 So today we were planning to have another speaker, so I'm just going to give a small replacement talk and it's going to be rather informal. Some ideas that are around in things Pierre and I have been doing. So this is partly collaborative work. And I just wanted to present these ideas and see what you think of it. Whether it's, I think, rather intuitive, but of course these intuitions are not necessarily shared. So it would be really useful to hear your thoughts on this. And it's all work in progress very much. So I wouldn't count as an official instance of the seminar and you're free to leave if you think if you were expecting something finished or proper. But nevertheless it might be of some interest and it might surprise, the topic of it might be surprising given that it's just about inter-synestic logic, which is a fairly standard logic for my standards. But there will be a connection with other stuff I've been presenting here. So there is some overlap. So it's going to be informal, as I said. It's about some recent discovery of semantics for inter-synestic logic that seems correct. I mean, I haven't gone to all the details of the proof, but the proof strategy is there. And well, in every case I have gone through, seems to be right. But I haven't written everything down and the devil is always in the details. So there might be problems with the formal semantics we've developed. But on the other hand these probably will, if existent, will be minor and the interpretation seems to go beyond that. But it's an interpretation of the semantics, which as a semantics for inter-synestic logic is also an interpretation for inter-synestic logic. So of the formal details there might be some minor mistakes, hopefully minor. But the content is mainly informal. But of course given that it's an interpretation of this formal semantics I have to show the formal semantics to. Which is a relatively simple semantics, at least from my taste. So what is inter-synestic logic? I assume most of you have heard about it. It's a logic that is created by Brower in 1907 already and then it's formalized by his student Heiting in 1930. And this is the basic basis for constructivism in the foundations of mathematics. So this idea that mathematical objects are not in some platonic universe but are constructed by people. And this requires a whole different way of seeing the logic because now we cannot assume that everything is either true or false from the start because that means that they would already have true values out there. So we have to construct truth and falsity if we want to prove something. So I go to the specifics immediately. So this idea is that if you have a constructivist view of mathematics you need a different logic. And it's a very well-developed and studied non-classical logic the most well-developed. There's tons of mathematical and philosophical literature about it. It's a very formally interesting logic even at the propositional level which classical logic is not really. It's just truth tables. This is not characterizable by truth tables. And it's really a big field inside the field of logic. So it's useful beyond this constructivism in mathematics in the sense that it could give an account of verificationist, empiricist philosophy of science which is the view that all our knowledge is based on consequences that come from observations things that are verified by means of observations. And this is very similar to the idea of mathematics that everything has to be constructed. Everything has to be constructed, you'd say, from observation. And Dumbled among other people have worked out such a more general philosophy of empiricism based on interestingistic logic. It's also very useful in computer science because all these constructions are perfectly algorithmic. So if we have a proof for something in mathematics we need to also have an algorithm for the thing you're proving which makes it like that part of mathematical and logistic mathematics is directly useful for automatically and by necessity useful for making programs. And also, interesting logic can be seen as the logic of programs but I won't go into details there. So there's a lot of links with computational aspects, deep links and more and more people in computer science, in computer science are interested in interesting logic. But besides that it's been used in type theory, category theory, algebra. So serious mathematicians, they are willing to engage with a non-classic logic which is very unusual, I mean this is the only instance. And you can either see it as an object of study or really have a constructivist philosophy of mathematics in the background but the two are perfectly compatible. By the way, interrupt me at that stage. This can be pure conversation, doesn't have to be me talking all the time. You can even have the whole thing in conversation form rather than presentation and discussion. So how can we characterize this formally? Well, the usual thing is proof theory of course and actually it's surprisingly simple in proof theory of interesting logic. So you just take away, you took any usual account of proof theory in classical logic which is axiomatic or natural deduction or sequence calculator. The thing that you have, whenever you did in a basic logic course you can just eliminate some aspect, some rules in it and you get interesting logic. I mean of course you can set up a proof theory for classical logic that is not just amendable to interesting logic but usually that is considered a problem. I mean that's not a good proof theory. It means that there is some problem with modularity but you cannot just take away something important and still leave the other things intact. So it's kind of almost like a benchmark for a good proof theory that you also get interesting logic out of there by removing some rule. So specifically what do we want to remove? That's excluded middle of course. We don't want A or not A because you don't necessarily have construction for A and its negation. You don't want to eliminate double negation and this is directly related to the absence of the reductio at absurdum for positive conclusions. So if we want to conclude A from a set of premises then that's not the same thing or you cannot conclude that from the fact that adding not A to premises gives you absurdity which is a usual technique used in mathematics everywhere, reductio argument and it is perfectly okay to do that for negations if we want to confirm that something is not true. We can just assume that if it is true then we can get to absurdity but to affirm something positively you cannot do it in this way by enthusiastic standards. So this is very close to the intuitions behind the constructivist philosophy and it is extremely simple to characterize not more difficult than classical watching. On the other hand, the semantics is a whole different story surprisingly I have always thought. So there are many adequate semantics I mean this is a whole like almost a field you can people have seriously tried this it's not that this is just like underdeveloped or something like that and I would make a distinction between two sort of approaches, structural approaches that are still relatively doable in terms of complexity and objectual approaches that have their issues without being disrespectful to any of these but for the structural ones there are algebraic approaches with the hiding algebras which are very well known algebras and there is an inferentialist approach where you basically just say well whatever, well it uses meaning if you have the rules in the inferential system in the proof theory then that determines the meaning of the connectives and of course given that the proof theory is so simple also an inferentialist's semantics is going to be simple I mean that's not a problem but it doesn't give you much extra information of what these sentences mean beyond how they are used by definition almost same with the algebra you want to keep it as general as possible you just care about the relations between propositions which come with which are in clear correspondence with the proof theory so there is not so much added there either but it's also a very it's not more complicated than brilliant algebra which is the one for classical logic so also on that level we are I mean there are interesting approaches and there are not there's not much problems with that just that they don't give a semantics in the ordinary sense of the word of a saying which things correspond to our sentences like in a classical case of course if you don't have on a plantonistic view maybe you don't want from mathematics to have these things that correspond to it to be objects in a platonic universe with rather like constructions or something like that but you want you want something some object states or constructions or something like that in an objectial approach and these are given and it's not that they are not there but they are rather complicated and there's crooked frames which I always have found very weird from constructivist point of view so these crooked frames or like all crooked frames just like sets of possible worlds but these possible worlds are classical or they could be seen as classical and it's like a sort of the picture you get is well that we have a sort of temporal view of proofs in in mathematics that always give you more information and so the worlds get more populated the whole time temporally and you have some restrictions on these frames but the notion of a possible worlds or even frame and the temporality of it doesn't really correspond well with the idea of constructions in itself it seems to me it seems like you already to capture the meaning need to think of possible futures in which you will establish stuff so to see whether something is true have to look at all possible futures and so on I mean it seems like almost inherently a non-constructive approach but this is I don't have time to give further arguments against it but it's a bit weird to say it's like a translation to another system which is super interesting that this translation is there but it's more complicated in mathematical semantics in topology in in enthusiastic arithmetic and so on but they are they evolve really substantial mathematical machinery and I would say that none of the objective ones have clear and attractive philosophical interpretation this is a very harsh blunt statement but well if you try to study them I think you might be convinced that they do of course these people have given philosophical interpretations but they are like difficult to get your head around then there is the BHK interpretation which is very well known Robert in the world of interpretation that is not really a semantics it doesn't work as a formal semantics but it's a good way to interpret the ideas behind the system intuitively so we say that the proof for A conjunction B is a combination of a proof for A and a proof for B or you might say construction for A and B's combination of construction for A and construction for B it's just either a proof of A or a proof of B and this is the very non-classical aspect here like we can very well this junction like for example A or not A by proving that it's converse this is absurd so A or not A holds in classical logic because we cannot have A and not A but here you really need to have a proof for both either A or proof of B and then a proof of A in YB is a procedure that takes you from a proof of A to a proof for B so it's a purely procedural approach where you just morph one proof into another and for negation that's just a proof for A and YB a bottom which means absurdum so it's possible to transform any hypothetical proof of A into a proof for the absurdum that's proving false but so this is a very interesting interpretation but it doesn't it's not formal you can have a version of this about programs and so on but it doesn't give you semantics in itself and here what I try to do is give an intuitive semantics that's or I hope it's kind of intuitive that is rather close to the BHK interpretation but nevertheless it's like yeah, you'll see so I first get the semantics but I'll go here very quickly because I'm mostly interested in the interpretation of it and for the interpretation we'll have to come back to the clauses of the semantics anyway so without the interpretation we will not be able to make much sense of it either way but I think I can argue with a favor of every single clause as being justified by some inhibitions so what is an enthusiastic model as we find it, I mean it's not it's not a concept that exists already maybe I should look for another word like enthusiastic it's a combination of a set of findings and a function simply that assigns to each member of the primitive formulas and to bottom a set of sets of findings what is a finding intuitively I will get into more detail there finding is just a concrete inquiry into the world or into what is true it could be a mathematical construction it could be an empirical scientific study it could be conceptual analysis but it's important that this is something, I mean for the interpretation it's important not for the formal semantics it's important that this is a concrete thing, a material thing we're not interested in like the proof for, like in an abstract sense, but like some person at some time having given a proof that's a finding or somebody some scientific study published in a paper done by some people the material stuff is the finding it's not something like the concept of a proof or it's not something like that it's something touchable at least in my interpretation but this is just a semantics any set is a model and any function that assigns to a primitive sentence a set of establishers so establishers are sets of findings so that's like scientific studies brought together like a bunch of scientific studies for example that establishes a sentence so and sentences can be established by several branches of scientific studies together so for example if we have a sentence a then that could be established by this sole scientific study over here and it could be established by these two other scientific studies over there one of the two would be enough but you might have two of course so that's the idea and it's not restricted to mathematics or to any specific epistemic task it's just like you have an epistemic action that establishes something is this like is this more or less clear so we have scientific studies and together they might establish a certain conclusion of course on their own they also establish some sentences but if you have multiple you can establish more for example if we have a a sentence that that the vaccine for covid is effective this might be established by a bunch of studies that all focus on some aspects of the vaccine and if you take them together you get an establish for the sentence the vaccine is effective yeah I have a question about these findings I suppose to be like positive the positive conclusion about something that does I suppose to be like just an event whatever could be something that doesn't exist that's a very good question and I think actually above effective reason reading and a non-effective reading works here so it's about an internal logic and that doesn't really matter whether you're right in your findings but of course and this kind of anti-realist perspective, empiricist perspective works pretty well with the original project of enthusiasm constructivism but in fact they could also be sort of facts like effective findings where we only have to look at if there is a finding then it is actualized finding then it's really also effective this is related to that question should I expect the set map to bottom to be empty or should I expect it to be the set of like the finding that the earth is flat and the finding that water is made of aliens yeah that's a good question I think sometimes it's the same question if we're only talking about facts it's just empty I guess they are concrete findings but they are not necessarily actual to your point they also have hypothetical findings and there will be no actual findings or no actual effective findings for the bottom for the absurdum but you could imagine that somebody had come up with a finding of the absurdum so it's concrete in the sense that we are thinking about them as we are thinking about this whiteboard something we can touch but of course we can speak about the possibility or the non-actualized blackboard that could have been there and a bottom a finding for bottom for the absurdum will never occur but nevertheless we can imagine hypothetically that somebody had found a proof for the absurdum this is like a potential object one step farther toward the fringe then how about the finding of a round square where maybe it's metaphysically impossible this is even worse it's logically impossible it's just like it's very important that there can be stuff that is not actualized like the bottom particle hopefully not be realized but you have to do subjunctive reasoning here so suppose that there were a proof for A even if A is false then we can give a proof for B this is in the BHK interpretation the IGBI implication but this is like a hypothetical presence of a finding suppose that we had a finding for A then we had a finding for B even though A might be false it's two accounts for and I will talk more about this later it's to account for this subjunctive reasoning in science where sometimes a scientific study does not confirm anything like direct observable about the world but like if if we had a vaccine with these properties then it would be useful for these groups or these diseases or something like that even though we might not have a vaccine to see this sort of subjunctive conditional reasoning is important and that's why we have findings that are not actualized but actually it doesn't matter whether they are actual or not because we are giving a semantics for logic which is all about hypothetical reasoning so which ones in the actual world are actualized and it doesn't have to be characterized by the semantics we have to add that on top of it so particular reasoning should think about findings as events or as a process or in a kind of concrete so they are an event of finding something or they can be I like to see them as acts of agents I guess you could also see them as events or even as just facts in the world facts not about the real world but facts about like epistemic activity or something like that or they can be like touchable objects simply like a scientific study in the sense of the I mean, scientific studies may be not touchable but a construction, a mathematical construction that a person writes on paper I mean, it is material stuff so there are all levels of characterization that you might have but it doesn't matter points that it's concrete and it establishes something and it can establish something together with other findings so we call the sets of findings bunch just for short, a bunch of findings collection of findings set of scientific studies something like that is a bunch it's not a very if you have a better word than it might be too so if we have a set of bunches we can call we can characterize its upper set which is basically just every bunch that extends is included in it so a sentence can be established by several bunches of findings and then we might be interested in richer bunches I mean, ABC the H for example is an extension of this thing and so this thing together I would not say establishes the sentence but it nevertheless gives you enough information for the sentence, right? because the writes you have A and B in here and A and B establishes the sentence there's too much here but nevertheless it will be interesting we will say that this one constructs A2 it doesn't establish it but it constructs it so we are interested in getting all the supersets of all these bunches that are establishing sentences and so this upper set just takes all these supersets of bunches we call a proposition just a set of bunches that is closed under supersets so if we take all extension this is a set of bunches you see, now we take all the superset extensions of this we get a big set of bunches and we call such a such a big set of bunches a proposition and we say that sentences or the bottom particle express the proposition that is the upper set of the bunches that establishes it so the proposition that is expressed by A is not just these bunches but all the larger bunches too that's the proposition expressed by A so the members of the proposition expressed by bottom remember that bottom also just got bunch of things that establish it hypothetical findings of course behold that's the word or or an anti-realistic notion so we call these consistent bunches so not just the establishers of bottom of the absurdum are consistent bunches but also everything that is bigger than that if you have an inconsistent finding than all sorry an inconsistent bunch of scientific studies than every richer bunch of scientific studies of findings will also still be inconsistent of course we call two bunches incompatible if their union is inconsistent so if you have a finding that establishes that this chair is black and blue and we have a finding a hypothetical finding that establishes that it's cream well then that's going to be a finding that a bunch of findings that establishes the contradiction the absurdum and we call it an inconsistent finding inconsistent bunch but if we add still other findings to that the bunch remains inconsistent if we add to that a finding for 1 plus 1 equals 2 it will still be an inconsistent bunch okay we use F and G for findings and B and Z for bunches as variables are we okay so far yeah I have a question that's interesting your findings are supposed to be acts but what if you have a new finding that you have to pray after one that contradicts the findings a used finding so this is a completely monotonic view on science so then the first one was not a finding in reality like as we said we have a clear physical finding but our new finding as well so what you might do in that case is say that the thing that that we thought was a finding because we were wrong was actually just hypothetical finding and not actualized I mean it seemed to be an actual finding but in fact that nothing is found that's wrong so but this aspect is completely not taken into account the whole falsive monotonic approach that we find in science it's like at a moment in science given the views we have what is a finding and what is not but that's an excellent point and there might be something to do with it in the sense of developing some sort of monotonic interesting logic but here it's like playing old fashioned interesting logic thereafter so that's why also we have this idea and proposition richer bunches are still that constructive formula are still in the proposition while in principle this finding contradicts this finding sorry this finding contradicts one of these two then adding it to the original establisher seems to be not anymore evidence in favor of this sentence because so then you might have other kind of propositions which are much more complicated but here everything is simple but thanks a lot for an excellent question so now we have two relations this is really crucial so usually people define just one relation of how semantic stuff verifies let's say sentences but here we have a notion of establishing and a notion of constructing this thing here constructs A because the information given by these findings is enough to construct the truth of A you have everything to get to A but it doesn't establish it there's all this stuff that has nothing to do with it so that's a big crucial difference and there you see the relation but all the work we've been doing is not relevant to the notion here the notion of construction is deeply irrelevant while the notion of establishing is deeply relevant at least as I read it so the notion of establishing will not be closed under upper sets while the notion of construction has to be so if you look at enthusiastic logic the conclusions we get arrive at by enthusiastic logic are about what can be constructed it's not about what is actually being constructed but about what can be constructed it's not about what has been established but what is establishable constructable from anything in a given pool of findings either possible or actual so the construction is the irrelevant notion and this is the notion we're first going to define so if we wonder whether a permitted formula is constructed by a bunch of findings that is true if and only if the bunch is inside the bunch there is a member of the establishers of like here this one is not an establisher so our semantics doesn't give it directly but there is an establisher inside of it is included in it so we will say that this one construct despite it not establishing it we just say that B constructs bottom if B is inconsistent where inconsistent I remind you is also a proposition is also closed under strength and extended bunch or extending of bunches that are so I guess none of these are really remarkable here is nothing it is very similar to the BHK interpretation which I remind you is the idea that you can construct A implies B is constructable if you have an algorithm to go from a proof of A to a proof of B here is the same thing except now in bunches so if you have a bunch strengthen sorry if you have a bunch B prime that establishes A then then adding the bunch that constructs A implies B to this hypothetical bunch that establishes B prime the union of those two gives you the conclusion of B so concretely when we say that that A implies B is okay if a bunch is for example in this case where you know that A and B together establish A oh I used the same letters sorry for that I guess I'll just call this B I'm sorry because this is sentence level I don't care about that so if A is a finding a singleton bunch that that constructs A implies that constructs well suppose that I need to add something here that A is an establisher for Q then A is an establisher for Q so if we add an establisher for Q to finding A for example B then we get B you see it's like this this A finding alone constructs A implies B because if you add for Q to I's P because if you add if you look at the establishers for B you can construct an establisher for P out of there so this is the same idea you basically put bunches together in order to see what can be concluded from them here we have conjunction this is just if something if some bunch constructs a conjunction that is true if it constructs both a conjunct and this junction if it constructs either of the conjuncts so nothing surprising there but it's very interesting because you don't allow that some other bunch to construct A so we have to add a bunch of A or a bunch of B in order to construct a bunch of constructs A or B so this is all pretty natural given BHK interpretation so and it seems to be like all the symbols so are we done do we have a semantic storage since it's logic unfortunately it's not as simple as that otherwise people would have already come up with that for a very long time the crucial thing is here that this here is another symbol it's not our construction relation it's our establishing relation so we have a hypothetical A and from that we should get a construction from B so it's you're not going to hypothesize there being a mere construction if you're trying to prove that A implies B A implies B means in this semantics that if I had a bunch of scientific studies that actually do the work I wanted to do and nothing more then from that I can construct that's my A implies B and that's doing the whole world so it's a strict notion it's an exact notion the first layer here that is going to give us the premise meaning while the conclusion meaning remains fairly standard so we have a kind of an an instability or an asymmetry between how we interpret the things we conclude from to the things that we conclude to and I think this is pretty natural given the fact that if we start concluding something we have to need something either hypothetically or abstractly but there being a construction it's like we assume that somebody has given to us the scientific studies that make that thing constructable it's like the the ask for if I want to prove something I don't have the task to assume that there is some big finding I don't have to take into account all of these things at the lower levels just have to look at the assumption that some findings actually establish this and see what follows from there and then of course getting the B here the thing that follows the consequence we are just interested in what is constructed by means of the establishes for A so establishes for A construct B that's more or less the idea behind this and the most consequence relation will also be defined by that before showing you how establishing works because now I've just said what construction is but of course given that we need for antecedents the establishing idea which is the novel idea here I also have to define what that is but maybe first like a preview for the definition of the consequence relations so the gamma intersensically entails A if and only if in each model a bunch constructs A for each consistent bunch that establishes gamma and what does it mean to establish gamma that means that you have for each member of gamma a bunch and take the union of all these bunches so we have an establisher say we have three sentences in there PQ and R only gamma so we have to give a bunch that establishes P at least one bunch that establishes Q at least one bunch that establishes R CD and then a bunch for gamma is just the union of all these things of all these bunches without anything else right it's still about establishing things that are irrelevant but if you want to have a constructive argument you have to assume for each premise there to be a hypothetical establisher of it and whatever establisher you choose you always have to be able to construct the conclusion so again there premises are interpreted in terms of establishing and conclusions in terms of construction there's this asymmetry going through I still didn't say what establishing needs except of course for primitive formulas and I think that's where whole originality lies and gamma will be more or less though so for establishment situation is maybe a bit a bit weirder but still simple clauses so of course for primitive formulas it just whatever is established by that formula that's given by the model right and a bunch that is the establishment for the sentence will establish that sentence at primitive formula level same for the bottom symbol the absurdum it will just be established by whatever the model says it's established or again this will not be actualized but it doesn't matter we will skip this for a while the implication conjunction and distraction are the weird ones out and are influenced by non-classical semantics specifically get fines and other people's exact semantics and what is the conjunction of what is an establishment for a conjunction of amd let's just think intuitively here how can we establish that this chair is blue and black and that Nivella N'Ava is in Belgium well we have to a finding for the one or a bunch of findings because we might not have enough with one finding a bunch of findings for the chair a bunch of findings for the state of Nivella N'Ava the state in which Nivella N'Ava is namely being in Belgium and then we just have to conglomerate them just have to take the union it's not that just like we need a bunch that is both a bunch that establishes the property of Nivella N'Ava and the property of the chair of course not there will not be such a thing there will be no set of findings that establishes both of them they are completely different sentences so we just need to have the union of the two just have the findings that together give you that Nivella N'Ava is in Belgium and the second finding is that the chair is black and blue and enjoying them so if A and B is true in a bunch if the bunch is the union of two bunches where the first one establishes A and the second one establishes B I find that the only possible clause you might give for establishing a conjunction I don't know maybe I'm too biased by the fact that I've been working on this for the last two weeks for disjunction the situation might be a little less completely obvious but still I would like to strongly defend it given this interpretation by the findings so when is a disjunction realized or established by a certain finding when can we say that a scientific study establishes a disjunction is it just because it is one of the disjuncts that is established by that scientific study I say no why not because you need a link I mean if I have a scientific study that establishes that the vaccine is safe that scientific study does not establish that the scientific that the vaccine is safe or ineffective or something like that that just the scientific study establishes only the strongest link neither does that scientific study that the vaccine is effective against COVID does it it does not satisfy either that either Belgium is a country or the vaccine is effective what are these findings if you are alive or dead two are you don't know which one but you have a finding that even for one year you want a bunch of findings to do post-establishing you want to if it's not one then you should get the other so this is just not A not A implies B and not B implies A that should be necessary for to be able to say that a scientific study establishes something and this implication is this notion of subjunctive findings that I've been talking about and because of the lack of reductio one doesn't just give you two and three for free you have to add them explicitly to the cement took me a second to catch that yeah exactly so you have to remember that this arrow is not just a material implication as we have in classical logic it really means that you can construct an establishment and do another establishment of a sentence actually construction and do an establishment so it's it's really something extremely strong it's a modal property it's an intentional disjunction that is here one that says that whatever of the two it is you get if somebody gives me a counter example for the left thing like for example a scientific study could very well show that that either the antidepressant works by arousing the patient or by decreasing the stress levels or something then what you actually confirm is that the antidepressant if it works that assume that this patient is not aroused then it's the stress level that makes the patient no longer depressed I don't know whether it's a good example but it's this intuition that you can only say that the scientific study establishes a or b if whatever proof or hypothetical proof or a would show up later you can also have a b whatever I put a little proof against a show up later would automatically give you b of course this might be a strong thing to ask but maybe it's not evident to establish stuff but in full degree there's not no probabilistic notions here or something you might build in some sort of subjective probability or probability or something that makes or establishing like less than one in degree of beliefs that's perfectly reasonable to do but here we just talk about really established stuff maybe that's something that never occurs that's something else or maybe a mathematics but not in actual science I don't know I don't think that's so that's all there is nothing else in the semantics and I think I have gone through every clause and showed that it was plausible the interpretation I mostly have gone through findings of the material of the STEM inquiry by agents at a specific time and place they need not to be actual they can be of several natures just whatever you interact whatever way you have to interact with reality out there or with whatever reality you consider so bunch simply like some findings bunch can establish and construct really the talk is basically over but I just want to see that I didn't so just to repeat establishing is a radically exact notion of the efficacy what exactly did just confirm this sentence while construction itself and this has some some concrete cases so K or B is not necessarily established by a study that establishes A or a study that establishes B as I said and a bunch that establishes A does not entail that extension the bunch also establish A I think I have also clarified that and a bunch that is a consequent a bunch that establishes A and B is not necessarily a bunch for A of course because it might be too big like the case of a bunch for chair being black and blue and the phenomena of being in Belgium that bunch is not a bunch that establishes that chair is black and blue so I think this corresponds to natural language idea of showing or establishing a scientific study one two and three together show that A is true I think according to my natural language situations you cannot just add a fourth study here and still get a true sentence if it's just an arbitrary one of course if it's one that like gives extra evidence and so on then you might have a story but if it has nothing to do with it seems very impossible so there is a little subjunctive establishing that is important so findings cannot only just work directly by confirming something but could also confirm subjective stuff confirm not facts in the world let's say but say that if a finding were found then sorry if a sentence were established then you might be able to construct some other sentence so that's when scientific studies show relations rather than categorical properties of the world okay so it seems that this, well this is a new interpretation it's based on a combination of exact and inexact motions usually you have exact semantics and inexact semantics here is the two working together I think it gives a pretty simple intuitive semantics for interesting logic if the semantics is adequate which seems to be the case and maybe this is kind of a revised approach to empiricism or to verificationism where you can build up a view on the logic of these philosophies that actually directly gives you full enthusiastic logic without further semantic machinery on top of it by a small detour via relevance that is like heading way out of it okay, thanks a lot I just, I may have a tiny question it's kind of a weird question like so I like the intuitive argument that I like the intuitive argument for the notion of establishing for disjunctions like actually I'm with you I guess my yeah it's kind of an awkward question I'm not sure what this looks like so I assume you're doing this now or you've already done it the flip side of that argument is like and those conditions give you what you want when you put them into action in the logic because that's I mean what I'm going to say is I'm almost applying so you can show that those conditions give you what you want for disjunction and that those are the only conditions that give you what you want for disjunction that's the one reasonable way to do disjunction given the rest of the constraints of the framework that you put into action yes, yes, yes because after all you want things like derivative syllogism to be perfectly okay right so if you have a bunch that establishes A or B and you have a bunch that establishes not A you're done, you have nothing else if you don't have all those clauses built in exactly so the establishment relation then should give you B because B should be constructable out of that which is not natural at all I think this is actually one of the problems why ordinary solutions that are not asymmetric and premises and conclusions are so notoriously difficult to get because it's not because you have either a construction of B or a construction of B of A or a construction of B that you would have something like disjunctive syllogism out of there I mean they might be completely independent so why why would the absence of the one or the other so this is really a crucial thing and it wouldn't work without and it's mainly also the most important difference in the interpretation between premises and conclusions establishing and constructing you establish stuff as this modal character sometimes of there being link between stuff while constructing is just whatever comes out of it whatever can be concluded on it the whole modal nuances are lost if you ask what can be constructed and shouldn't be touchable either construction is just like an abstract notion so it's always going from something concrete and material to something a bit in the air abstract logical consequence in that sense it's really a logical empiricism here like it's from the establishment which is like a relation of of empirical evidence to combined with logic as like a non material observable thing you can choose the consequence for yourself ah yeah gamma is a set of sentence of feeling of bench because this side we stipulate that be established gamma bench be established a set of sentence so the set set of sentence is an abstract of bench or a specific study on you say in your definition that an abstract object gamma construct no so the same so it is a union so be be establishes gamma gamma is an abstract thing but be is a concrete bunch of findings establishes this abstract state the union of a set of bunches so there are still the same same between gamma and gamma a it's not a structure of Russia it's a consequence of Russia so it's not the same symbol this symbol now it's hard to say but this is just consequence relation and this is the construction to say I use the same symbol but it's quite common in logic to have it's a bit awkward but consequence sometimes people use a nice symbol and later but the point is to establish on two things consequence symbol only you have universal consequence between gamma and gamma why do you use it's a bit confusing yeah I agree maybe I should just something that is quite conventional and logic to do maybe I should have followed that example you can add to the line or something like that it's really a different notion but it's pretty close nevertheless this means that a is constructed from gamma and this means that a is constructed a can be constructed from gamma let's say a can be constructed by the bunch so it's like from and by it's not completely different but that maybe makes it more counterintuitive or more problematic even so you're right that I probably should use another symbol there just a semantic consequence this one this one is a verification a sense of construction so we went back to what Charles was saying earlier if a is a correct it's a consequence of your establishing relation for this a very specific case there is no possibility to establish knowledge that's not exhaustive of all possibilities because if you need to have your own but establishing is not a case so in this case every such event can only be exhaustive of all possibilities right maybe also if you don't you cannot establish knowledge right because if a is a but is left out you can prove that negating one's stuff will authenticate together yeah you should indeed so in this case it's from each bunch that most establishes a that we should have a construction establishing that's the only clause I didn't talk about when you talk about this yes yes yes I used it I didn't explain it but here I don't explain what it means to establish an implication I wasn't talking about vocation I was talking about this term this is defined in terms of you know don't start I'm sorry if we're not explaining this right so indeed I went a bit quick here so here you have an and in order to do the establishing of that sort of thing from not B to A it's something, it's an implication you need to establish not just construct the implication because constructing implication we have seen what it means it's from an hypothetical establisher any establishing of it you have to be able to to construct the B here the implication has to work differently and that's maybe the only tricky aspect in the semantics so for it to work I think I can justify it but it's a bit weird nevertheless so here we also have this kind of asymmetry but in the other direction so now we say that whatever construction there is for A the state or the defining B should be with that construction should establish B so for construction you should go to establishing and so this is a kind of heavy task to do so from the fact that something can be constructed you should be able to establish B but this is a matter of definition I guess because establishing is just what a scientific study gives you so just like you have a set of so it's the art of making putting hypotheses of taking hypotheses in a scientific study or in mathematics let's say to keep it a bit more touchable so it's like suppose somebody else would construct this formula A for example or some proof that are available in literature exists and together they give they give me arguments to construct a sentence that's the hypothesis so why is my study an argument for A implies B or an establishment or A implies B well because my study on top of these other studies establishes B so suppose that they really do construct A then my extra argument on top of it is is enough proof is all the proof you need and not more than that to go to B so that's an establishing relation it's not about oh maybe only a part of it is needed or something no no it's really those things that together construct plus my study establishes the sentence so it's like a bit of a downgrading sort of I just want to jump in on this this is related to a question that I had about the same clause if I am I'm trying to put on a hat that I do not wear I'm trying to imagine that I am like a super cranky intuitionist like old school I might be worried about that whenever each is introducing some kind of implicit universal quantifier over something that I'm not going to be happy that you're quantifying universally over right like how am I supposed to know about all the constructions even the ones that I don't even have that I've never done the proofs I've never written the approaches that I've never thought about am I going to be mad about that I'm trying to impersonate I'm misimpersonating the constructiveness too but I'm worried about this right I see where you're coming from but on the other hand it's purely hypothetical right so maybe there or as a mathematician you don't care so much about or not only about what is true but also what is relations between proofs or what is construct what is a relation to what is construct you have to give an algorithm to transform a certain proof into another proof now it might be enough that there might not be a proof for this very outrageous universal claim but if there were a constructable proof right it's not established in relation but it's not the established in relation but it is the constructive so there is a constructive proof I mean you hypothesize that there is one then my algorithm the new bunch that is the algorithm together from the algorithm gives you the an argument and how do you say that I have distinct memories of reading like reading in in our papers and stuff like the constructivist do reason like this all the time say that we had a constructive proof for this so we would know exactly what the members of the set were so then we could take those members of the set and we could blog with them and then you use that to go somewhere else that's right they do talk like that they talk like that all the time so maybe they won't be mad yeah yeah that's helpful I still have my concern you can understand some of all the rich you can't establish that because it's not an exhaustive of the world of the total sum of knowledge and you can you can't say that either one thing of your own right it's the junction of all the knowledge that you have you can start establishing you can construct it but not establish yes that's true yeah because you need some exhaustion of all the above I need that you will understand But these big disjunctions are never supposed to be established, it is not interesting to establish them. And they are typically the kind of thing that we conclude from establishments, from findings. So there we are really at the level of logic and not at the level of empirical findings. So yeah, no, no, these big disjunctions are never going to be established or I mean of course the model does not, I mean there might be models that do that, this is just an abstract construction. But in the interpretation it would not, the model of the world, it corresponds to the world is not like that. Yeah, it also means that I don't want to be doing the modeling stuff, but still we need to prioritize that there is some kind of progress in the science right, that if you have a good position, you guys are doing the progress and the evolution changes what you are going to do on the average, you can't establish this kind of... Well you can still do it at each time, but you need another mechanism to account for the fact that... Diachronic logic and revolutions, which we don't have. I don't think it's contrary to Kunian approaches, but it doesn't give logic or something for those. It's just to be very careful about putting findings together and not believe that this will still keep the first thing of finding because it might be contradiction. One more question that's totally wild, so you're absolutely free to say, no, I haven't thought about that. But I wonder, because I've played with a lot the Kripke frame interpretation back in the day, so now that you have this, have you thought about looking at so what these relations look like, if you cast them back into the Kripke frame kind of perspective, are they going to analogize or is that just not going to make any sense? I don't think so. I wouldn't think so either, which is kind of cool, right? Because there's something interesting about having almost like incommensurable interpretations of the same semantics. Is it kind of wild? Yeah, it's really weird and it's kind of almost a miracle that it's the same. And that I should maybe have said. So this gives you prima facie a non-transitive relation because you have a very strong interpretation of the premises, namely they have to be established and a weak interpretation of the conclusions, they can just be constructed. So if you want to chain them, of course, you don't have a construction for, I mean, if you have A implies B or A entails B, you have B entails C. And you go down, you have to strengthen, I'm not thinking myself right here, I'm not thinking here. So if you have A implies B and B implies C, well this means that A is from an establishment of A you get construction of B and here from an establishment of B you get construction of C. So this doesn't say that you have, if you want to go from A to C you have to go from an establishment of A to construction of C, by an establishment of A you get construction of B. But there's no changing because you don't have an establishment of B, you just have a construction of B. So the fact that this holds, which of course it does if it's a good semantics for logics, it's kind of a miracle and it's a miracle of cut elimination. So this is basically the cut rule which Genson proved for his statistics calculus that it can be eliminated, you can do everything without ever using cuts. And that is exactly the reason why the semantics works, like you can't get all the Genson rules. So every proven interest is a goal you can perfectly build and afterwards you know that you have proven everything. So like in specific bunches if you're going to look at when you already have fines and so on, then cut is not going to work. But in like the empty bunch where you start from nothing at all and you have to really presuppose A, a bunch for A and then construct the C, B and then it will be transitive. So like under the roots it's non-transitive, deeply non-transitive and relevant and so on. But the relation you get out of it, like in the empty bunch, like the interesting thing there is, why miracle as Pierre says it, transitive. And this is a thought behind every single semantics I've seen that interested in is about a transitive notion. It doesn't take into account the delimitation theorem by Genson. So that is why I am fairly honest, fairly sure that like in Kripke frames you will always look at same interpretation for premise and conclusions and notions that are not hyper-intentional, well establishment is obviously an hyper-intentional notion. Practically no laws of logic, ordinary laws of logic apply for it. And these are typically in impossible world semantics, not characterisable at all. You need like more fine-grained semantics for it. So I'm fairly confident that it doesn't work at all. That's cool. That's really cool. Hopefully it's correct too. I mean the proof seems to work. I would understand if you did it so it doesn't matter. Oh it's pretty simple. Because it's so close to the Genson rules. It's really derived from the proof theory rather than like trying to thanks a lot for all these nice questions. But I'm asking you whether you find this reading intuitive. I find it intuitive. And the clauses, at least for this junction and conjunction, which are usually the most difficult to argue for for transitive approaches, they seem to be the only thing you can get given these interpretations of establishment and constructing. Maybe I'm extremely biased. The only word I do vaguely share, kid, is worry about the metaphysics of findings. But partly I think it's just funny because I'm not used to living in this world in some sense that in that sense it's how to put this. It's actually potentially advantageous that it's vague. In the sense that there's more than one, there being more than one kind of thing that you do your job is actually like in no sense a problem. That would be the end of it. Yeah and I'm not actually used to that. So I guess in some sense maybe I should kind of quash my unnies because like actually it's good. It means more people can agree with you because you're not forcing them to adopt a particular metaphysics of findings. Yeah so that's why I like it because it's it's it's both usable for mathematical purposes and construction or more the untake approaches or philosophy of science like usages of interstitial logic. Because a finding might also be like concrete like punishment given or something like that. I mean if you have the untake leading but that would take its way to forward. Like I haven't properly thought that through but like I really like the people have been using interstitial logic in many different domains. And I kind of think that a good interpretation should also take these applications seriously to the maximum. That's why a finding is sufficiently vague but of course if you give substance to it it might become... I guess the one other thing that I have the one thing that I have a little bit of trouble kind of intuitively interpreting is so finding find upper set or sorry finding find set of findings fine. Upper set of set of findings. That's a bit of a weird object to play with. Right because it seems to kind of imply I guess that maybe that this is and this is a bit related to this kind of synchronic diachronic notion maybe that's a bit think of the universe of think of it there being a kind of closed universal findings such that the upper set is like a reasonable thing for you to play with. It's a little awkward. Now I'm not saying it's perfectly within the realm of like modeling assumptions that you could make to try to construct a logic to play with these objects like I'm not saying that I think it's weird once you're already in that perspective but it does it does have that implication it's a bit funky. Yes. It's at a hyper intuitive level. I do agree and it's also non-constructive because it's an infinite set. Right right of course it is. But you don't need it really it doesn't occur in the in the in it's just to characterize this notion of proposition of of every of what is constructed. Sorry of which bunches construct the same formula. If you have a given a sentence and you wonder like how can we construct it. That would be the proposition. If you want a notion of proposition like every possible way which you can construct it. Then obviously that will be an infinite notion. I mean it's not just like actual but also like potential ways to construct. Of course it's going to be. But I don't use that notion. You don't like it. Don't use it then. Yeah that's it. So all the clauses I mean you have the clause of course for for what is the construction for primitive formula. And that is that some bunch in it explicates it establishes it. Included in it some bunch included in it. And this is very close to the notion of construction of this this upper set. But I mean that this just a property right it's not a set as such you need in that case. Just need that one of things I am a bunch one of the things inside of me establishes it. You don't have to look at in the other way around it up you have to look down. And that does not that is not a non-constructive notion. So it's only if you want to get a notion of construct of proposition out of it. Of non-hyperintentional proposition because there's also hyperintentional propositions namely the potential establishes of it. But if you want this notion of proposition I don't know why. Then that's a non-constructive notion but you don't need to think that seriously. Sort of potential infinity. Okay then there's no more comments than that. Thanks. Thank you so much for the attention and for the great questions and comments that was extremely useful.