Five Propositions on the Evental Fallacy

•February 15, 2009 • 2 Comments

eventhorizon

Proposition 1: The Event (whether named Event, Other, Difference, Differance, Ereignis, etc.) is the necessary and sufficient condition for philosophy today.

Proposition 2: The Event (whether named Event, Other, Difference, Differance, Ereignis, etc.) provides for the construction of the philosophical world.

Proposition 3: The philosophic world is delineated by a normative factor, i.e. a political or ethical project made obvious by the Event (whether named Event, Other, Difference, Differance, Ereignis, etc.).

Proposition 4: The Evental Fallacy is operative within all contemporary philosophical worlds (i.e. worlds conjured by the event).

Proposition 5: The Evental Fallacy erases the operation of philosophical thought (i.e. the thought conditioned by the Event).

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Non-Cantorian Theory of the Multiple

•February 15, 2009 • 3 Comments

counting2

Over at Object-Oriented Philosophy, Graham Harman and (at Larval Subjects) Levi Bryant have been carrying on a discussion of Badiou’s idea of the count (and its related concepts, i.e. inconsistent & consistent multiple etc.) and what effect it has on the status of objects (real or subjective, etc.).Here is a taste:

if I were to start saying: “I’m a Badiouian, but I think that rocks and earthworms are also capable of invoking the generic through art, politics, science and love,” what do you honestly think Badiouians would say in this case? Would they say: “Cool. Badiou never specifies that it has to be a human”? You know full well that they would dismiss such a position as vitalist crap. The whole spirit of Badiou’s philosophy is of a militant human subject disrupting given states-of-situations in truth events.

The discussion strikes something that I have been bouncing around in my readings of Badiou. First, I must admit that there is something that I like about the language of the count, which is to say that I like it better than say Deleuze & Guattari’s discussion of infinite speeds and slowing down. But, second, I have found myself troubled by it in a way that is similiar to Graham and Levi’s discussions of it. It strikes me that Badiou and his discussion of the the count-as-one sounds remarkable close to Plato’s description of the demiure in the Timeaus. Specifically, the demiurge (the rational ordering God) is confronted by the choas of pure nothingness; a nothingness without void, but excessive becoming without order. The demiurge then “orders” the nothing under the function of same and different (following Plato’s understanding of order). Doesn’t this sound an awful lot like Badiou’s conception of the movement from the inconsistent multiple to the consistent multiple via the count. Badiou’s willingness to accept his “platonism” aside, this can be quite troubling for a materialist.

This brings us to the second point. Taylor Adkins (Speculative Heresy) and myself will soon publish a translation of Francois Laruelle’s “review” of Deleuze & Guattari’s What is Philosophy? in the next issue of PLI. In this article, Laruelle states that what is needed is a “Non-Cantorian Theory of the Multiple.” This is a curious concept indeed. Since finishing the translation, I have been trying to figure out what this would entail. Obviously, it would be an understanding of the multiple without count, a given without givenness. We have here a first step in a non-philosophical critique of Badiou (with Meillassoux being guilty by association). I wish I had more to say but I have quite got it down yet. It will come. Eventually.

UPDATE: Albert Lautman – “Mathematics and Reality”

•September 15, 2008 • 3 Comments

UPDATE (12/17/2012): There is a much better translation available in Albert Lautman’s Mathematics, Ideas and the Physical Real. Go buy that.

 

(Here is a translation of Albert Lautman’s “Mathematiques et Réalité”. Its a bit rough but I will try to clean it up over the next couple of days.)

“Mathematics and Reality”
Albert Lautman

The logicians of the Vienna School maintain that the formal study of scientific language must be the only object of the philosophy of science. It is a difficult thesis to accept for those philosophies that consider it their sole task to establish a coherent theory of the relationship between logic and the real. There is physical reality and there is the miracle of explanation, it is only that it needs more developed mathematical theories in order to interpret it. There is in the same way mathematical reality and it is similar object of admiration to see certain domains resist their exploration in a way, so that one must approach them with new methods. It is just as when analysis is introduced into arithmetic or topology into the theory of functions. A philosophy of science that would be unable to carry the entirety of the study into this solidarity between domains of reality and the methods of investigation would be singularly without interest. Philosophy is not the effect of mathematical nature. If logico-mathematical rigor can seduce it, it is certainly not because it permits the establishment of a system of tautological propositions, but because it shines a light on the tie between rules and their domain. It, at the same time, produces a curious event that is for the logicians an obstacle to eliminate, becoming for the philosopher the highest object of interest. It is a question of all the “material” or “realist” implications that the logicist is obliged to accept: this is the well-known axioms of Russell, the axiom of infinity and the axiom of reducibility. It is above all, following Wittgenstein, the affirmation that all true propositions correspond to an event of the world, which involves the entire procession of restrictions and precautions for logic. In particular, any proposition relating to the set of propositons, every logical syntax, in the sense of Carnap, is impossible since it would need for itself the power to consider, correlatively, the world as a totality, which is impossible.

The logicistian of the Vienna School continuously affirm their complete accord with the school of Hilbert. Yet nothing could be more questionable. In the logicist school, in the style of Russell, one strives to find the atomic constituents of every mathematical proposition. The arithmetic operations are defined starting from the primitive notions of elements and classes and the concepts of analysis are defined by extension starting from arithmetic.  The concept of number plays here a central role and this role is again augmented by the arithmetization of logic in the style of the work of Gödel and Carnap. The primacy of the concept of number, nevertheless, seems not to be confirmed by the development of modern mathematics. Poincaré had already indicated in relation to the theory of dimension that the arithmetization of mathematics does not always correspond to the true nature of things. Herman Weyl has, in the introduction to his work Gruppentheorie und Quantenmechanik, established a distinction that appears fundamental to us and will have to be accounted for in all future philosophies of mathematcs. It distinguishes two currents in mathematics. The first, issuing from Indian and Arab thinkers, puts the concept of number into relief and succeeds in the theory of functions of complex variables. The second is the Greek point of view in which every domain carries with it a system of characteristic numbers. It is primacy of the geometric idea of the domain over that of whole numbers.

The axiomatics of Hilbert and his students, far from wanting to restore the mathematics of sets as only a promotion of arithmetic, tend, on the contrary, to free for every domain of study a system of axioms as such, the union of conditions implied by the axioms emerging at the same time as the domain and the valid operations within this domain. It is also that one is axiomatically constituted in the modern algebraic theory of groups, of ideas, of the system of hypercomplex numbers, etc.

The consideration of a purely formal mathematics must therefore give order to a dualism of a topological structure and the functional properties in relation to this structure. In a similar way, the formalist presentation of theories: axiomatization is only a question of greater rigor. The object studied is not the set of propositions derived from the axioms, but organized being, structures, totality, having an anatomy and physiology proper. Citing as an example Hilbert’s space defined by the axioms that confer on it a structure appropriate to the solution of integral equations. The significant point of view here is the synthesis of the necessary conditions and not this analysis of first notions.

This same synthesis of the domain and the operation finds itself in physics under a slightly different point of view. Carnap, sometimes, seems to consider the relationship between mathematics and physics as that between the form and the matter. Mathematics would provide the system of coordinates in their inscription of physical givens. This conception hardly appears defendable seeing that modern physics, far from maintaining the distinction of a geometric form and a physical matter, combines, on the contrary, spation-temporal and material givens in the common framework of a mode of synthetic representation of phenomena; that this were by the tensorial representation of the theory of relativity or by the Hamiltonian equation in mechanics. Thus, one witnesses for every system with a simultaneous and reciprocal determination of container and contents. It is again a determination proper to every domain, the interior of which no longer subsists for the distinction between matter and form. Carnap seems, it is true, to have another theory of the relationship between mathematics and physics, conforming much more closely to his logistic tendancy. It considers physics, not as the science of real facts, but as a language in which one expresses experimentally verifiable statements. This language is submissive to the rules of syntax, of mathematical nature when it is uniformly valid in all their defined domains, of physical nature when their determinations vary with experience. There is the new affirmation of the universality of mathematical laws by opposition to the variation of physical givens. It seem to us that this conception does not reflect that the variation of physical givens only make sense in relation to the presumed variety of varying sizes, and this variety is physics. Carnap’s example from page 131 of his book The Logsche Syntax der Sprache is characteristic. When, we say, the components of the fundamental metric tensor are constants, it is a mathematical law; when they change they observe laws of physics. The real philosophical problem would rather be of the knowledge of how a differential geometry could become the theory of gravitation. This relation between geometry and physics is the proof of an intelligibility of the universe. It results in the development of the mind, in a way, to structure the universe in profound harmony with the nature of this universe. One conceives that this penetration of the real by human intelligence has no meaning for those excessively formalistic. These, in effect, would rather see in these pretensions of the mind to know nature a relevant approach to the studies of Lévy-Bruhl. Understanding would be for them a mystical belief analogous to the participation of the subject to the object in the primitive soul. The term of participation has in philosophy another origin more noble and Brunschvieg has justly denounced the confusion of the two senses. The participation of the sensible in the intelligible in Plato can be identified. If the first contacts with the sensible are not sensations and emotions, the constitution of mathematical physics gives us access to the real via the knowledge of the structure it has been given. Similarly, it is impossible to speak of the independent reality of the modes of thought according to which it leaves it to dread and disparage the merely mathematical language indifferent to the reality that it decries, philosophy engages itself through an attitude of meditation where it should achieve the secrets of nature. There is, therefore, no reason to maintain the distinction, made by the Vienna School, between rational knowledge and intuitive experience, between Erkennen and Erleben. In wanting to remove the relationship between thought and the real, as in refusing to give science the value of a spiritual experience, one risks only having a shadow of science, rejecting the spirit in the search for the real towards the violent attitudes where raison plays no part. This is a resignation that the philosophy of science must not accept.

3rd: Alain Badiou – Logic Course (1982 – 83)

•July 9, 2008 • 4 Comments

[Here is the final part of Badiou’s logic course from 1982 – 83. I find the brief discussion of the infinite and infinitesimal interesting. Enjoy.]

Alain Badiou – Logic Course (1982 – 1983)

Kant: What can I know? → the phenomena and the thing-in-itself. What must I do? → conform my action to the categorical imperative and ensure that my maxim is universalizable without contradiction. What must I hope for? → The immortality of the soul.

Gödel: the not-all of the calculable in relation to the rational.
Lowenhein-Skolem: the not-one of the rational in relation to the calculable.
Tarski: the impossibility of restoring to the calculable the crucial predicate of the rational that is said of this truth. The rational concept of truth is not representable in calculation: what is not representable are true statements that should be true.

Metatheorem of equivalence

Let C be a theorem of L. Let A be a part of a correct formula C. Suppose that A/B. Let C′ be where A is replaced by B. Then C′ is also a theorem of L.

Language LP of propositional calculus

1) alphabet =
• propositional variables: A, A′, B, B′ …
• logical operators: negation (¬), implication (→) [ The operators ‘or’, ‘and’, ‘equivalence’, are logically deduced from these two operators]
• punctuation signs: ‘(‘ ‘,’ ‘[‘ …

2) grammar

3) three axioms
• p → (q→p)
• [p → (q → r)] → [(p → q) → (p → r)]
• (¬p → ¬ q) → (q → p) = transposition

4) rule of deduction
• Modus ponens (MP): (p→q / p // q)

5) Theorem of deduction
• Theorem of Kalmer (1922): theorem of the completeness of propositional calculus that is said of all tautologies/contradictions: syntacto(decidable)-semantic(being a tautology). Where the language LP is coherent, complete, and decidable.

Language L of the predicate calculus of the 1st order

In addition to LP:

1) alphabet =
• individual constants or proper names: a, a′, b, b′ …
• individual variables: x, y, x′, y′ …
• unary predicates: P, P′, Q, Q′ …
• binary (R, R’ …) and tertiary (T, T′) relations
• Logical connectors:
• Universal quantifier (∀) [ Cf. ∃ is logically deduced from ∀]
2) grammar

3) two axioms
• [∀a (A→B)] → [A→(∀a) B] if ‘a’ is not free in A
• (∀x)A → Subst(y/x)A: one can replace in A x with y if it does not have a particular role in A
4) rule of deduction
• rule of generalization: If A then (∀a)A (one can universalize a statement) [The rule of existentialization is logically deduced from the precedent as ∃ is logically deduced from ∀]

→ (Meta)theorem of equivalence
→ Lowenheim-Skolem’s Theorem (1st)
→ Lowenheim-Skolem’s Theorem (2nd)

Gödel’s Theorem (1930): says the completeness of 1st order logic which is introduced in universally valid statements. Where the language L is coherent, complete but decidable. Cf.:
• Church’s Theorem: 1st order logic is not decidable
• Zermelo’s Axiom of Choice is formulable.
• Herbrand, then Henkin, have also worked on the completeness of 1st order logic for universally valid statements.

Logic of arithmetic by Peano = theory of identity + theory of successor + axiomatic schema for demonstration by repetition: {A[0] and (∀x)(∀y)[S(x,y) → A[x] → A[y]} → (∀x)A[x]

Properties of a Theory

a) consistency and coherence: one cannot deduce all at once A and ¬A; one cannot, therefore, deduce, only introduce. Cf. if I am → A and →¬A then:
• ¬A → (A → B): cf. “ex facto”
• ¬A and ¬A → (A → B) brings about (A → B) by Modus Ponens
• then A and (A → B) brings about (by MP) B!
• An inconsistent theory is a theory where at least one statement is not deducible.
• Lowenheim-Skolem’s Theorem (1st): If (closed formula) A is not deducible in (consistent theory) T the T + ¬A is consistent.
• Lowenheim-Skolem’s Theorem (2nd): If T is coherent, the theory obtaining an additional list of individual constants (= proper names), eventually becoming infinite, is equally coherent.
b) completeness (syntax): all statements are deducible. [semantic completeness is where all true statements are demonstrable. This forbids Gödel’s Theorem.]
• more precisely: a theory is syntactical complete if all of the closed universal formulas are decided. A complete theory is maximal: one cannot add them without rendering them incoherent.
• Cf. Closing the universe of open formulas: one adds to a formula a ∀ for every free variable. An open statement can be true or false depending on this or that interpretation. A closed statement, however, is constrained by a single interpretation. Cf. the open statement: (∃x)P(x) → (∀x)P(x) is indecidable. The open statement P(x) → P(x) is transformed into universally closed statement: (∀x)[P(x) → P(x)]
• A universally closed statement:
ß does not have a free variable
ß is closed by the universal quantifier (≠ existential closure)
• Gödel’s Theorem → incomplete in this sense
• Lindenbaum’s Theorem → ∃ extension completes all theory
• One does not have a complete theory of completeness. One cannot know why a theory is complete or not.
• Cf. Complete theories:
ß Additive arithmetic
ß Boole’s non-atomistic algebra
ß Naïve set-theory
• Most of the grand theories are incomplete.
• Lindenbaum’s Theorem: every theory admits a syntaxically complete extension
• Remarks: this extension is monstrous in three ways:
ß This extension is entirely ineffective: one cannot know whether the given statement is or is not.
ß This extension is not given by axioms but in a single package. It is very difficult to axiomatize. Cf. there exists a tension between completeness and axiomatic presentation.
ß One cannot know at all models of extension taken into consideration by the initial model
ß Extension is also ineffective, axiomatically inaccessible and pathological in terms of the model.

c) decidability = There exists a mechanical procedure that permits the verification for all formulas if it is a theorem.
• Cf. idea of a mechanical procedure that is a decidability machine. Cf. concept of recursive function, Turing machine, Markov algorithm…for these verifiable procedures (and not demonstrable)
• It is a calculable on the calculable.
• Example of decidable theory: the formal theory of commutative groups.
• Why a theory is decidable? One cannot know. Its not a question of simplicity.
• One shows that there is no decidability as decidability!
• One shows that in common decidable theories (cf. naïve arithmetic is additive arithmetic) there always exists formulas where numbers approach the gigantic.

Concept of the Model

Semantic: that yields 5 things, minimally:
• a domain of objects
• proper names distinguishing these objects
• there can exist properties of these objects
• there can exist relations between these objects
ß Intensive relations: given in their definition
ß Extensive relations: given in a list of these n-uplets
• rule of the true and the false

The interpretation of a theory T according to a model M makes:
• variable of T → object of M
• individual constant T → fixed element of M
• predicate in place of T → property of M
• relation of T → relation of M
One translated in this way all elementary formulas of T and one cuts on true/false.

non p        p and q   v     f          p or q    v   f        p → q    v    f          p ↔ q    v    f
v       f             v      v    f              v          v   v            v        v    f             v         v     f
f       v             f          f     f              f         v   f               f          v    v             f          f     v

The logical operators are not translated but one has directly the rules of evaluation. The logical operators are impossible to translate. Towards them, one has a disposition of evaluation, not translation.
Critique of modal logic (with possible, necessity, etc.): one can formalize the theories of probability with classical logic.

N.B. Formalization is the experimental moment of mathematics. It is not true to say, from this point of view, that mathematics is a formal science.

Mathematics exemplifies, in the simplest way, the fact that reality is an impasse. And all experience of the reality is a tense experience, extreme,

History

Boole (1850): 1st version of propositional calculus (with ¬, ø; but without quantification)
Frege, Peirce (1870): idea of Frege that everything returns ideography, that is to say, a writing where the concept would be transparent.
Whitehead and Russell: cf. criticize an inconsistency in Frege’s system holding to the possibility of applying the predicates to the themselves, holding quantifications over properties. Where stratification of syntax and the theory of types: what is to the left and to the right of the sign ∈ is not the same type. Cumbersome and useless (One needed an axiom of reducibility to destratify.
Hilbert: project of the auto-foundation of mathematics over calculus; to seek that everything true be demonstrable = reinstate without rest the truth of demonstrability. Auto-founding the demonstrable.
1915: Lowenheim’s 1st version
1920: Skolem’s recaptures this
1930: Gödel’s theorem

Universally valid formula = valid formula in all interpretive domains.
In logical language, all theorems are universally valid.

Kalmer’s theorem (1920) = theorem of completeness of propositional calculus is said of all tautologies/antilogies. Where language LP is coherent, complete, decidable.

Gödel’s Theorem (1930): says of the completeness of 1st order logic that introduced into universally valid statements. Where the language L is coherent, complete, but not decidable. Cf.:
Church’s Theorem (1937): 1st order logic is not decidable.
Cf. in logic, one cannot know what one says. Cf. mathematics ≠ logic. In mathematics, contrary to Russell, one knows of what they speak.

Logicist School (cf. Russell): logic and mathematics indistinguishable, under the order of a single syntax and argument that if two things are axiomatizable, they are the same.

Therefore:
• Propositional calculus → tautologies → language LP with “¬” and “→” → three axioms and rules of deduction.
• 1st order logic → universally valid formulas → Language L with another ∀, an axiom and rule of deduction.

All coherent theories have at least one model (theorem of Léon Henkins) = anti-idealist position: coherence allows the reference. There is not uninterpretable coherence. All coherence makes sense. Cf. Hegel: “all rationality is real”
All theory that have a model are coherent.

Lowenheim-Skolem’s (1st): all coherent theory has a numerable model
Lowenheim-Skolem’s (2nd): all coherent theory has a model of poor infinity

Categoricity of a theory:

The models of a type infinity given are isomorphs? If yes for a given infinity α the theory is said α-categoricity. It is not of an absolute categoricity since two different models of infinity are not isomorphs.

Examples:
• theory categorical (and therefore univocal) in numeration but not categorical for the superior types of infinity.
• theories α-catergorical for α > numerable but not for the numerable; ex. group theory communitive to unique divisor.
• theories categorical for all infinites separately; ex. theory of vectorial spaces of the entire modulo 2.
• theories never categorical; ex. group theory
Conjecture of Los: it would not have to have these 4 cases.

Morley’s Theorem: has demonstrated this conjecture! It is the longest and most difficult theory of all of logic.

In this way, categoricity does not distinguish the two types of infinity: the numerable, all that is beyond and which makes for categoricity. It is close to the numerability of writing.

Infinites, transcendence

Indefinite ≠ infinite, as potential infinite ≠ actual infinite

The infinite is not connected to the one in mathematics for Cantor. Before this connection was theological. (Uncreated) God was infinite and the (created) world was finite. Their one takes precedence over their infinity: It is the united transcendence of God that brings about their infinity.

One has therefore, before the 17th-century:
• a theological infinite (God)
• a physical finite (world)
• a purely potential logico-mathematical infinite.

Shaking the 17th-century: introduce the actual infinite in the physical world (cf. dialectical dramatization of this point by Pascal) but still not the concept of the actual infinite in mathematics.

The mathematical shaking comes instead from infinitesimal quantities. One progressively eliminates (between Newton = end of 17th and Cauchy = beginning of the 19th) these quantities by the introduction of a calculus of limits: “tightening toward” restoring the potential infinite. (Cauchy ≡ Russell more than Brouwer in the crises of sets)
Cantor introduces the radical revolution in secularizing the infinite by calculating the actual infinite. The introduction of the Hebrew alphabet!
The mathematical concept of infinitesimals is not far behind. Cf. Abraham Robinson (1960) and non-standards analysis. Nevertheless this has little repercussions: one redeems these known results for a long tome to come!
In this way, the mathematical revolution of the infinite has more than a century whereas the infinitesimal only has 25 years.

Pascal makes these two infinites symmetrical. Cantor does not believe in the infinitesimal.

Large Cardinals
• inaccessible
• ineffables
• of Ramsey
• low compacts [faiblement compacts]
• compacts
• super compacts
• enormous
• measurable

2nd: Alain Badiou – Logic Course (1981 – 82)

•July 6, 2008 • 3 Comments

[Here is the second translation that I promised. It is a little more interesting than the first, but the two do flow together quite well. Enjoy!]

Alain Badiou – Logic Course (1981 – 1982)

Lacan: “the real is the impasse of formalization” as well as the symbolic. Its nature pierces
the symbolic.
Theorems of limitation: theorems of the formal real
Gödel: impossibility of nomology, of completeness, of the decidable. Every formal
system is pierced. There is an excess of the true in the calculable.
Tarski: impossibility to formalize the notion of truth (of a statement): The true is
unrepresentable in arithmetic.
Beth: impossibility to demonstrate that such a notion is definable in a given system.
Lowenheim-Skolem: superiority of the infinite in relation to the numerable. There is an
excess of the calculable in the true.
However, this theorem = theorems also of excess! There is something of the truth that
exceeds the formal.

General Crises

Greeks: Crisis of irrationals. The idea that being is given in numbers, that all are counted,
is called into question by a “being” that is no longer a “rational” number.
17th – 18th century: differential and integral calculus. They can have a mathematics of
infinity and no longer only of finitude. They can therefore have a mathematics of movement.
Beginning of the 19th century: Non-Euclidian geometry. Doubt over the absolutivity of
Euclidian space taken by Kant as a structural category of experience.
End of 19th century: Paradoxes of set-theory. Attained through the reliability of the
intuition of the multiple.

Gradual Intersecting

Complete result of signs → correct formulas → true formulas → demonstrable formulas
The indecidable that limits calculation is wandering and cannot be assigned.

Theorem of Lowenheim-Skolem (TLS or LS):
if a formal system has an infinite model, it necessarily has a demonstrable (pathological)
model. All formal theories necessarily have an infinite demonstrable model. The infinite non-demonstrable does not have a specific formalization! All formalization is in excess in relation to possible interpretations. There is an over-powering logic in regards to that which one might formalize.
Gödel: true (and rational) > calculable; or: the semantic exceeds the syntax.
LS: the contrary: There is a point that flees formalization.
Torsion = this double excess of something at an indecidable and non-conforming
(pathological) time.

Deviations:
• true → calculable → the only indecidable → reversal in the irrational!
• purely productive insight by overlooking the point fleeing formalization → reversal by ruining the rigor of calculus at the same time

Metatheorem of deduction:
1. ((S+A)→B)→(S→(A→B)) ≡ deduce B from S+A again by deducing (A→B) from S ≡ joining A to S and in deducing B again from by deducing B from A in S alone.
2. This demonstrates a connection between the rational notion of the hypothesis (cf. S+A) and this deduction of implication! It’s a connection between the rational and the calculable.
3. This demonstrates this by recurrence along the length of deductions.
4. The calculable → the finite law where the infinite is on this side of reality. Theology posits the contrary: the world is finite but the law of God is infinte.
5. Grand mathematical thought is not secondarily calculating: it is conceptual. Hegel despised mathematics because he believed it was reducible to calculation. He criticized mathematics because the negative was not operative (because the negation of negation = affirmation).
6. Calculation is not the absence of the subject but its withdrawal. In calculation the subject does not lack but comes to be missing.
7. Formalism is mechanisable. The machine, if it deals with the real, has no relation to the real. Only the subject has a relationship.
8. Axiomatic schema: in a deduction, one can replace any variable by any concrete expression.
9. The logic of predicates of the 1st order: one only quantifies the individuals and not the predicates or properties. There is not, therefore, properties of properties.
10. Most of the algebraic theories are axiomatizable in this logic. On the contrary, topological structure necessitate, in general, a superior order logic.
11. Existence defined by double negation. But ∃ and ¬¬; are they equivalent? Existence is then posed as a limit to the total power of negation. And yet the universal does not have the ontological range that the existential does. Example: All gods are infinite → There exists an infinite God. This last statement commits me, but not the first where the verb has no existential value. The equivalence of ∃ and∀ is purely logical and would not know to infer on the ontological.

3 transformations for substituting ∃ and ∀:
1. replace (A → B) with ¬(B→A)
2.  substitute ∃ and ∀
3. put a negation (¬) in front

An erasure is always a marker, but on its borders.
TLS shows that the homogeneity of the inscription does not arrive to homogenize the
semantic.
Univocity does not exist, therefore, more for artificial languages (langues) than it does for
natural languages. It does not exist as an absolutely univocal language (cf. the failed dream of Leibniz). All languages are poorly made. The subject is effaceable from all language.
TLS is a massive return of the repressed.

Formal theory:
0. underlying logic
1. list of individual constants
2. list of individual predicates
3. additional specious axioms

Theory of identity:
1. no individual constants
2. predicative constants at 2 places: I
3. Three axioms:
∀xI(x,x) =  reflexivity
(∀x)(∀y)I(x,y) → I(y,x) = commutativity or symmetry
(∀x)(∀y)(∀z)I(x,y) and I(y,z) → I(x,z) = transivity
This theory gives 1st-order predicate calculus with equality. One can sometimes consider it not even being mathematical and remaining purely logical.

Theory of the strict partial order:
1. not individual constants
2. predicative constants at 2 places <(x,y) or x<y
3. three axioms: irreflexivity, anti-symmetry and transivity

Theory of the strict total order:
Supplementary axiom: ∀x∀y x<y or y>x

Theory of successor:
1. no individual constants
2. binary relation S(x,y) = y is the successor of x
3. four axioms:
¬∃x S(x,0) ≡ 0 is not succeeded
S(x,y) → [S(x,z) → y = z] ≡ there is a single successor.
S(x,y) → [S(z,y) → x = z] ≡ there is a single predecessor.
All numbers have a successor.

1st: Alain Badiou – Course on Logic (1980 – 81)

•July 5, 2008 • 7 Comments

[What follows is a translation of lecture notes from a course on Logic delivered by Alain Badiou in 1980 – 81. It is rather short and deals specifically with Gödel and his theorem of incompleteness. I am not sure who would be interested, but here is it. There are two more course notes (1981 – 82 and 1982 – 83) that I will be posting in the realitively near future. Comments would be greatly appreciated. Enjoy!]

Course on Logic (1980 – 1983)
Alain Badiou

1980 – 1981

A formal system S is composed of 4 things:

1) Alphabet = stock of signs, either
• individual constants or proper names (0, for example)
• individual variables: x, y, x’, y’ …
• predicate  constants (“=”, etc.)
• function constants (“+”, “*”, “succession”…etc.)
• statement of propositional variables: p, q, p’, q’ (Not predicate variables)
• logical signs (negation, implication, and, or…and quantifiers ∃ (there is at least one), ∀ (all)); cf.:
i. operator “or”: A or B ≡ ¬A→B
ii. operator “and”: A and B ≡ ¬(A→¬B)
iii. operator “equivalence”: A/B ≡ (A → B) and (B → A)
• signs of punctuation: “(“ “,” “[“ …

2) Grammar = rules used to form the correct     formulas

3) Axioms
Cf. Three axioms of the calculus of propositions (= axiomatic schema)
• p → (q→p)
• (p → (p → r)) → ((p → q) → (p → r))
• (¬p → ¬q) → (q → p) = transposition
Cf. also axioms of calculus of predicates
• [∀a (A → B)] → [A → (∀a)B]
• (∀a)A → Subst(x/y)A: one can  replace Ax for y if y does not play a particular role in A (Universal instantiation)

4) Rules of deduction
• rules of substitution: if A ↔ B, on can replace A by B in a formula.
• Modus Ponens [p →q/p//q]
• Existential Instantiation: if p then (∃x)p
• Universal Generalization: if A then (∀x)A (One can universalize a statement)

Model M constitutes an interpretive domain: all systems of formulas are interpreted as true/false.

In the system, there is not true/false.
In the model, there is not demonstration.
Where the relation of to truth to demonstration (= as in proof)
S → M ≡ Syntax → Semantics = interpretation
M → S ≡ Semantics → Syntax = formalization

Gödel’s Theorem: There is always a formal statement p that is not demonstrable or refutable in S and where it is translated in M it is true.

There are three limitations where one demonstrates that there are equivalents:

1) incompleteness (semantics): ∃ a true statement not demonstrable in S.
2) indecidable (or incomplete syntax): ∃ an indecidable statement (neither demonstrable nor refutable)
3) irreflexibility as in coherence: impossibility to demonstrate in S its coherence (non-nomology)

1+5 conditions authorizing Gödel’s Theorem:
0) Countable alphabet (discreets!→writing) and final statements (= one can number the statements!)
1) The system must receive as a possible model the integers (the system must have an arithmetic capacity): the translation of the symbols must be capable of being the integers (condition for the function crossing translation and numerization).
2) The system must receive a diagonalized numerization: ∀P ∃a as Num(P)=Tran(a): “a” is translated as the number “n” at the same time as the statement P(a) has “a” for a number; “n” is then in a diagonal position for the predicate P (double function!).
3) The system must receive a predicate D support under the demonstrability: D(i) interpreted as Tran(i)∈Dem that is true if n=Tran(i) is the number of a demonstrable statement (= reflexive condition of the system or expressive condition as far as the question of demonstrability).

These three conditions converge ≡ a single condition: that the system is capable of representing all of the recursive functions. For any elementary arithmetic model, it has these properties.
4) The system must have a symbol translating the negative.
5) Being given a predicate P, ∃ the predicate Opposed-P(i) that corresponds to not-P(i) = condition of complementation.

In fact, three conditions arise:
1. the language is finite or numberable (not the design!)
2. the system comprises a symbol for negation
3. the system admits an elementary arithmetic model (whole numbers)

Principles of demonstration

1) Let D be a predicate demonstrably defined by:
Condition 3 = ∃ Opposed-D contrary of D (cf. Gödel Theorem 5); or
Condition 2 = ∃a such that Num[Opposed-D(a)]=Tran[D(a)]=n (It states that a statement “a” is indemonstrable.
2) One can demonstrate then that Opposed-D(a) (i.e. that “a” is not demonstrable in S) is not demonstrable (→incomplete):
If it was demonstrable, on could have D[opposed-D] and therefore Tran[Opposed-D(a)]∈Dem; or the translation is precisely “ ‘a’ is not translatable” therefore, it would not know it belongs to Dem! → Contradiction. Where incomplete: a true statement (in case the statement “ ‘a’ is indemonstrable”) is indemonstrable.
3) One demonstrates that more than that it is not refutable (→ indecidability)
(a) If it was refutable, then the model would no longer be a model because one could have decided that it would be false!
(b) If not-[Opposed-D(a)] is demonstrable, its translation is not-not-Tran[D(a)]∈Dem if Tran[D(a)]∈Dem. Therefore, n=Tran[D(a)] translates demonstrable statement; however n=Num[Opposed-D(a)]. Therefore, one could in a system demonstrate a statement Opposed-D(a) and its contrary not-[Opposed-D(a)]! Inconsistence of the system.
4) One then deduces the irreflexivity of the system:
If one could have demonstrated in the system a predicate affirming the consistency of the system, then one could demonstrate in the system that one could not have at the same time D(a) and Opposed-D(a); therefore one could deduce in the system Opposed-D(a) since one knows that D(a) is true; but then, by modus ponens, the indecidable statement would be decidable in the system and the system must be incoherent!

In this way, the formal does not exhaust the true.

Double function. Canonical example: a[P(a)] if, for example, “ ‘a’ says that it has a property P”
Husserl → nomological theories say that it is its own law.
Program of work defined by Hilbert.
Gödel indicates the failure of the nomological program.

Critchley & Simmons

•June 21, 2008 • 1 Comment

It is said that Nietzsche, during his fascination with Wagner, sent some of his compositions to Franz Liszt. Liszt’s first response was to believe that it was some kind of joke (which, of course, it wasn’t). With this in mind, I give you Simon Critchley’s artistic expression.