed totality.
There is a slightly more complicated instance in Rosser (1936, p. 15) of an undecidable formula dispensing with the \(\omega\)-consistency assumption, yet the form of the new proposed undecidable formula is such that my remarks concerning the application of Cantor’s diagonal argument are still valid.
In this sense, redness as a species is associated, for instance, with the same redness of certain here and there lying red paper ribbons where their individual ‘rednesses’ are put next to other concrete constituting act-moments (extension, form, etc.).
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Livadas, S. The Transcendental Source of Logic by Way of Phenomenology. Axiomathes 28, 325–344 (2018). https://doi.org/10.1007/s10516-017-9367-x
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DOI: https://doi.org/10.1007/s10516-017-9367-x