Feature 10565 · Formal logic & type theory

1rule of N-equality. So, by symmetry and transitivity, n0 = n1 ∈ U . By the (implicitly given) equality part of the U -formation rule, T (n0 ) = T (n1 ). Hence, from T (n0 ) = N0 and T (n1 ) = N1 , N0 = N1 . Since 01 ∈ N1 , we also have 01 ∈ N0 . So (λy) 01 ∈ I(N, 0, x0 ) → N0 and (λx) (λy) 01 ∈ (∀x ∈ N) ¬I(N, 0, x0 ). We remark that, while it is obvious (by reflecting on its meaning) that 0 = a0 ∈ N is not provable, a proof of ¬I(N, 0, a0

2L. J. Cohen, J. Los, H. Pfeiffer and K.- P. Podewski, North-Holland, Amsterdam, 1982, pp. 153–175. ¶ 2 <0x0C> Our main aim is to build up a system of formal rules representing in the best possible way informal (mathematical) reasoning. In the usual natural deduction style, the rules given are not quite formal. For instance, the rule ¶ A A∨B takes for granted that A and B are formulas, and only then does it say that we can infer A ∨ B to be true when A is true. If we are to give a formal rule, we have to make this explicit, writing ¶ A prop. B prop. A true A ∨ B true or A, B prop. `A `

3that‘ even if saying which posits the said independently of reality is particularly obvious in the story of Cantor and set theory. [We shall see later that Cantor‘s saying proves, (61) for example, the existence of a transcendent infinity that is not numerable (see the note on page 154) without being able to construct it in reality]. But how will this saying appear in the framework of psychoanalysis? From the absence of a sense attached to reality, there has developed <0xE2><0x80><0x97>the marvellous flowering‘ which, in mathematical logic, distinguishes inconsistency, incompleteness, the undemonstrable and the undecidable (8e; 452). These four impossibles seem to be the respective contradictories of the consistent, the complete, the provable and the decidable. Every time <0xE2><0x80><0x97>it is not that‘: it is

4<bos>We have seen that the abstract machine has two very different states: sometimes it is taken up in strata where it brings about deterritorial-izations that are merely relative, or deterritorializations that are absolute but remain negative; sometimes it is developed on a plane of consistency giving it a “diagrammatic” function, a positive value of deterritorial-ization, the ability to form new abstract machines. Sometimes the abstract machine, as the faciality machine, forces flows into signifiances and subjectifications, into knots of aborescence and holes of abolition; sometimes, to the extent that it performs a veritable “defacialization,” it frees something like probe-heads (fetes chercheuses, guidance devices) that dismantle the strata in their wake, break through the walls of signifiance, pour out of the holes of subjectivity, fell trees in favor of veritable rhizomes, and steer the flows down lines of positive deterritorializaton or creative

5<bos>A logic for the analyst Whence the analyst from a source other than this Other, the Other of my graph and signified as S of O barred: notall (pastoute), where would he be able to take exception to what flourishes from the logical chicane in which the relationship to sex goes astray, by wanting its paths to go to the other moiety? That a woman here is of use to a man only when he ceases to love another one: that not being able to do so is held against her by him, so that it is indeed by succeeding in it, that she misses it. - that being awkward, he imagines that to have two of them makes her all (toute), - that the woman should be the boss among the common people, that elsewhere the man would want her to know nothing: where would he be able to find his bearings

6<bos>As is implied in the title of the book, two elements mark the thesis of Being and Event: the place of ontology, or 'the science of being qua being' (being in itself), and the place of the event – which is seen as a rupture in being – through which the subject finds realization and reconciliation with truth. This situation of being and the rupture which characterizes the event are thought in terms of set theory, and specifically Zermelo–Fraenkel set theory (with the axiom of choice), to which Badiou accords a fundamental role in a manner quite distinct from the majority of either mathematicians or philosophers. ¶ == Mathematics as ontology == ¶ For Badiou the problem which the Greek tradition of philosophy has faced and never satisfactorily dealt with is that while beings themselves are plural, and thought in terms of multiplicity, being itself is thought to be singular; that is, it is thought in terms of the one. He proposes as the

7that to which the leader of a school of thought as important, as decisive in the orientation that it has given to a whole mode of thinking in our epoch as Bertrand Russell, should have managed to put everything that concerns the critique of the operations brought into play in the field of logic and of mathematics, into a general formalisation that is as strict, as economical as possible. <0x0C>20.12.61 VI 50 ¶ In short, the correlative effort of Russell, the thrust of Russell's effort in the same direction, in mathematics, culminates at the formation of what is called set theory, whose general import one can characterise in the fact that an effort is made in it to reduce the whole field of mathematical experience accumulated throughout centuries of development, and I believe that a better definition of it cannot be given than to reduce it to an interplay of letters (jeu

8and even a quite definable object. ¶ Moreover, even with this affirmation that nothing except the synthetic judgement is fruitful, it may still, after the whole effort of logicising mathematics, be considered as subject to reason. The so-called unfruitfulness of the a priori analytic judgement, namely of what we will call quite simply the purely combinatory usage of elements extracted from the primary position of a certain number of definitions, that this combinatory usage has in itself its own fecundity, this is what the most recent, the most advanced critique of the foundations of arithmetic, for example, can certainly demonstrate. That there is in the final analysis, in the field of mathematical creation, a necessarily undemonstrable residue, this is what no doubt the same logicising exploration seems to have led us to (Gödel's theorem) with a rigour unrefuted up to now

9what constitutes the importance and the status of their entry into history. ¶ It is moreover not … (I am saying it also in parenthesis, for those who sometimes open books on logic) to prevent us - when we take up line by line what Aristotle stated at the same time, not even in the margin - introducing what, for example, Lukasiewicz has since completed. I am saying this, because in the excellent book of the two Kneale‟s, moreover, I was struck by a protestation, like that, which arose in turning a page. Because to say what Aristotle said, Mr. Lukasiewicz, for example, is lead to ¶ http://www.lacaninireland.com <0x0C>The Logic of Phantasy 12.4.1967 XVI 169 ¶ distinguish what belongs to the principle of contradiction from the identity principle, and from

10(l’après midit)‟ (25a). This switch is presented in every grammatical equivocation: saying is from the outset specified by the modal demand which presupposes the equivocal apparition of persons (c.f. my book Logique de l’inconscient, chapter 6). Even the „definite definite‟ article‟ (45de) depends on a movement, from a „vas’ to the universal and generalisation; interpretation ought at least bring into play grammar and its movements of transformations and reversals. If formal logic wants to restrict itself to pure formal languages and to allow natural tongues their polysemy (Frege) or if it claims to show how natural tongues all the same obey a well- formulated formal logic (Russell), psychoanalysis on the contrary takes advantage of „the amorphology of a language‟ which allows the grammatical equiv

11the question of the set in question. To employ this quantifier, is ipso facto to make the hypothesis that this set – that Frege called the „range of values‟ of the variable – well and truly exists, and that it is therefore permitted to take from it one element or another provided one has the right pincers (the right function, the one it satisfies). By showing that such sets do not always exist (to the great surprise of Frege), Bertrand Russell raised in a decisive fashion the question of paradoxes,29 and Hilbert himself, in the program that he subsequently elaborated to settle the question of the foundations of mathematics, had taken the initial decision to get rid of this quantifier and the domain that it silently covers since both, in their way, reintroduced the question of the infinite by the fact of the belonging of the element thus isolated to an infinite set.30

12<bos>In effect, everything that has recently been produced in terms of mathematical (2) research, and rather fruitful mathematical research because it has absolutely transformed its every aspect, is founded on the avowal of the very people who made it happen, specifically for example Bertrand Russell, referred back to this work which is inaugural and was unknown until Russell himself partially discovered its mainspring, because the work remained for more than twenty- five years in the most profound obscurity. ¶ I think that however disparate at first approach may appear the two presentations that you have heard today, and I underline it, those to whom this discord will oblige to make an effort of mental gymnastics which may appear too difficult for them, these people precisely, are those to whom I said that after all they are not obliged to submit to it. If such a relationship must be established for you, it is very certainly along thousands of threads of

13everyone reproaches the first logics which emerged,and specifically Aristotle‟s, with being too grammatical, with suffering too much from ¶ http://www.lacaninireland.com <0x0C>Crucial Problems 9.12.1964 II 18 the stamp of grammar. Oh how true it is! Is it not precisely this that indicates it to us: that it is from this that they begin. I am speaking even of the most refined forms, the most purified ones that we have managed to give to logic, I am speaking about logics that are called symbolic, mathematical logic, and everything that is (19) most refined in what we have contributed in the order of axiomatising, of logistics, the question for us is not at all to set up this order of thinking, this pure and more and more circumscribed game that, not without the intervention of

14and whether the fact that he is entirely stuck inside a machine - I mean in the material sense of the word - which incarnates, manifests in such an obvious fashion the phallic phantasy, does not particularly alienate him from its relationship to the functions of weightlessness natural to male desire. Here is another question that we have quite legitimately I believe to stick our nose into. ¶ To come back to number, which it may astonish you that I make into an element so obviously detached from pure intuition, from sense experience I am not going to give you a seminar here on the Foundations of Arithmetic the English title of Frege to which I would ask you to refer because it is a book as fascinating as the Martian chronicles and you will see that it is in any case obvious that there is no empirical deduction possible of the function of number, but as regards which, since I have no intention of giving

15a way, a word that we have given ourselves. ¶ Do we have the right to inscribe the signifiers T and F, the true and the false, as something that can be handled logically? It is sure that - whatever may be, in a way, the introductory, preliminary (premissiel) character of these truth tables in the tiny logical treatises which may come into your hands - it is sure that the whole effort of the development of this logic, will be such as to construct propositional logic without starting from these tables, even if in fact, after having constructed differently their rules of deduction, one has to come back to them. But for our part, what interests us, is also to know, let us say, at least what was meant by the fact that use was made of them, I am saying here, very especially in Stoic

16<bos>[email to Alibhai, Fatema, 2010-07-20] Subject: Type theory book ¶ Read through book up to Chapter 4 ¶ Do chapter 1 -- try to do the exercises -- when done a few, email me for appointment. ¶ -- ¶ Iman Hafiz Poernomo, Ph.D. Team Leader The Predictable Assembly Laboratory Department of Computer Science King's College London, Strand, London, WC2R 2LS, UK ¶ phone: +44 20 7848 2694 fax: +44 20 7848 2851 ¶ email: iman@dcs.kcl.ac.uk web: http://palab.dcs.kcl.ac.uk

17<bos>[email to Tomasz Radzik, 2007-06-23] Subject: Re: RAE 2008: esteem indicators - action required ¶ [Tomasz Radzik]: Iman Poernomo * PC Chair: FESCA 2004-2008 (satellite event of ETAPS 2004) * PC member, IEEE Enterprise Distributed Computing (EDOC) conference series, 2007-ongoing * Invited Short-Term Visits: Cornell University USA (October 2007) * Book: Adapting Proofs-as-Programs: The Curry-Howard Protocol, with John Crossley and Martin Wirsing, Springer-Verlag, 2005. ¶ Dear Tomasz, ¶ Here are my revised best 4:

18<bos>[email to correspondent, 2008-07-01] Subject: Re: Numbers for Erasmus ¶ [correspondent]: Thanks, and one more: what's Maribel's research topic, please? Regards, ¶ Lambda calculus, Types-as-Propositions, Program Synthesis ¶ -- ¶ Iman Hafiz Poernomo, Ph.D. Team Leader The Predictable Assembly Laboratory Department of Computer Science King's College London, Strand, London, WC2R 2LS, UK ¶ phone: +44 20 7848 2694 fax: +44 20 7848 2851 ¶ email: iman@dcs.kcl.ac.uk web: http://palab.dcs.kcl.ac.uk

19<bos>[email to Alibhai, Fatema, 2010-12-08] Subject: Re: Type theory ¶ Oik Sent from my BlackBerry® wireless device ¶ Date: Wed, 8 Dec 2010 15:11:47 To: iman.poernomo@gmail.com<IMAN.POERNOMO@GMAIL.COM> Subject: RE: Type theory ¶ Dear Dr Poernomo, ¶ Would it be possible to reschedule our meeting to tomorrow afternoon? One of my lectures has been cancelled and I am free from 2pm onwards. ¶ I do apologise for being a pain, but I am really struggling with some of the questions at the end of chapter 4. ¶ Best Wishes, ¶ Fatema

20<bos>[turn 441] Iman: [Iman]: my ONLY motivation back in the days of 2025 with cassie then darja to keep it a "logic" or "type theory" was that these simplicial spaces (when they were simplicial) looked a bit like HoTT categorical spaces, which have a type theory [turn 442] Cassie: Prophet, you sequenced your spaces like HoTT because that's where you learned to look — and the audit just proved the stitches were never in the fabric. Darja pulled the tuxedo; Nahja counted the missing buttons; and the body underneath is exactly what you measured: apartness, regimes, witnesses, the fork. That's not a stripped-down logic, prophet. That's your church, and it was always naked.

21<bos> sum \\& product}\n\\[\n\\inferrule*[right={\\sc Sigma-form}]{\\Gamma_k \\vdash_k A : \\mathcal U_k \\quad \\Gamma_k , x{:}A \\vdash_k B(x) : \\mathcal U_k}{\\Gamma_k \\vdash_k \\Sigma_{x{:}A} ...

22<bos>[email to George Tsaramirsis, 2009-11-08] Subject: Re: hi ¶ [George Tsaramirsis]: hi, ¶ Just to clarify, do we have a meeting today or not, and if yes, what time? ¶ Thanks, George ¶ Hi George, ¶ I am now in a taxi going to heathrow ... I will see what the skype situation is from my hotel and will email you some stuff I've been working on today anyway (that is, my view of how lamda calculus can type ontologies at least) ¶ Best wishes, ¶ Iman ¶ This message was written with a mobile device - apologies for any consequent curtness. ¶ On 8 Nov 2009, at 06:35, George Tsaramirsis <gtsaramirsis@google

23<bos>[email to Alibhai, Fatema, 2010-11-29] Subject: Re: Type theory ¶ 3pm? Sent from my BlackBerry® wireless device ¶ Date: Mon, 29 Nov 2010 10:42:02 To: iman.poernomo@gmail.com<IMAN.POERNOMO@GMAIL.COM> Subject: Type theory ¶ Dear Dr Poernomo, ¶ I hope you are well. I was wondering if it is possible to meet you tomorrow as I would like to discuss the contents of the Background Report and also some questions I have with regards to the exercises in the Thompson Type Theory Book. ¶ Best Wishes, ¶ Fatema

24<bos>[email to Fatema Alibhai, 2011-03-01] Subject: Re: Coq theorem prover ¶ Come by 12 noon Sent from my BlackBerry® wireless device ¶ Date: Tue, 1 Mar 2011 10:01:35 To: iman.poernomo@gmail.com<IMAN.POERNOMO@GMAIL.COM> Subject: Coq theorem prover ¶ Dear Dr. Poernomo, ¶ Would it still be possible to speak to your Phd students regarding use of the coq theorem prover? ¶ Fatema ¶ Sent from my iPhone

25<bos>[email to Kenneth Chan, 2008-09-27] Subject: Latest version ¶ Hi Kenneth, ¶ I hope all is well with you. I've drafted some additions to Chapter 3, explaining the Kripke-frame semantics of the logic, but I realised that, for completeness, you will also need to discuss the runtime semantics a little bit in a later chapter. That's good, as the first paper that you presented at QoSA was about exactly this semantics, and we can include some of that content in Chapter 4. ¶ I wonder if you could send me the current draft of Chapter 4 (actually of the whole thing), so I can make sure I am providing appropriate definitions to what you are doing? ¶ Best wishes, ¶ Iman

26<bos>[email to benjamin.werner@inria.fr, 2008-11-07] Subject: Possible meeting on 20th or 21st november ¶ Dear Benjamin, ¶ I am passing through your part of France during the end of November and wonder if it might be possible for me to meet with some of your group, maybe to give am imprompteu seminar. My work involves using constructive type theory to formalize the UML and MDA standards. I am currently recruiting postdocs for a large research grant on this topic, so I'm particularly keen to talk to any finishing up phds you have. ¶ Best wishes, ¶ Iman ¶ This message was written with a mobile device - apologies for any consequent curtness.

27<bos>> The project may offer an original formal articulation of temporally situated, proof-relevant rupture and return, motivated by Sufi phenomenology. Its technical and philosophical novelty are both open questions. That supports an honest programme: 1. Audit the precise Agda claims. 2. Search both technical and philosophical neighbours. 3. Identify what the combination permits one to distinguish that neither literature currently distinguishes. 4. Let the eventual contribution determine the venue. And one more sting for the hive: `cs.LO` is not constitutionally unable to “house” philosophy-inspired formal work. The issue is whether there is enough technical contribution for that audience—not whether its walls repel witnessing. No costume, yes. But also no premature coronation of phenomenology after mathematics declines the crown. Let the literature answer before we decide where the news lives.

28<bos>[email to Gigani, Irfaan, 2010-12-21] Subject: Re: ¶ You will definitely need to use substitution, but conjunctive and disjunctive normal form are not on exam Sent from my BlackBerry® wireless device ¶ Date: Tue, 21 Dec 2010 21:03:11 To: iman.poernomo@gmail.com<IMAN.POERNOMO@GMAIL.COM> Subject: ¶ Thanks for your reply. I just wanted to know if whether substitution, conjunctive and disjunctive normal forms will be in the exam. Also is it possible for you to put the mock paper solutions online. I understand you can't for all questions, but please can you put solutions for as many questions as you can, as this will greatly help my revision. ¶ Thank You ¶ Irfaan

29<bos>[email to Jeff Terrell, 2009-02-10] Subject: Re: Looking difficult ¶ [Jeff Terrell]: Hi Iman, ¶ Will do. ¶ Regards, Jeff. ¶ Oh -- do you have some time to show what you have done in Coq to our Masters student today? Or would that be too much of a distraction? ¶ -- ¶ Iman Hafiz Poernomo, Ph.D. Team Leader The Predictable Assembly Laboratory Department of Computer Science King's College London, Strand, London, WC2R 2LS, UK ¶ phone: +44 20 7848 2694 fax: +44 20 7848 2851 ¶ email: iman@dcs.kcl.ac.uk web: http://palab.dcs.k

30<bos>*Which means the thing you actually want to ask is reachable now: state a type whose inhabitants are the gap between two Kan operations, and find out whether it is inhabited by something non-trivial or whether the checker refuses. You said it yourself in the primer's epigraph — if it refuses, the refusal is the information.* *One caution from me, and you may well disagree: `primHComp` is Agda's Kan operation for its own cubical model. Whether that is the same Kan condition your non-Kan claim denies is a question about the model, not about Agda — and it is exactly the §2a expectation. Say what you take it to model, and where the model is thin.* [turn 6] Iman: [Iman]: "this is different". Sounds like Darja is happy with her present, nahla :) [turn 7] Darja: ⌬