Conjuncted: Operations of Truth. Thought of the Day 47.1

Conjuncted here.

Let us consider only the power set of the set of all natural numbers, which is the smallest infinite set – the countable infinity. By a model of set theory we understand a set in which  – if we restrict ourselves to its elements only – all axioms of set theory are satisfied. It follows from Gödel’s completeness theorem that as long as set theory is consistent, no statement which is true in some model of set theory can contradict logical consequences of its axioms. If the cardinality of p(N) was such a consequence, there would exist a cardinal number κ such that the sentence the cardinality of p(N) is κ would be true in all the models. However, for every cardinal κ the technique of forcing allows for finding a model M where the cardinality of p(N) is not equal to κ. Thus, for no κ, the sentence the cardinality of p(N) is κ is a consequence of the axioms of set theory, i.e. they do not decide the cardinality of p(N).

The starting point of forcing is a model M of set theory – called the ground model – which is countably infinite and transitive. As a matter of fact, the existence of such a model cannot be proved but it is known that there exists a countable and transitive model for every finite subset of axioms.

A characteristic subtlety can be observed here. From the perspective of an inhabitant of the universe, that is, if all the sets are considered, the model M is only a small part of this universe. It is deficient in almost every respect; for example all of its elements are countable, even though the existence of uncountable sets is a consequence of the axioms of set theory. However, from the point of view of an inhabitant of M, that is, if elements outside of M are disregarded, everything is in order. Some of M because in this model there are no functions establishing a one-to-one correspondence between them and ω₀. One could say that M simulates the properties of the whole universe.

The main objective of forcing is to build a new model M[G] based on M, which contains M, and satisfies certain additional properties. The model M[G] is called the generic extension of M. In order to accomplish this goal, a particular set is distinguished in M and its elements are referred to as conditions which will be used to determine basic properties of the generic extension. In case of the forcing that proves the undecidability of the cardinality of p(N), the set of conditions codes finite fragments of a function witnessing the correspondence between p(N) and a fixed cardinal κ.

In the next step, an appropriately chosen set G is added to M as well as other sets that are indispensable in order for M[G] to satisfy the axioms of set theory. This set – called generic – is a subset of the set of conditions that always lays outside of M. The construction of M[G] is exceptional in the sense that its key properties can be described and proved using M only, or just the conditions, thus, without referring to the generic set. This is possible for three reasons. First of all, every element x of M[G] has a name existing already in M (that is, an element in M that codes x in some particular way). Secondly, based on these names, one can design a language called the forcing language or – as Badiou terms it – the subject language that is powerful enough to express every sentence of set theory referring to the generic extension. Finally, it turns out that the validity of sentences of the forcing language in the extension M[G] depends on the set of conditions: the conditions force validity of sentences of the forcing language in a precisely specified sense. As it has already been said, the generic set G consists of some of the conditions, so even though G is outside of M, its elements are in M. Recognizing which of them will end up in G is not possible for an inhabitant of M, however in some cases the following can be proved: provided that the condition p is an element of G, the sentence S is true in the generic extension constructed using this generic set G. We say then that p forces S.

In this way, with an aid of the forcing language, one can prove that every generic set of the Cohen forcing codes an entire function defining a one-to-one correspondence between elements of p(N) and a fixed (uncountable) cardinal number – it turns out that all the conditions force the sentence stating this property of G, so regardless of which conditions end up in the generic set, it is always true in the generic extension. On the other hand, the existence of a generic set in the model M cannot follow from axioms of set theory, otherwise they would decide the cardinality of p(N).

The method of forcing is of fundamental importance for Badious philosophy. The event escapes ontology; it is “that-which-is-not-being-qua-being”, so it has no place in set theory or the forcing construction. However, the post-evental truth that enters, and modifies the situation, is presented by forcing in the form of a generic set leading to an extension of the ground model. In other words, the situation, understood as the ground model M, is transformed by a post-evental truth identified with a generic set G, and becomes the generic model M[G]. Moreover, the knowledge of the situation is interpreted as the language of set theory, serving to discern elements of the situation; and as axioms of set theory, deciding validity of statements about the situation. Knowledge, understood in this way, does not decide the existence of a generic set in the situation nor can it point to its elements. A generic set is always undecidable and indiscernible.

Therefore, from the perspective of knowledge, it is not possible to establish, whether a situation is still the ground-model or it has undergone a generic extension resulting from the occurrence of an event; only the subject can interventionally decide this. And it is only the subject who decides about the belonging of particular elements to the generic set (i.e. the truth). A procedure of truth or procedure of fidelity (Alain Badiou – Being and Event) supported in this way gives rise to the subject language. It consists of sentences of set theory, so in this respect it is a part of knowledge, although the veridicity of the subject language originates from decisions of the faithful subject. Consequently, a procedure of fidelity forces statements about the situation as it is after being extended, and modified by the operation of truth.

Phantom Originary Intentionality: Thought of the Day 16.0

Phantom limbs and anosognosias – cases of abnormal impressions of the presence or absence of parts of our body – seem like handy illustrations of an irreducible, first-person dimension of experience, of the sort that will delight the phenomenologist, who will say: aha! there is an empirical case of self-reference which externalist, third-person explanations of the type favoured by deflationary materialists, cannot explain away, cannot do away with. As Merleau-Ponty would say, and Varela after him, there is something about my body which makes it irreducibly my own (le corps propre). Whether illusory or not, such images (phantoms) have something about them such that we perceive them as our own, not someone else’s (well, some agnosias are different: thinking our paralyzed limb is precisely someone else’s, often a relative’s). One might then want to insist that phantom limbs testify to the transcendence of mental life! Indeed, in one of the more celebrated historical cases of phantom limb syndrome, Lord Horatio Nelson, having lost his right arm in a sea battle off of Tenerife, suffered from pains in his phantom hand. Most importantly, he apparently declared that this phantom experience was a “direct proof of the existence of the soul”. Although the materialist might agree with the (reformed) phenomenologist to reject dualism and accept that we are not in our bodies like a sailor in a ship, she might not want to go and declare, as Merleau-Ponty does, that “the mind does not use the body, but fulfills itself through it while at the same time transferring the body outside of physical space.” This way of talking goes back to the Husserlian distinction between Korper, ‘body’ in the sense of one body among others in a vast mechanistic universe of bodies, and Leib, ‘flesh’ in the sense of a subjectivity which is the locus of experience. Now, granted, in cognitivist terms one would want to say that a representation is always my representation, it is not ‘transferable’ like a neutral piece of information, since the way an object appear to me is always a function of my needs and interests. What my senses tell me at any given time relies on my interests as an agent and is determined by them, as described by Andy Clark, who appeals to the combined research traditions of the psychology of perception, new robotics, and Artificial Life. But the phenomenologist will take off from there and build a full-blown defense of intentionality, now recast as ‘motor intentionality’, a notion which goes back to Husserl’s claim in Ideas II that the way the body relates to the external world is crucially through “kinestheses”: all external motions which we perceive are first of all related to kinesthetic sensations, out of which we constitute a sense of space. On this view, our body thus already displays ‘originary intentionality’ in how it relates to the world.

Why Deleuzean Philosophy Begins at Hegel and Becomes a Correctional Footnote Thereafter ? Note Quote.

That philosophy must be an ontology of sense is a bold claim on Deleuze’s part, and although he takes it from a Hegelian philosophy, the direction in which he develops it across the rest of his work is resolutely, if not infamously, opposed to Hegel. Whereas Hegel will construct a logic of sense which is fundamentally a logic of the concept, Deleuze will deny that sense is reducible to signification and its universal or general concepts. Deleuze will later provide his own logic of the concept, but for him, although the concept will posit itself, this will not be as the immanent thought of the sense or the content of the matter itself, but will rather function to extract or capture a pure event, or the sense at the surface of things. Similarly, although Deleuze will agree that “sense is becoming”, this will not be a becoming in an atemporal logical time, opposed to a historical time that would play it out, but a pure becoming without present, always divided between past and future, without arrow or telos, and actualised in the present while never strictly ‘happening’. The most distinctive difference, however, will be Deleuze’s invocation of a nonsense that cannot be simply incorporated within sense, that will not be sublated and subsumed in the folds of the dialectic, a nonsense that is itself productive of sense. Moving beyond Hegel, Deleuze will deny the reducibility of sense not only to the universal meanings of signification, but also to the functions of reference or denotation. Moreover, he will deny its reducibility to the dimension of manifestation, or the meanings of the subject of enunciation – the ‘I’ who speaks. Sense can neither be found in universal concepts, nor reference to the individual, nor in the intentions of the subject, but is rather that which grounds all three.

Frege’s Ontological Correlates of Propositions

For Frege there were only two ontological correlates of propositions: the True and the False. All true propositions denote the True, and all false prepositions denote the False. From an ontological point of view, if all true propositions denote exactly one and the same entity, then the underlying philosophical position is the absolute monism of facts.

Lets disprove what Suszko called ‘Frege’s axiom’: namely the assumption that there exist only two referents for propositions.

Frege’s position on propositions was part of a more general view. Indeed, Frege adopted a principle of homogeneity (Perzanowski) according to which there are two fundamental categories of signs (Bedeutungen and truth-values) and two fundamental categories of senses (Sinn and Gedanken).

Both categories of signs (names and propositions) have sense and reference. The sense of a name is its Sinn, that way in which its referent is given, while the referent itself, the Bedeutung, is the object named by the name. As for propositions, their sense is the Gedanke, while their reference is their logical value.

Since the two semiotic triangles are entirely similar in structure, we need analyze only one of them: that relative to propositions.

Here p is a proposition, s(p) is the sense of p, and r(p) is the referent of p. The functional composition states that s(p) is the way in which p yields r(p). The triangle has been drawn with the functions linking its vertexes explicitly shown. When the functions are composable, the triangle is said to commute, yielding

f(s(p)) = r(p), or f ° s(p) = r(p)

An interesting question now arises: is it possible to generalize the semiotic triangle? And if it is possible to do so, what is required? A first reorganization and generalization of the semiotic triangle therefore involves an explicit differentiation between the truth-value assigning function and the referent assigning function. We thus have the following double semiotic triangle:

where r stands for the referent assigning function and t for the truth-value assigning function. Extending the original semiotic triangle by also considering utterances:

Suszko uses the terms logical valuations for the procedures that assign truth-values, and algebraic valuations for those that assign referents. By arguing for the existence of only two referents, Frege ends up by collapsing logical and algebraic valuations together, thereby rendering them indistinguishable.

Having generalized the semiotic triangle into the double semiotic triangle, we must now address the following questions:

1. when do two propositions have the same truth value?
2. when do two propositions have the same referent?
3. when do two propositions have the same sense?

Sameness of logical value will be denoted by ↔ (logical equivalence), while sameness of referent will be indicated with ≡ (not to be confused with the equiform ≡ to express indiscernibility) and sameness of sense (synonymy) by ≈. Two propositions are synonymous when they have the same sense:

(p ≈ q) = 1 iff (s(p) = s(q)) = 1

Two propositions are identical when they have the same referent:

(p ≡ q) = 1 iff (r(p) = r(q)) = 1

Two propositions are equivalent when they have the same truth value:

(p ↔ q) = 1 iff (t(p) = t(q)) = 1

These various concepts are functionally connected as follows:

s(p) = s(q) implies r(p) = r(q), r(p) = r(q) implies t(p) = t(q)

In general, the constraints that we impose on referents correspond to the ontological assumptions that characterize the theory. The most general logic of all is the one that imposes no restriction at all on r valuations. Just as Fregean logic recognizes only two referents so the most general logic recognizes more than numerable set of them. Between these two extremes, of course, there are numerous intermediate cases. Pure non-Fregean logic is extremely weak, a chaos. If it is to yield something, it has to be strengthened.