Abstract
As a multi-layered theory of the foundations of “‘mathematicizing’ logic”, Husserlian pure logic is stratified on three levels (sub-theoretical, theoretical, meta-theoretical), which are then themselves transversally split in two sides (apophantic and ontological). This paper investigates how symbolic knowledge works in this framework—viz. in terms of ‘How can the subjective operating with symbols be justified in the process of obtaining objective contents of knowledge?’ To do so, it innovates in showing how Husserl’s theory of semiotic intentionality provides the epistemological-transcendental foundation for the syntactic-semantic correspondence of the meaning relationship that is operative at the kernel of pure logic—i.e. on the theoretical level, between systems of axioms and formal manifolds. And thereby, it bridges the gap between the two volumes of the first edition of the Logical Investigations (namely, the Prolegomena and the six Investigations).
Now in fact, there exists today the beginning of a science which claims to possess a systematic method for such a clarification of meaning, and that is the phenomenology founded by Husserl.
Kurt Gödel (1961/?)
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- 1.
N.b. sans serif font refers to the Glossary in Appendix (with asterisks for first occurrences).
- 2.
Cf. the first two components of Peirce’s “trivium of conceivable sciences”, namely: formal grammar, which “would treat of the formal conditions of symbols having meaning, that is of the reference of symbols in general to their grounds or imputed characters”; and logic, which “would treat of the formal conditions of the truth of symbols” (Peirce 1931, 1.559 [also: 4.116 and Peirce 1958, 8.342]).
- 3.
See Casari (2000), Correia (2004), Fine (1995) (on the theory of wholes and parts), (Centrone 2010, Chap. 3), da Silva (2000a, b), Hill (1995), (Hill and Rosado Haddock 2000, Chap. 9), Hartimo (2007), Majer (1997) (on the theory of manifolds), and (Centrone 2010, app. 5) (on Husserl’s logical calculus).
- 4.
- 5.
Indeed, while it conveys assumptions about the maximality as well as the descriptive completeness of axioms systems, Husserl’s definiteness can be interpreted as involving a form of deductive completeness stated as dually syntactic and semantic. (cf. Centrone 2010, Chap. 3; da Silva 2000a, b; 2016; Hartimo 2007; Isaac 2015).
- 6.
In fact, quite few studies approach pure logic from this standpoint (for suggestive insights, see Hill 2015; Peucker 2019; Smith 2002a, b; Tieszen 2008, 2010), whereas conversely, Husserl’s semiotics and theory of intentionality are usually disconnected from the system of pure logic in the literature.
- 7.
- 8.
Cf. Peirce’s distinction between index “which refers to the Object that it denotes by virtue of being really affected by that Object” and symbol “which refers to the Object that it denotes by virtue of a law, usually an association of general ideas, which operates to cause the Symbol to be interpreted as referring to that Object” (Peirce 1931, 2.249 [also: 2.306, 2.297]).
- 9.
Although there is no reference to Peirce’s semiotics in the First Investigation, this criticism should also concern his 1902 definition of representamen as the “subject of a triadic relation TO a second, called its OBJECT, FOR a third, called its INTERPRETANT” (Peirce 1931, 1.541 [also: 1.346, 2.274]), and furthermore, as “related to its object only in consequence of a mental association” (Peirce 1931, 3.360).
- 10.
“Was in einer Vorstellung vorgestellt wird, ist ihr Inhalt; was durch eine Vorstellung vorgestellt wird, ist ihr Gegenstand” (Twardowski 1894, p. 18).
- 11.
E.g.: “Relational talk of ‘intimation’, ‘meaning’ and ‘object’ belongs essentially to every expression. Every expression intimates something, means something and names or otherwise designates something. In each case, talk of ‘expression’ [(sic) ausgedrückt] is equivocal” (Husserl 1970a, p. 199).
- 12.
See Appendix.
- 13.
In short: “An expressive sign emerges in the combination of the two intentions; the sign refers to its meaning by virtue of the signitive intention and refers to its object by virtue of the significative intention. The meaning to which the signitive intention points and the object to which the significative intention refers are in an important sense the same” (Drummond 2007, p. 191 [cf. Moran and Cohen 2012, pp. 299–300]).
- 14.
In some sense, as in Peirce, “A sign therefore is an object which is in relation to its object on the one hand and to an interpretant on the other” (Peirce 1958, 8.332)—and yet, the order of the triadic relation is opposite (see notes 17, 21).
- 15.
Cf. example of arabesques becoming meaningful in the 1894 Psychological Studies (Husserl 1994, p. 163).
- 16.
Cf. Peirce’s definition of “A Third [namely, the interpreting thought] [a]s something which brings a First into relation to a Second” (Peirce 1958, 8.332).
- 17.
- 18.
Such ontological autonomy is precisely the purpose of introducing the term objectuality (see footnote 46), and the collateral risk here is that of linguistic idealism.
- 19.
That combination is coined in the genitive ambiguity (either object, or subject) of the German Bedeutungsintention: “The word intentional is so framed as to permit application both to meaning and the object of the intentio. Intentional unity does not therefore necessarily mean the intended, the objective unity” (Husserl 1970a, p. 322 [LI I: § 30, note]).
- 20.
“[A]n expression only refers to an objective correlate because it means something, it can rightly said to signify or name its object through its meaning” further on: “In meaning a relation to an object is constituted. To use an expression significantly, and to refer expressively to an object (to form a presentation of it), are one and the same”; and in this regard: “To be an expression is rather a descriptive aspect of the experienced unity of sign and thing signified” (Husserl 1970a, pp. 198, 201, 193).
- 21.
Again and in other words (see footnote 17), the signification of a sign (a.k.a. ‘interpretant sign’ in Peircean parlance) does not proceed from and is not determined by the sign (cf. Peirce 1931, 1.541, 2.274, 2.275, 2.228, 2.303, 5.473)—a fortiori, when defined as a sign representamen grounded on or caused by its object (Peirce 1931, 2.228, 5.473); and no infinite process of semiosis stands here as proper significant outcome of a sign (cf. Peirce 1931, 2.274, 2.303, 5.473)—namely, infinite insofar as the interpretant“must have a second triadic relation in which the Representamen, or rather the relation thereof to its Object, shall be its own (the Third’s) Object, and must be capable of determining a Third to this relation. All this must equally be true of the Third’s Thirds and so on endlessly” (Peirce 1931, 2.274) (so that, in short: “Every thought is a sign” [Peirce 1931, 1.538]).
- 22.
Cf. the Peircean motto: “to clear the sign of its mental associations” (Peirce 1931, 5.492) (yet, see note 9).
- 23.
Thus: “To mean is not a particular way of being a sign in the sense of indicating something” (Husserl 1970a, p. 183) (indeed, indications are always grounded on signs given as intuitive contents).
- 24.
See also paragraph 35: “There is, however, no intrinsic connection between the ideal unities which in fact operates as meanings, and the signs to which they are tied, i.e. through which they become real in human mental life. We cannot therefore say that all unities of this sorts are expressed meanings” (Husserl 1970a, p. 233).
- 25.
Such a reduction of the semiotic triangle to a pure process of meaning contrasts with Peircean semiosis as a “tri-relative influence” irreducible to “actions between pairs” (i.e. to dyadic relations alone) (Peirce 1931, 5.484 [also: 1.345]), for the object of the interpretant is indeed the triadic relation of the sign-representamen to its object (Peirce 1931, 2.274).
- 26.
Cf. the formalist texts from the 1890s (e.g. “On the Logic of Signs (Semiotic)” (Husserl 1994, pp. 20–51)) and the 1913 Drafts for the Sixth Investigation Husserl 2005 (in the literature: Bernet 1988, Melle 1999, Münch 1993); such an indispensability of signs for thought is also part of Peirce’s credo (Peirce 1931, 2.302 (also: (Peirce 1958, 8.332))) (cf. note 21)—indeed for Peirce, Thirdness involves Secondness and Firstness (Peirce 1931, 1.530), and thus symbols may always have indexes or icons as constituents (Peirce 1931, 2.293) (cf. footnote 28).
- 27.
For more details on nonsense and counter-sense in Husserlian pure logic, see Isaac 2016a.
- 28.
In other words, the existent replica (or ‘token’) of the abstract units of meaning conform to and are governed by such types these units (or ‘types’) to work meaningfully (cf. Peirce 1931, 2.292, 2.293); and conversely in Peirce’s view, “In order that a Type may be used, it has to be embodied in a Token which shall be a sign of the Type, and thereby of the object the Type signifies” (Peirce 1931, 4.537).
- 29.
Cf. the meaning-intentions as concrete subjective acts of aiming at something (Sect. 5.2.2).
- 30.
Since meaning is the necessary condition for intentionality, only the upper sphere of the pure morphology of meanings is here a concern (see Remark 1)—i.e. the distinction at stake is only between sense and counter-sense (while the exclusion of non-sense has already been performed).
- 31.
On this process, cf. the Sixth Investigation, Chap. 4 (Husserl 1970b, §§ 40, 43, 48, 58).
- 32.
As Husserl puts it: “The ideal conception of the act which confers meaning yields us the Idea of the intending meaning, just as the ideal conception of the correlative essence of the act which fulfills meaning, yields the fulfilling meaning, likewise qua Idea” (Husserl 1970a, p. 200).
- 33.
“[The fulfilling meaning qua Idea] is the identical content which, in perception, pertains to the totality of possible acts of perception which intend the same object perceptually, and intend it actually as the same object. This content is therefore the ideal correlate of this single object, which may, for the rest, be completely imaginary” (Husserl 1970a, p. 200).
- 34.
- 35.
Only the object’s ideal correlate is of concern here (i.e. not the object itself): “[T]he object never coincides <(excepted in a quite exceptional and logically worthless case)> with the meaning” (Husserl 1970a, p. 197—in angle brackets: first edition, not translated (see also Husserl 1970b, LI IV: §§ 2, 8a)) (Sect. 5.4.1).
- 36.
The purpose of these laws is to secure the possibility of the material singularization of a referential meaning; owing to their categoriality, they are valid for every value of the categorially typed variables of a meaning-category (see Appendix) (Husserl 1970b, LI IV: § 13).
- 37.
- 38.
- 39.
In Husserl’s view, the possibility of existence (not the existence as such) is an absolute condition for the possibility of thought (as intentionally directed toward an object), and not a mere category of thought (Husserl 1970b, LI III: §§ 6–7).
- 40.
This correspondence is conjectured as nomological in the Fourth Investigation (paragraph 14): “In part, but only in part, relations of equivalence link the two sets of laws [viz. ontological-analytic and apophantic-analytic], but we cannot go into this further here” (Husserl 1970b, p. 72).
- 41.
In the Sixth Investigation (paragraph 45): “If we are asked what it means to say that categorially structured meanings find fulfillment, confirm themselves in perception, we can but reply: it means only that they relate to the object itself in its categorial structure” taken as the “categorial forms in the [...] objective sense” (Husserl 1970b, pp. 280/307).
- 42.
Indeed, since the domain of meanings always surpasses the domain of intuitions (thus, of possible fulfillments), impossible fulfillments of meaningful intentions are always possible.
- 43.
In other words, the basement of pure logic (viz. the two categorial morphologies) proceeds from the epistemological application of intentionality to the relation of meaning embedded at its theoretical level (cf. (Husserl 1913, pp. 38-9)).
- 44.
Cf. Husserl’s conception of logic as, in short, the science of “the pure ‘laws of thought’, which express the a priori connection between the categorial form of meanings and their objectivity or truth” (Husserl 1970a, p. 225), with that of Peirce (1908 [1867]) as “the doctrine of the formal conditions of the truth of symbols; i.e., of the reference of symbols to their objects” (Peirce 1966, p. 402) (see footnote 5).
- 45.
In short, as Husserl puts it in 1913, it is the “logic of propositional meanings” (Logik der Aussagebedeutungen) (Husserl 1970b, p. 352 [LI IV: § 13, note]).
- 46.
The vaguer expression objectuality was precisely introduced by Husserl to get rid of the object in the narrower sense of something real, simple, etc. (Husserl 1970a, p. 321 [LI I: § 9, note]).
- 47.
Formalizing abstraction is therefore expressly distinguished from the simple term of abstraction used to mean “the emphasis on a non-independent ‘moment’ of content” (Husserl 1970b, pp. 350–1 [...]) (e.g. the redness of some concrete visual datum).
- 48.
When the operations of a formal domain are used on the components of formal manifolds as species, the formal domains produce formal states of affairs as outputs.
- 49.
When the laws of a system of axioms are used for deriving logical consequences from the axioms, they produce a necessarily consistent collection of propositions.
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Appendix: Short Husserlian Glossary
Appendix: Short Husserlian Glossary
- Apophantics :
-
(and its derivations) designates the side of pure logic that deals with formal meanings in view of their possible truth or falsity (namely, in contrast with the ontological side of formal objects and states of affairs).Footnote 45
- Category :
-
designates the basic units of form that articulate the structures of complex apophantic or ontological compounds.
- Categorial :
-
See Category.
- Categoriality :
-
See Categorial.
- Categorial morphology :
-
See Categorial and Morphology*.
- Designation (Relation of—) :
-
concerns the relation of an expression to an objectuality (Gegenständlichkeit), and links either a name to some object (in the paradigmatic case) or a proposition to some state of affairs.Footnote 46
- Epistemology :
-
(and its derivations) signifies both theory of knowledge and theory of science, owing to the focus on the link “between the subjectivity of knowing and the objectivity of the content known” (Husserl 1970a, p. 2).
- Epistemological-transcendental :
-
See Epistemology and Transcendental.
- Formalizing abstraction :
-
refers to formalization conceived as a process of abstracting the form of some content, and turning that form into a (categorially typed) variable, with categorial (versus sensible) intuition as intuitive counterpart.Footnote 47
- Formal domains :
-
are based on “formal manifolds” (i.e. totalities of individually distinct formal objects inserted in a network of relations) and furthermore comprise some operations (Husserl 2003, p. 486).Footnote 48
- Intensionalization :
-
refers to the process of becoming intensionally determined (as opposed to any extensionalist prerogative).
- Intuitive (—intention) :
-
operates as the act of giving an intended object as present.
- Mereological :
-
refers to ‘mereology’ as the doctrine of the laws for connections of apophantic or ontological categories based on whole-part relationships.
- Model (Semiotic—) :
-
signifies the schematization of the structure that links a sign to what it designates* (viz. its referent via/or its signified, when it works as a signifier).
- Morphology :
-
designates the theories of the forms of the categories (apophantic or ontological).
- Semiotic intentionality :
-
designates intentionality on its semiotic meaning mode (i.e. the intentionality of a sign working as signifier).
- Significative (—intention) :
-
operates as the act of constituting the sense of an object as expressible by/in an expression.
- Signitive (—intention) :
-
operates as the act of using a sign to express a meaning and aim at some intended reference.
- Systems of axioms :
-
comprise a finite totality of formal axioms (“mutually consistent and independent of one another” [Husserl 2003, p. 410]) plus some laws of operation (viz. rules of derivation).Footnote 49
- Transcendental :
-
means in epistemological terms the “critical understanding” (Husserl 1970a, p. 2) (in the Kantian sense) of something.
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Isaac, M.G. (2019). Symbolic Knowledge in Husserlian Pure Logic. In: Shafiei, M., Pietarinen, AV. (eds) Peirce and Husserl: Mutual Insights on Logic, Mathematics and Cognition. Logic, Epistemology, and the Unity of Science, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-030-25800-9_5
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