The Third Trichotomy. Thought of the Day 121.0

peircetriangle

The decisive logical role is played by continuity in the third trichotomy which is Peirce’s generalization of the old distinction between term, proposition and argument in logic. In him, the technical notions are rhema, dicent and argument, and all of them may be represented by symbols. A crucial step in Peirce’s logic of relations (parallel to Frege) is the extension of the predicate from having only one possible subject in a proposition – to the possibility for a predicate to take potentially infinitely many subjects. Predicates so complicated may be reduced, however, to combination of (at most) three-subject predicates, according to Peirce’s reduction hypothesis. Let us consider the definitions from ‘Syllabus (The Essential Peirce Selected Philosophical Writings, Volume 2)’ in continuation of the earlier trichotomies:

According to the third trichotomy, a Sign may be termed a Rheme, a Dicisign or Dicent Sign (that is, a proposition or quasi-proposition), or an Argument.

A Rheme is a Sign which, for its Interpretant, is a Sign of qualitative possibility, that is, is understood as representing such and such a kind of possible Object. Any Rheme, perhaps, will afford some information; but it is not interpreted as doing so.

A Dicent Sign is a Sign, which, for its Interpretant, is a Sign of actual existence. It cannot, therefore, be an Icon, which affords no ground for an interpretation of it as referring to actual existence. A Dicisign necessarily involves, as a part of it, a Rheme, to describe the fact which it is interpreted as indicating. But this is a peculiar kind of Rheme; and while it is essential to the Dicisign, it by no means constitutes it.

An Argument is a Sign which, for its Interpretant, is a Sign of a law. Or we may say that a Rheme is a sign which is understood to represent its object in its characters merely; that a Dicisign is a sign which is understood to represent its object in respect to actual existence; and that an Argument is a Sign which is understood to represent its Object in its character as Sign. ( ) The proposition need not be asserted or judged. It may be contemplated as a sign capable of being asserted or denied. This sign itself retains its full meaning whether it be actually asserted or not. ( ) The proposition professes to be really affected by the actual existent or real law to which it refers. The argument makes the same pretension, but that is not the principal pretension of the argument. The rheme makes no such pretension.

The interpretant of the Argument represents it as an instance of a general class of Arguments, which class on the whole will always tend to the truth. It is this law, in some shape, which the argument urges; and this ‘urging’ is the mode of representation proper to Arguments.

Predicates being general is of course a standard logical notion; in Peirce’s version this generality is further emphasized by the fact that the simple predicate is seen as relational and containing up to three subject slots to be filled in; each of them may be occupied by a continuum of possible subjects. The predicate itself refers to a possible property, a possible relation between subjects; the empty – or partly satiated – predicate does not in itself constitute any claim that this relation does in fact hold. The information it contains is potential, because no single or general indication has yet been chosen to indicate which subjects among the continuum of possible subjects it refers to. The proposition, on the contrary, the dicisign, is a predicate where some of the empty slots have been filled in with indices (proper names, demonstrative pronomina, deixis, gesture, etc.), and is, in fact, asserted. It thus consists of an indexical part and an iconical part, corresponding to the usual distinction between subject and predicate, with its indexical part connecting it to some level of reference reality. This reality needs not, of course, be actual reality; the subject slots may be filled in with general subjects thus importing pieces of continuity into it – but the reality status of such subjects may vary, so it may equally be filled in with fictitious references of all sorts. Even if the dicisign, the proposition, is not an icon, it contains, via its rhematic core, iconical properties. Elsewhere, Peirce simply defines the dicisign as a sign making explicit its reference. Thus a portrait equipped with a sign indicating the portraitee will be a dicisign, just like a charicature draft with a pointing gesture towards the person it depicts will be a dicisign. Even such dicisigns may be general; the pointing gesture could single out a group or a representative for a whole class of objects. While the dicisign specifies its object, the argument is a sign specifying its interpretant – which is what is normally called the conclusion. The argument thus consists of two dicisigns, a premiss (which may be, in turn, composed of several dicisigns and is traditionally seen as consisting of two dicisigns) and a conclusion – a dicisign represented as ensuing from the premiss due to the power of some law. The argument is thus – just like the other thirdness signs in the trichotomies – in itself general. It is a legisign and a symbol – but adds to them the explicit specification of a general, lawlike interpretant. In the full-blown sign, the argument, the more primitive degenerate sign types are orchestrated together in a threefold generality where no less than three continua are evoked: first, the argument itself is a legisign with a halo of possible instantions of itself as a sign; second, it is a symbol referring to a general object, in turn with a halo of possible instantiations around it; third, the argument implies a general law which is represented by one instantiation (the premiss and the rule of inference) but which has a halo of other, related inferences as possible instantiations. As Peirce says, the argument persuades us that this lawlike connection holds for all other cases being of the same type.

Mathematical Reductionism: As Case Via C. S. Peirce’s Hypothetical Realism.

mathematical-beauty

During the 20th century, the following epistemology of mathematics was predominant: a sufficient condition for the possibility of the cognition of objects is that these objects can be reduced to set theory. The conditions for the possibility of the cognition of the objects of set theory (the sets), in turn, can be given in various manners; in any event, the objects reduced to sets do not need an additional epistemological discussion – they “are” sets. Hence, such an epistemology relies ultimately on ontology. Frege conceived the axioms as descriptions of how we actually manipulate extensions of concepts in our thinking (and in this sense as inevitable and intuitive “laws of thought”). Hilbert admitted the use of intuition exclusively in metamathematics where the consistency proof is to be done (by which the appropriateness of the axioms would be established); Bourbaki takes the axioms as mere hypotheses. Hence, Bourbaki’s concept of justification is the weakest of the three: “it works as long as we encounter no contradiction”; nevertheless, it is still epistemology, because from this hypothetical-deductive point of view, one insists that at least a proof of relative consistency (i.e., a proof that the hypotheses are consistent with the frequently tested and approved framework of set theory) should be available.

Doing mathematics, one tries to give proofs for propositions, i.e., to deduce the propositions logically from other propositions (premisses). Now, in the reductionist perspective, a proof of a mathematical proposition yields an insight into the truth of the proposition, if the premisses are already established (if one has already an insight into their truth); this can be done by giving in turn proofs for them (in which new premisses will occur which ask again for an insight into their truth), or by agreeing to put them at the beginning (to consider them as axioms or postulates). The philosopher tries to understand how the decision about what propositions to take as axioms is arrived at, because he or she is dissatisfied with the reductionist claim that it is on these axioms that the insight into the truth of the deduced propositions rests. Actually, this epistemology might contain a short-coming since Poincaré (and Wittgenstein) stressed that to have a proof of a proposition is by no means the same as to have an insight into its truth.

Attempts to disclose the ontology of mathematical objects reveal the following tendency in epistemology of mathematics: Mathematics is seen as suffering from a lack of ontological “determinateness”, namely that this science (contrarily to many others) does not concern material data such that the concept of material truth is not available (especially in the case of the infinite). This tendency is embarrassing since on the other hand mathematical cognition is very often presented as cognition of the “greatest possible certainty” just because it seems not to be bound to material evidence, let alone experimental check.

The technical apparatus developed by the reductionist and set-theoretical approach nowadays serves other purposes, partly for the reason that tacit beliefs about sets were challenged; the explanations of the science which it provides are considered as irrelevant by the practitioners of this science. There is doubt that the above mentioned sufficient condition is also necessary; it is not even accepted throughout as a sufficient one. But what happens if some objects, as in the case of category theory, do not fulfill the condition? It seems that the reductionist approach, so to say, has been undocked from the historical development of the discipline in several respects; an alternative is required.

Anterior to Peirce, epistemology was dominated by the idea of a grasp of objects; since Descartes, intuition was considered throughout as a particular, innate capacity of cognition (even if idealists thought that it concerns the general, and empiricists that it concerns the particular). The task of this particular capacity was the foundation of epistemology; already from Aristotle’s first premisses of syllogism, what was aimed at was to go back to something first. In this traditional approach, it is by the ontology of the objects that one hopes to answer the fundamental question concerning the conditions for the possibility of the cognition of these objects. One hopes that there are simple “basic objects” to which the more complex objects can be reduced and whose cognition is possible by common sense – be this an innate or otherwise distinguished capacity of cognition common to all human beings. Here, epistemology is “wrapped up” in (or rests on) ontology; to do epistemology one has to do ontology first.

Peirce shares Kant’s opinion according to which the object depends on the subject; however, he does not agree that reason is the crucial means of cognition to be criticised. In his paper “Questions concerning certain faculties claimed for man”, he points out the basic assumption of pragmatist philosophy: every cognition is semiotically mediated. He says that there is no immediate cognition (a cognition which “refers immediately to its object”), but that every cognition “has been determined by a previous cognition” of the same object. Correspondingly, Peirce replaces critique of reason by critique of signs. He thinks that Kant’s distinction between the world of things per se (Dinge an sich) and the world of apparition (Erscheinungswelt) is not fruitful; he rather distinguishes the world of the subject and the world of the object, connected by signs; his position consequently is a “hypothetical realism” in which all cognitions are only valid with reservations. This position does not negate (nor assert) that the object per se (with the semiotical mediation stripped off) exists, since such assertions of “pure” existence are seen as necessarily hypothetical (that means, not withstanding philosophical criticism).

By his basic assumption, Peirce was led to reveal a problem concerning the subject matter of epistemology, since this assumption means in particular that there is no intuitive cognition in the classical sense (which is synonymous to “immediate”). Hence, one could no longer consider cognitions as objects; there is no intuitive cognition of an intuitive cognition. Intuition can be no more than a relation. “All the cognitive faculties we know of are relative, and consequently their products are relations”. According to this new point of view, intuition cannot any longer serve to found epistemology, in departure from the former reductionist attitude. A central argument of Peirce against reductionism or, as he puts it,

the reply to the argument that there must be a first is as follows: In retracing our way from our conclusions to premisses, or from determined cognitions to those which determine them, we finally reach, in all cases, a point beyond which the consciousness in the determined cognition is more lively than in the cognition which determines it.

Peirce gives some examples derived from physiological observations about perception, like the fact that the third dimension of space is inferred, and the blind spot of the retina. In this situation, the process of reduction loses its legitimacy since it no longer fulfills the function of cognition justification. At such a place, something happens which I would like to call an “exchange of levels”: the process of reduction is interrupted in that the things exchange the roles performed in the determination of a cognition: what was originally considered as determining is now determined by what was originally considered as asking for determination.

The idea that contents of cognition are necessarily provisional has an effect on the very concept of conditions for the possibility of cognitions. It seems that one can infer from Peirce’s words that what vouches for a cognition is not necessarily the cognition which determines it but the livelyness of our consciousness in the cognition. Here, “to vouch for a cognition” means no longer what it meant before (which was much the same as “to determine a cognition”), but it still means that the cognition is (provisionally) reliable. This conception of the livelyness of our consciousness roughly might be seen as a substitute for the capacity of intuition in Peirce’s epistemology – but only roughly, since it has a different coverage.

Noneism. Part 1.

Meinong

Noneism was created by Richard Routley. Its point of departure is the rejection of what Routley calls “The Ontological Assumption”. This assumption consists in the explicit or, more frequently, implicit belief that denoting always refers to existing objects. If the object, or objects, on which a proposition is about, do not exist, then these objects can only be one: the null entity. It is incredible that Frege believed that denoting descriptions without a real (empirical, theoretical, or ideal) referent denoted only the null set. And it is also difficult to believe that Russell sustained the thesis that non-existing objects cannot have properties and that propositions about these objects are false.

This means that we can have a very clear apprehension of imaginary objects, and quite clear intellection of abstract objects that are not real. This is possible because to determine an object we only need to describe it through its distinctive traits. This description is possible because an object is always chacterized through some definite notes. The amount of traits necessary to identify an object greatly varies. In some cases we need only a few, for instance, the golden mountain, or the blue bird; in other cases we need more, for instance, the goddess Venus or the centaur Chiron. In other instances the traits can be very numerous, even infinite. For instance the chiliedron, and the decimal number 0,0000…009, in which 9 comes after the first million zeros, have many traits. And the ordinal omega or any Hilbert space have infinite traits (although these traits can be reckoned through finite definitions). These examples show, in a convincing manner, that the Ontological Assumption is untenable. We must reject it and replace it with what Routley dubbs the Characterization Postulate. The Characterization Postulate says that, to be an object means to be characterized by determined traits. The set of the characterizing traits of an object can be called its “characteristic”. When the characteristic of an object is set up, the object is perfectly recognizable.

Once this postulate is adopted, its consequences are far reaching. Since we can characterize objects through any traits whatsoever, an object can not only be inexistent, it can even be absurd or inconsistent. For instance, the “squond” (the circle that is square and round). And we can make perfectly valid logical inferences from the premiss: x is the sqound:

(1) if x is the squond, then x is square
(2) if x is the squond, then x is round

So, the theory of objects has the widest realm of application. It is clear that the Ontological Assumption imposes unacceptable limits to logic. As a matter of fact, the existential quantifier of classical logic could not have been conceived without the Ontological Assumption. The expression “(∃x)Fx” means that there exists at least an object that has the property F (or, in extensional language, that there exists an x that is a member of the extension of F). For this reason, “∃x” is unappliable to non existing objects. Of course, in classical logic we can deny the existence of an Object, but we cannot say anything about Objects that have never existed and shall never exist (we are strictly speaking about classical logic). We cannot quantify individual variables of a first order predicate that do not refer to a real, actual, past or future entity. For instance, we cannot say “(∃x) (x is the eye of Polyphemus)”. This would be false, of course, because Polyphemus does not exist. But if the Ontological Assumption is set aside, it is true, within a mythological frame, that Polyphemus has a single eye and many other properties. And now we can understand why noneism leads to logical material-dependence.

As we have anticipated, there must be some limitations concerning the selection of the contradictory properties; otherwise the whole theory becomes inconsistent and is trivialized. To avoid trivialization neutral (noneist) logic distinguishes between two sorts of negation: the classical propositional negation: “8 is not P”, and the narrower negation: “8 is non-P”. In this way, and by applying some other technicalities (for instance, in case an universe is inconsistent, some kind of paraconsistent logic must be used) trivialization is avoided. With the former provisions, the Characterization Postulate can be applied to create inconsistent universes in which classical logic is not valid. For instance, a world in which there is a mysterious personage, that within determined but very subtle circumstances, is and is not at the same time in two different places. In this case the logic to be applied is, obviously, some kind of paraconsistent logic (the type to be selected depends on the characteristic of the personage). And in another universe there could be a jewel which has two false properties: it is false that it is transparent and it is false that it is opaque. In this kind of world we must use, clearly, some kind of paracomplete logic. To develop naive set theory (in Halmos sense), we must use some type of paraconsistent logic to cope with the paradoxes, that are produced through a natural way of mathematical reasoning; this logic can be of several orders, just like the classical. In other cases, we can use some kind of relevant and, a fortiori, paraconsistent logic; and so on, ad infinitum.

But if logic is content-dependent, and this dependence is a consequence of the Ontological Assumption’s rejection, what about ontology? Because the universes determined through the application of the Characterization Postulate may have no being (in fact, most of them do not), we cannot say that the objects that populate such universes are entities, because entities exist in the empirical world, or in the real world that underpins the phenomena, or (in a somewhat different way), in an ideal Platonic world. Instead of speaking about ontology, we should speak about objectology. In essence objectology is the discipline founded by Meinong (Theory of Objects), but enriched and made more precise by Routley and other noneist logicians. Its main division would be Ontology (the study of real physical and Platonic objects) and Medenology (the study of objects that have no existence).