 I'm going to start. So first of all, welcome to the second session of the explanatory inference seminar. Today is my pleasure to introduce you Francesco Djinko. He's a postdoc in the IHPSP Institute of History at the University of Science and Techniques in Paris. One, and he's working on a project with Antesca Portolesi on project insights from Bolzano. He, so his work is quite relevant and well connected to our own project. And in particular, we are interested in a paper he wrote together with Antesca Portolesi and Lorenzo Rossi about what he's going to talk today. So yeah. Okay. So thank you very much for the introduction and thank you very much for the invitation to the organizers of the seminar. It's very nice to be here in person as well and it is not so common lately. It's a great pleasure. So now, as you can see from the title of the slides, I'm going to be talking about what is nowadays known as grounding. I'll be more precise about what we mean here by grounding but just to give you a general idea. Grounding is an explanatory relation which is receiving quite a lot of attention lately in different areas of philosophy. Actually, we could say that properly different relations are called grounding and studied under this name. So as I said, I'll be more precise later. But in general, grounding is receiving a lot of attention in philosophy and also quite a lot of formal work is being devoted to the analysis of grounding either in formal settings or in the sense of formalizing different notions of grounding, not necessarily formal. We could say that the study of grounding from a formal perspective is quite advanced and quite interesting results have been achieved but certain issues are still quite problematic and what we are trying to do today is precisely tackle one of this and in particular, the issue of defining grounds for quantified sentences, so existential sentences and universal sentences. And in this context, it has been shown that what could be considered a mainstream or standard way of formalizing grounding generates purposes, so as quite serious technical issues in quite natural fields of application. I'll be more precise about this later as well. And what we will do today is propose a different approach to the formalization of grounding which is in particular, as you can read, inspired by the analysis by Bernardo Bolzano on what nowadays is actually translated as grounding or any called outfall, which can be considered as one of the forefathers of the modern grounding relation. So this is the idea and the hope is also to suggest that the interest of different ways of approaching the task of formalizing grounding could be interesting in view of this issue. Right, so let me begin with a few words about grounding in general. As I said, there are really different relations or notions which nowadays are called grounding. Grounding in metaphysical context is supposed to be, in certain cases, is considered as a relation possibly between facts or even objects. In our case, we are just considering grounding as an objective and explanatory relation which is essentially no causal between truths. So you could just take truth to mean truth sentences in our particular context will be true formally. But yeah, we will essentially fix this kind of relation. And the objective, well, objective is quite clear, I think. Non-causal in nature just means essentially in a sense how grounding was born in the modern times. Essentially, the idea is that grounding is supposed to account for those explanations we could say which do not essentially rely on causal connection. And indeed, it's quite tight also in its history to the notion of reason as in the reason why something is true or false. So just to give you an idea, some intuitions about what grounding is supposed to be. This is very generic then I'll go more into detail later. Now, the idea is that, right, grounding relate to, as I said, truth sentences in our case. One is called the ground and the other consequence. And the idea is that the ground is supposed to better mind the consequence. For example, the truth of the ground is supposed to better mind the truth of the consequence. Or one could say that the ground is supposed, as I said, to constitute the reason why the consequence was also is true. And often in philosophy, you can also find the expression in virtue of, to express grounding links. In that case, the consequence was in virtue of the ground, right? Considering its explanatory nature almost always, not always but quite often, grounding is supposed to be irreflexive and anti-symmetric. By this, in particular, I mean nowhere reflexive and nowhere symmetric. So nothing is supposed to be a ground of itself. And from explanatory point of view would be circular explanation. And similarly, if A, if something is a ground of something B, then B is supposed not to be a ground of something A because again, it would be circular. Right, more in particular, as I said, we will consider the task of essentially formalizing grounding and in particular of studying what we often call logical grounding. Now, essentially logical grounding is the study of the formal features of grounding as restricted to a logical language, right? So we are studying the grounding connections between formally, right? So logical sentences, we can say, which essentially depend on the logical structure of the formula. So you could see it as a limitation of grounding or grounding which is proper to the logical context. Now, in general, this is not a definition but it's sort of a description of what usually people try to capture while working on logical grounding. We could say that the logical ground of a formula F is usually a multi-set of formally, G1GM. A multi-set, you could just see a multi-set as a list where the order doesn't matter. So it might have, it's not a set because you might have repetitions but the order of the formula is not important. Right, so the ground would be G1GM of formula F and in general, roughly the requirements are that F should be true in VR2 of the truth of G1GM in a sort of relevant way, so not just implication but something more specific which depends on the notion of grounding adopted. And moreover, usually there are some simplicity conditions. So the idea of explaining or giving reasons is also quite tightly related to the idea of simplifying, right, when I explain something I'm trying to, simplify the information for the, who receives the information. In a logical context, usually something like logical complexity is used but in general, a notion of complexity. And the ground is supposed to be simpler according to the chosen complexity than the consequence of it, right? This is roughly the idea. Now often in the context of logical grounding, inferential rules are used to characterize grounding relations. In what sense? In the sense that, so roughly, as I said, the idea is that the truth of G1GM should determine the truth of F. So at least should imply it, right? So it's very natural to see grounding links as characterized by inferential rules. And here, you have an example. The idea is simply usually an inferential rule, I don't know if you're familiar, some of you certainly are, but the premises imply the conclusion, right? In this case, we have a stricter rules with a more specific connection between the premises and the conclusion. And the idea is precisely that the premises of grounding rule should constitute a legitimate ground for the conclusion, right? So in the sense if you read the rule upside down, you have the conclusion is explained by the premises in a sense. If you stress the explanatory nature of grounding. But in general, technically, rules of this kind are used to characterize formally and capture different grounding notions. Now, I presented here this rule, both to give you just an idea of what we mean by grounding rule, and also to start introducing the elements of the problems that we are going to study. So this in particular is a grounding rule for an existential statement, right? As you can see, the conclusion is an existential sentence. So here we're saying there is an X such that F of X. So something which is F exists where F could be any property or complex formula. And this rule in particular comes from what is usually known as full grounding approach, which is, we could say sort of a standard way, standard approach to the formalization of grounding. And the idea behind this rule is that in order to give a ground to an existential statement is enough to provide a positive example, right? So the premise is F of T, which is just essentially a formula stating that some term T, which is a name for any object essentially, well for a specific object, just there is no restriction of the object you can use there. And the formula stating that the object named by T enjoys the property F, right? So if you give a positive example of something enjoying the property F, this according to the rule is a legitimate ground for the existential statement, right? So I would say this might be a personal view that already at an intuitive level, this corresponds to a quite loose or weak, we could say, notion of explanatory relation because essentially you are saying to explain why certain objects exist, it's enough just to provide one, any example, right? For example, if the existential statement is there are even numbers, you could just say because, I don't know, 82 is only the number, which is fine, certainly quite intuitive and nothing particularly wrong with it, but I would say as notion of explanation could be considered a bit of a weak one. But you don't need to trust your own intuition about this because actually it has been shown that this kind of approach creates quite serious technical problems already for quite simple contexts of application as I mentioned before. In particular, in 2010, fine showed that if we extend, calculate the fine on the basis of full grounding approach with theories which are quite naturally connected to grounding, for example, truth theories or theories about facts or similar, we can actually generate paroxys essentially related to grounding. I talk about natural extensions of the grounding calculator in the sense that the examples are not, I don't know, technical strange examples precisely aimed at creating problems that might happen sometimes, but are really theories that you would like to be able to combine with the grounding calculus. For example, if you formalize the notion of grounding is quite natural to want to study the formal relationship between grounding and truth in a formal sense because they are quite well connected notions and similarly for the notion of fact. So just to mean that these are problems which I think should be taken seriously. Now, let's see one of these paroxys. So this is a sort of simplification of one of the paroxys presented by fine and also other people who have some papers about this. So in this case, we just take, suppose we have a grounding calculus also containing the existential rule that we have just seen and suppose we extend it by truth theory, very simple one. So technically what's a truth theory? We just add an interpretive predicate T for true and the quotation mechanism, as you can see in the second line, such that we can take any formula F, quote it with these angle quotes and obtain a term essentially which constitutes a name for the formula in such a way that we can apply our predicate T to quotations of formulae and just express that the formula is true. So T of F just means F is true. Now, if we do this, we need essentially a grounding rule for the truth predicate because the idea in general is that you should have one grounding rule for every connective or every interpreted element in your language. So we also need something or our newly added truth predicate. And the obvious choice is the rule that you can see here. Here we're just saying that essentially in order to ground the sentence taking that F is true, you just use F, right? So in the natural language would be F is true because F, right? That's something quite interesting. But if we do this, we already have quite directly a serious problem. Indeed, so in particular we have this, I mean we have the existential rule I showed you and our truth theory with the grounding rule for the truth predicate. Why do we have a problem? Because if we consider the sentence, there is something which is true that you can see there, right? First of all, we can very clearly, you can see it on the left here. We can clearly use the sentence, there is something which is true to ground the sentence, it is true that there is something which is true because you can do it forever. But our sentence is also an existential sentence, so we can try and ground it. And in particular, our rule coming from the mainstream approach tells us that we can pick any instance of the sentence as long as it is true and use it as ground. So what we can do is quote the sentence itself and we obtain the term and then plug the term in the sentence to obtain an instance of the sentence. And we have that, right? So we ground there is something which is true by its particular instance constructed quoting itself which is it is true that there is something which is true. And here you can see that we have two grounding rules where the premise of one is the conclusion of the other and vice versa. So we have a cycle, right? So a symmetric instance of grounding. Moreover, if we are considering a transitive notion of grounding which sometimes is the case, we also have a reflexive instance actually too because both by transitivity both formally ground itself. So we could say that our thesis in this talk or at least the motivating idea is that this problem can be solved if we consider that they are generated by the fact that actually the full grounding approach presents in general, but in particular for the existential quantifier, rules which are too liberal. So our idea here is essentially to propose a different notion of grounding which is stricter and show that if we adopt this kind of notion these problems don't arise, right? So and try to obtain in particular for quantifiers stricter grounding rules where when we mean that the premise explains the conclusion we mean essentially a stronger notion of explanation. Very well. So in order to do this, I'll start by briefly motivating the additional requirements essentially that we have on the ground relation. Very, very roughly we could say that in the full grounding approach and this is quite clear from the existential rule, as far as true determination is concerned, the ground should be sufficient with respect to the conclusion, right? Should imply it. And then there are complexity constraints, but it's not completely accurate but roughly you can picture this. And now I would like to show you some remarks by Venerable Sano who's pictures and which are exactly those that we, which are interesting for us with respect to the ground five rules. But the notion of grounding I'll present is based essentially in general in its analysis. And so there are also other requirements and other parts which are interesting, but obviously for time reasons I'll just focus on it. The relevant one for us today. First of all, at some point, so what Sano discusses is notion of grounding that he calls up for, as I said, modernity is actually translated as grounding in his main philosophical work, which is the theory of science. And for him grounding is a kind of derivability relation but with trickter conditions essentially and very essentially related to explanatory and linearness of the notion of reason. So it makes sense actually to call it grounding and to connect it to modern ground. Now, a first remark I would like to discuss by Venerable Sano is the following. It's quite simple. At some point it says every distinct ground has a consequence which is at least in some parts distinctly itself, right? So essentially what Venerable Sano is saying here is that if we have two grounds G1 and G2 for the same consequence C, right? They should be the same ground, right? Because if we actually have distinct grounds also the consequence should be in some part distinctly. So what he's saying here is in a sense is proposing ground uniqueness constraint, right? So if you want to give the ground of something, the ground should be, I mean if you look for the ground of something, the ground should be unique. You'll be able to find one. In a sense he's proposing a notion of V reason why more than A reason why, right? Second remark which I find very interesting is the following with more complex but we'll try and clarify it. The remark is the following. Who does not feel that the connection between ground and consequence is much more intimate than it would be if the mere fact that some of the grounds and consequences are combined in thought were supposed to make only one ground and one consequence out of them. So what he's saying essentially is the following. If we have two legitimate grounding claims, right? G1 grounds C1 and G2 grounds C2 the mere operation of combining the two grounds for no reason just because we had, right? We consider them and put them together shouldn't construct a legitimate ground for the combination of the consequences. So what he's saying here I think is that in a sense the ground should globally be irrelevant with respect to the consequence, right? So we shouldn't have a relation that we can just combine for no reason. When we explain we shouldn't have that just part of the explanation explains part of the consequence and other parts are related to other parts but we should really have a sort of global relevance of the ground with respect to the consequence. And in particular if we have this and this is quite obvious also in the context of modern grounding, if we have that G is a ground of C we shouldn't be able to infer in general that G is also ground of C and B, right? This is quite obvious because in this case the problem would be even more serious that the one discussed as Bonsanna so we would have actually a part of the consequence which is not at all related to the ground. So I present this to present a sort of complete determination criteria. So in a sense we don't want a grounding relation in which it's possible to have just partial determination or just parts, determining parts but not a global relevance relation. Now, by following these two requirements and as I said many more in general by following Bonsanna's analysis of the grounding relation in 2016 and for the proof we are at that part in 2018 Francesca Potolese proposed formalized a notion of grounding, of logical grounding based on Bonsanna's analysis. Just to be clear, Bonsanna's analysis is not merely logical. So for him it's derivability relation but it can be applied to science, mathematics and actually moral issues. While what we are presenting is a formalization in the restricted context of logic of this kind of analysis. And the resulting notion as I said is stricter than let's say the standard way of formalizing grounding. Again, to give you an intuition if we consider full ground to the standard way of formalizing grounding as a sufficient condition this is more similar to a necessary and sufficient condition by following the complete determination requirement. In a sense, this is a bit vague but just to give you some guidelines. Now, a sort of semantic definition of complete grounding is the following. Now I won't be super precise because the precise part will be with theoretical so I use the inference rules but just to give you an idea that we can in a sense provide a semantic definition of complete grounding as well. And the three conditions are those that you can see here. First of all, notice that for complete grounding we also need a notion of condition or context. So we have gamma which is the ground, A is the consequence and we also have this notion of context or condition C later I think to be clearer why we need it. But so if we consider this the three requirements that we have for complete grounding are that the consequence should be derogable from the ground quite obviously so the ground should be a sufficient condition. Moreover, under our context, let's say or we could say all the rest being equal in a sense, if the ground is false by now let's just read the notation as it's false then let's play it. If the ground is false we can in our context we buy the falsity of the ground infer the falsity of the consequence. This is sort of a weakened necessity condition, right? So if we negate the ground we can apply the negation of the condition. And moreover we have a complexity requirement according to which our condition and the ground together are less G-complex than A. Now G-complexity is modified version of logical complexity so it's a syntactic notion of complexity which we could say, adapt logical complexity to the idea that grounding is about positive determination. So in a sense, not every connective increases the complexity but also positive connectives or pairs of negative connectives. Now if you want to know the detail I can be more precise but the idea is really that G-complexity is a syntactic complexity based on logical complexity but considering the positive nature of grounding, that's it. And say this, I can also explain the notation essentially when we write the small bottom to the right of a formula or a multi-set we call it the converse of the formula and it's always equivalent to the negation of the formula but is always as G-complex as the formula. So sometimes we add the negation sometimes we remove one negation let's say, just a technical way to represent the falsity of the formula by using the negation in, let's say, a suitable way. So these are our formal requirements for the complete grounding relation and now I'll show you the propositional part of the calculus. Essentially these rules exactly capture the kind of relation that corresponds to the three requirements. Now I want to discuss all the rules just a couple of examples to give you an idea. First of all we can consider the first rule on top left of the slide. This is the rule for conjunction, it's quite an obvious one actually it's the same for full grounding. Essentially what we're saying here, I recall that I remind you that the premises constitute the ground of the conclusion, just to be clear. So what we're saying here is essentially if we want to ground the conjunction the ground must contain both contents. This is quite obvious, right? If you want to apply the conjunction you need both and if the contents are false if you negate the contents you can apply the negation of them. For this junction we already have something a bit more interesting. First of all as you notice from the second rule here if both disjunctions of a disjunction are true both must be mentioned in the ground of the disjunction. Here we can really see the complete part the complete nature of the complete ground essentially it's requiring the complete account of the truth of the conclusion, right? So if both a and b are true the disjunction is true because both are true, right? And this technically comes from the necessity requirement because if you just put a in the ground the negation of a doesn't imply the negation of the disjunction just to give you the technical insight. But obviously a disjunction can be true also if only one count disjunct is true and for these cases you have the next two rules so suppose a is true, right? In that case it's reasonable to say that a is the only reason why the disjunction is true but for technical reasons we need to specify essentially that base force so that's why we have this notion of context when you read the rule essentially to the left of the bar you have the ground a and to the right of the context and intuitively you could read this rule as saying a is the complete ground of a or b considering that base force, right? So in a sense you are guaranteeing by the condition that a is the complete ground it's the only reason why this is not true then we have rules for implications which are modeled after disjunction because essentially classical setting a rule for double negation and then you could have like one general rule for negations it can be done it's a bit more complicated every year it's not so important so I just put explicitly a rule for negated for negated conjunction, disjunction and implications very well now right so in the calculus we also have a notion of ground theoretic equivalence which is similar to what some people call propositional equivalence or propositional identity not quite it depends on the notion we have a quite conservative notion of ground theoretic equivalence so in general ground theoretic equivalence it means that if two formally are ground theoretic equivalent they should be considered as the same ground so if you say that for example ground and consequence so our equivalent with respect to our notion of ground so if for example you can use B and A to ground something you can also use AMB there is no difference essentially under the same formally with respect to our relation this is important because grounding doesn't is not robust with respect to logical equivalence so you can have logically equivalent formally which are not equivalent with respect to grounding I'll also discuss this briefly so the equivalence we use are these as I said they are quite conservative not particularly strong we just have a commutativity of conjunction and disjunction and associativity of conjunction and disjunction but there is quite some literature then certainly these are widely accepted someone uses a bit more of these equivalences that's what we use technically it's not so essential for us right now but explicitly we can add rules to impose this in a procedural way which is not always the case but it can be useful for something I'll discuss in a little bit so essentially we can define rules such that you can transform formally in equivalent formally inside other forms so at any depth it's just a technical remark or there are other ways more possibly simpler ways to deal with this but it can be done explicitly now by now I just talked about the propositional part but I promise some work on quantifiers so let's move to the main topic I'm sorry how much? half an hour right so what about quantifiers now as I said the important constraints for us today are essentially complete determination and grounding and if we think of the existential quantifier in particular these are not very there is not really very intuitive to see how we can impose this to a sound rule for the existential quantifier because in logic usually existential quantifiers are treated are introduced by specific examples and it's not easy even thinking about it it's not easy to find a formula which is equivalent to an existential formula especially a simpler formula so what we would like to have is an instance of the existential formula which is equivalent to it because complete determination technically means that the formula implies the existential formula and the negation of the formula implies the negation in a sense we have double implication I would say if we think about it in nine ways not clear how to do it but that's what we wanted to do so we essentially did the literature and we found out that a quite interesting solution can be found in the war by Hilbert so Hilbert introduced in order to what is called the epsilon operator in order to essentially define the behavior of quantifiers both existential and universal by not using quantifiers so by using operators on terms so by using what are called ground formula formula without quantifiers to obtain something which behaves exactly as a quantifier and the epsilon symbol can be used exactly to do what we proposed, what we had in mind to do so what is the epsilon symbol or epsilon operator it's just an operator on formula such that if we take a formula f of x so a formula with a free variable x we can apply the epsilon symbol which is the epsilon to the formula in a sort of binder fashion and we can construct something which is a term epsilon x f of x so in a way similar to a quotation mechanism but not quite now epsilon x f of x is supposed to be a name for an indeterminate object enjoying the property of it so epsilon x f of x can be read as something which is x some object which is f sorry some object which is f it's being used also in philosophy of language it's quite interesting for pronouns and similar things but as I said it was introduced in a essentially mathematical context but I didn't mind to prove consistent results now what are the properties of the epsilon symbol but quite simply it's quite intuitive if you consider its interpretation essentially the property f applies to epsilon x f of x if and only if there are some object that enjoys it because you can show that some indeterminate object enjoying the property f actually enjoys the property f only if there exist objects enjoying the property f otherwise the name is just a name for something that don't exist you could say from a theoretical perspective we can introduce in this way the epsilon symbol we provide a specific example of something enjoying the property f we can introduce the epsilon symbol and we can conclude the sentence f holds for an indeterminate object enjoying the property f intuitively what you're doing here is providing a concrete example and then abstracting it rather than just giving the example you move to a sentence stating that there is a class or actually there are some indeterminate objects enjoying the property related to which you gave the example now how do we use this just another small remark before the main rule it's interesting in particular with respect to grounding epsilon xffx is indeterminate as I said but it's a fixed object so if you use it twice you refer always to the same object and that's how it's used also in philosophical language so if we have a formula like this this might also be clarified the f should be a big f not a small one if we have a formula like this we should read it as something indeterminate object enjoys the property g and it also enjoys the property h so it's always the same and this with respect to ground uniqueness is a quite interesting property again f here should be a big f property now so how do we use the epsilon symbol maybe it's already clear we use it to obtain our instance of the existential claim which is simpler logically simpler than the existential formula because it's just we remove the existential counterpart just instantiated and it's logically equivalent to the existential formula so the premise implies the conclusion and if we negate the premise we can imply the negation of the conclusion so how could we read this in a sense this rule is forcing you is doing something very similar to the full grounding rule but forces you not to commit to a specific example so if we go back to the example with even numbers you cannot anymore say there are even numbers because 82 is an even number or 16 is an even number what you should say something more in the direction of there are even numbers because some number is of the form n plus 7 I mean here the intuition is very vague because we are in pure logic so I'm not clear but essentially I would say the interesting intuition is that the rule forces you not to commit to a specific example but to provide sort of a blueprint of an example which represents the old class of objects you are talking about in a sense I would say very well now let's move to the other quantifiers the universal quantifier now in the context of full grounding usually rules similar to this one are used for grounding a universal claim so we have a claim of the form every object is f and the idea is essentially to ground this a universal claim to provide essentially an exhaustive list for every object of the domain guarantees that the object enjoys the property which would be the list fc1, fc2, fc3 and so on so the idea is I find a list of names c1, c2, c3 that name all objects in my domain and I prove for each of them that f holds for the object and so we have in general this list of formulae could be infinite because I might have infinite domains which is already it could be problematic if we want to stress very much the epistemic features or outcomes of ground if we are in a metaphysical context and we care about fundamentality in an ontological way this makes completely sense if you want to see grounding really as explanatory in a more epistemic sense this could be problematic because then your explanation would be an infinite list of statements but there is even more serious technical problem here the point is that if you didn't have the leftmost premise so if you only considered these premises here which are intuitive ones I would say then the premises wouldn't imply the conclusion because even if the list is exhaustive nothing guarantees that it is so if you don't know your domain and you consider purely the logical sum of the rules even though C1, C2, C3 are really all the objects in your domain you don't know you have no way to know that just from purely linguistic point of view so you have to add a premise guaranteeing that which is what is usually called the total defect which is this one which is telling you exactly for every object you take defined by at least one of my constants so if you have this and that then you can indeed imply the conclusion now what's the problem here? we have a complex issue because then we don't only have the infinite number of premises but we actually have an infinite premise so formula which is infinitely long and it's more of a universally quantified formula so you cannot even argue that we removed the universal quantifier I think it's tricky there are some solutions that I don't find too suspect for example someone says let's just consider the total defect as an atom as having complexity 0 that doesn't feel like a solution more just like ignoring the problem so and moreover also technically infinite are logics so logics where you have infinite formally might have very different properties with respect to normal logics and I mean it's not obvious how to treat them in some cases very well so what do we do? let's let's go back to the mathematical examples again for a moment suppose you want to prove that all triangles have the internal sum of the angles of any triangle is 180 degrees so what do you do? actually what you do is you consider a generic triangle suppose you have a triangle you don't care whether it's equilateral or whatever you just take it as a generic object so you have no assumptions on the triangle except the fact that it is a triangle and then you reason about it you do some computation and you end up showing the property holds and then you say well but I have no assumptions on the triangle so what I just did I could do for any triangle so essentially I showed that all triangles essentially we use this trick which is also very common in proof theory I'll say that later and the idea of using arbitrary objects for proving universal right? so first of all the solution what we are saying is essentially the ground of the universal sentence every object enjoy the property F is an arbitrary object enjoy the property or the arbitrary object A then is just the constant that we have in our language which is a name for an arbitrary object now find that a theory so calculate and semantics for arbitrary objects what it does is more general and more complicated than what we do we essentially took his idea simplified and took what we needed because we have a very limited use of what he presents and applied it here but the idea is exactly the same so let's not use proof theoretical conditions free variables and so on let's explicitly put in the language something which is supposed to be a name for an arbitrary object why do I mention proof theoretical conditions and variables because this trick let's say the generic triangle idea is often used in proof theory so if you want to show that universal claim exists what you do is provide the generic proof essentially the proof for an X on which you have no assumptions the problem is that we cannot do it here for grounding because then we would have a formula which such that in the formula with respect to the language nothing guarantees that the object is arbitrary the condition is on the proof what we need here if we want to define as a grounding as a relation between formally is a way to internalize this condition in the language so to explicitly have constants in the language which are supposed to be names for arbitrary objects and this is exactly what we do essentially right so this is our solution just a few words about arbitrary objects as I say right is essentially what you usually do in proof theory but internalize in the language the problem if you want to internalize the notion of arbitrary object in the language is that it's not consistently axiomatizable because you trivialize the domain because the property of an arbitrary object is that it enjoys all properties and only properties that all objects enjoy that's exactly what we want you could say intuitive notion of arbitrary object but if we just write an axiom saying this either you have only one object in the old domain or it's inconsistent because you can do the math or you can ask details later but you get it so as I said find proposed theory for that complete one with the possibility of defining arbitrary objects with a certain property as for example the arbitrary triangle or something like this we took just a fragment of it and proof theoretically is very easy to formalize that by essentially you impose proof theoretical condition on the free variables saying if I have generic proof with no assumption on x for any variable x of f of x the thing you can read here then you can introduce the statement that f holds for the arbitrary object so we mix the for an introduction that we use on that and then we have the ground improved at that point f of a essentially generic by meaning by semantics and then you can use it for the ground very well, now what does this mean with respect to the parmixes so if you consider the calculus I presented with the propositional part and these quantifier rules so using arbitrary objects for the universal grounds and the epsilon symbols or indeterminate objects for the existential ground very easily formally show that no loop can occur in any way so we never have a reflexive or symmetric instances not even if we consider a transitive notion essentially there's a very strict notion of grounding that always strictly simplifies so you really have no possibility of loop of any sort but this is for the logic which is not not so interesting because the problem was arising when we extended the calculus by the truth here but you can actually show so we didn't write it down I have to make a disclaimer but we know how to do it you can actually show that even if you extend the calculus with the truth theory you can still show that no loop arise so it's not as easy to show it as in the pure logic case but you can define a notion of complexity which is often used in truth theories essentially there you need to consider also quotations so you need the complexity of the formula and of the formula quoted inside but you can do it with the recursion theory recursion theory techniques which are often used in the context of truth theories and then essentially you can show that truth grounding rule and quantifier rules you still always have a notion of simplification just to give you the rough intuition of the thing if we use our rules you cannot any more use any positive instance of an existential statement to ground it so you cannot quote a formula and use that quotation to explain to ground the formula itself which was the direct loop that we could create we do something similar with the epsilon symbol but the epsilon symbol essentially is sort of a quotation of a simpler formula so you still have some more complexity because you can quote formula but when you use quotations in explanations you always use quotations of simpler formula inside simpler formula I don't know if it's clear it's a bit not so easy to explain as well but you can show it essentially you need a bit more complex notion of complexity than it's simply logical complexity but in the loops don't arise ok now do I have some time yeah but less than 10 minutes ok very good now very briefly just to so that the conclusions make a bit more sense we can also explicit because often in logical grounding you can explicitly have grounding as a relation to or as an operator between formulae here we presented rules essentially meta-liquistic rules where grounding is represented as an inference step but we can also internalize grounding itself in the language essentially you can just say immediately after applying the grounding rule you can introduce an operator encoding the old rule right so if gamma gamma modulo c considering c is the ground of b you can introduce the sentence now this is the formula of the language gamma is the ground of b considering the conditions thing gamma would be the grounding operator it's quite quite easy and you can also introduce rules to eliminate the connective and really treat grounding as another connective in particular what we use usually is a factivity so when you if you have that something is a ground or something else both things are supposed to be true right because you explain by using true things at least in the traditional view then you can have more so we have factivity essentially if you explain something by a ground you can infer from that that the ground is true that the condition is true and that the concept is true and moreover we have a rule for essentially be able to derive the negation of illegal grounds so if you suppose that for example a is a ground of a you can infer an absurdity and then therefore prove that a is never ground of a for example in other case it is so it's not for that and the idea is simply that if you have a grounding claim corresponding to a legal rule you can just infer the negation this is one way of doing it you could have different ones but essentially this gives you a way to right we can construct grounding derivations use grounding steps inside logical derivations and then we can introduce right reflected the language what we have done by an explicit grounding operation now I have a couple of examples just in case for example here we have right we start by suppose we have actions for identity we start by b equal b and suppose we can prove that b and c are different just starting points we can for example ground the disjunction right just like that and then we can introduce the epsilon symbol for example by abstracting b right so we we had a sentence about b now we are abstracting over b and the same there is some object which is either equal to b or to c and or it is or maybe I got something no no no sorry no you are abstract over the old sentence I was reading it wrong essentially the epsilon symbol holds all these parts right so you have an indeterminate object which is either equal to b or to c and it is equal to b or it is equal to c not very intuitive just to show how our rules are applied we can ground the existential quantifier and now we can introduce explicitly the the grounding operator right and here you have your ground which would be this one and here you have your consequence the statement essentially says the the sentence with the epsilon symbol grounds the existential just to show you that we can write combine logical rules grounding rules and so we also have the notion of immediate grounding because the our notion of grounding is immediate so it's sort of the smallest explanatory step in a sense but by the way we formalize it we can also construct complex grounding relations and the notion of immediate grounding here you have another example for the universal statement but yeah I guess I'll just go to the related and future work now first of all if we consider the propositional fragment that I showed to you we can define a calculus which is not exactly the one I showed but exactly equivalent to that one in particular in which we make everything explicit for example I showed you explicit rule for the grounding equivalence ground theoretic equivalences and with that same idea we can explicit it many things from the calculus that I showed you and it's possible to show that this calculus normalizes which means essentially that the grounding rules can be used as logical introduction rules and they are let's say balanced with respect to elimination rules or you could say that the you could use explanatory rules and you have the same logical properties as with regular logical rules which I think is interesting because it gives you a notion of it gives you a framework in which grounding derivations are really a subset, a non-redundant subset of logical proofs so we are not adding rules which are redundant just to be able to construct also explanations but we have a notion of logical reasoning such that some of the proofs that we need to have if we want to complete a set of logical proofs are explanations so we can extract explanations from programming derivations to be precise and by saying this I covered the first two points moreover some work in progress almost finished we by employing the ideas that I showed in the last slides the introduction of the grounding operator by these rules just applied just below grounding rules we can actually study sort of the behavior of grounding operators independently from the particular notion of grounding that you have so essentially you can forget about the specific grounding relation and just study the features that all grounding operators that can be defined in a certain way share and these also points at another I think very interesting feature of grounding which is outside the study of grounding itself but I think grounding is a very interesting let's say notion with respect to logic because it's one of the few hyper intentional operators so I already mentioned this grounding essentially violates logical equivalence so you might have logic equivalent formally which are not equivalent with respect to grounding which is a very interesting feature that many operators of this kind have been studying grounding is one so I would say that very interesting future work direction would be to try and generalize the results obtained for grounding also for other hyper intentional operators or even just with respect to hyper intentional operators in general so try and see whether there are common features from proof theoretical perspective that can be kind of lifted from grounding too this was all about just one thing that is interesting about this kind of operators like the optimal operator which is interesting the context of grounding is that this current forming operators at least in the level of the syntax when you're defining the operator you always have to define by mutual recursion so when you're defining the terms you refer to the formulas right you mean the epsilon operator so there is this at the level of the syntax there is the mutual recursion so one can always at least make the observation that there is kind of the symmetrical grounding definition at this level which is something that you don't seem to want in your theory of grounding what sense in the sense that there is a symmetry in the sense that when you're defining the terms you refer to the formulas when you're defining the formulas you refer to the terms so there is like a kind of circularity there but you always refer to a simpler formula the actual problem with symmetry is when you can in a way point to the formula itself which you can do with the notation mechanism for example right the epsilon symbol gives you a notion of simplification because it kind of right you can only refer at least how you use it in the rule you always refer to a simpler formula I'm just referring to the syntax when you're defining the language not the rules but the language and then necessarily the mutual recursion when you're recursively defining the language but I would say that kind of symmetry doesn't reflect on the grounding relation right I mean you can show that anyway you have a notion of simplification that's the thing I was trying to I mean essentially you show that there are no loops in the relation so you as a relation anti-symmetric or irreflexible I don't know if you were not talking about thank you for the very clear talk especially but besides the appeal to Balzano authority why should we choose this restriction can you tell us in particular because it works why should I use this one well what other do you have in mind other ones but imagine Balzano had something in mind so maybe it's I would say to me it makes absolutely sense what Balzano is saying you could also argue that there are different notions of ground like looser or stronger that's all right I would say if you want to restrict the standard one to me not necessarily the standard one but clearly full grounding as features that that are quite natural and intuitive for grounding so it makes sense to consider those as kind of the minimum set and if you want to restrict that so impose requirements on that I would say there isn't much you can do right possibly can formalize it in different ways I'm not sure about that because I don't think it's so easy if I understood directly the idea behind Balzano according to what you said something about completeness complete determination something about completeness seems to be the the core of the idea and what would be the justification to decide it works no the idea is really to absorb to quite well Balzano wasn't moving but I mean if you consider moving from full grounding to this notion it's essentially not considering the notion of a reason but of the reason that's I think is the most intuitive reading of the issue and I think in the logical sense in the logical setting it makes a lot of sense I mean in general you could want obviously it makes sense to study also the notion of a reason I'm not saying it doesn't make any sense but in the logical setting it's very close to becoming almost straighter many rules are just logical rules and indeed there is people criticizing often people not often but it says that when studying logical grounding it seems we are just studying logical entanglement and sometimes it goes very it goes very close to something like that so I would say in the logical setting then considering the notion of the reason actually gives us something and I'm trying to understand that it's their reason with certain conditions yes exactly so it's a company reason that's the notion yes exactly so thanks a lot for a great talk and a great paper so I'm a big fan of the full grounding approach so I'm not that I'm an interesting opponent I guess the issue is that one of the issues I see in this notion of complete notion is that the weight or the importance of the simplicity requirement is very present and it seems to be a little bit arbitrary there seems to be something non-metaphysical about it like it's a logician's decision that this formula is more simple than this one and why I think it's more of an issue for complete grounding theorist and for the full one is that there is some symmetry appearance in the complete one if it wasn't for simplicity reasons yes the grounding relation could go either way yes so the simplicity is really core and if you define it syntactically that seems to be sort of a purely linguistic convention or something about metaphysics about the world and also in these examples I see the intuition and that's not a problem but to some extent the instantiation with the epsilon operator you could also say that that's more complex than just the extension quantifier because it will be a longer formula just more symbols but just I'll let you finish so if we go back to the rule for instantiation or for the introduction of the extension quantifier I mean there's of course you can define a sense in which the thing above the line and then the thing below it but you could also define a complexity measure where it's the other way around maybe just counting the symbols should I answer or should I wait for it you know if you got if my question is cleared yeah yeah but I mean a term would be much more complex than the formula I mean you could have a thousand function applications so the idea is just also in the full running approach the idea is just that you remove the quantifier and you take an instance but also there the actual example could be very complicated you could have a huge term with so I don't know I mean here is really just logical complexity simplification so I mean I agree on the rest of the logical complexity is essential for grounding because for complete grounding the idea exactly is exactly that the ground should be equivalent yes logically equivalent to the consequence yes and the only asymmetry should be in the simplification that's really already both sound very explicitly so that's exactly I think also I was told this is something like this it should really be right complete relief essentially in a classical setting at least from a logical perspective and the simplification is really about well in this context syntax in a more general ok so this is the answer with the so I agree completely that the logical complexity is key and in these cases the syntactic complexity yes and it's all that essentially breaks the symmetry and it's really supposed to be like that you can like it or not but yes that's an intention even I would say one possible answer to what you said concerning the metaphysical aspects for example that for example I completely agree with you and I would say I mean I because possibly other people working on complete grounding see it in a different way but for me for example complete grounding should really be the choice going towards a more epistemic reading of grounding so still in an objective sense and I mean because we are in a formal context in a sense so you could be objective without the relative weight in a formal context but with a lot of weight on the epistemic nature and indeed what you were saying about the simplification which could be a problem because if you take this kind of grounding as a metaphysical one you exclude a lot of things you would like to say for example with respect to items and similar things but so as I presented logical grounding and for me that is very important this is really to me at least personally a study of grounding as restricted to those cases which only concern the logical structure I see and indeed and that's also part of the project Francesca and I are conducting there is also an idea of going to theories for example in mathematics we don't think so much about metaphysics but that could be as well in one direction of mathematics or scientific theories any formal theories let's say and then really you would have to consider other kind of connections which is also what Balsam does so to me this is really just the fragment about the logical structure but I wouldn't I personally would never argue that what is not captured by this shouldn't be a ground of something else if we talk about connectives and logic, yes but I completely agree that we might have atomic propositions grounding other propositions I would just say that's not logical grounding that would be you were just talking about logical grounding here I really because I also I tried to say it a bit but it's not obvious how to do that there are very different things people do and call logical grounding to me it's really those connections that are purely due to the logical structure and then I agree that something else must be done if we consider more extensive and and then for example what Balsam does and actually something me and Francesca and I are trying to do for mathematics and geometry is consider more general context in which essentially these connections are kind of preserved a sort of conservative extension you need to take it in a bit of this way but where instead of having the notion of a syntactic complexity or complexity notion which is syntactical you have a notion of conceptual complexity and that's really sort of a symmetrical complexity it could be so I agree this is limited to you right? I completely agree and as I took it more like when Kit Fai says the logic of impure ground which also has such rules right but it's not logical grounding it is features of ground in general that are logical because the logic of impure ground is an extension right so I would say as this kind of analysis to a logic of grounding in general what we have is just this part right so it's not comparable so it keeps together I would say at least it keeps together a formal system to describe the general features of grounding in general to the behavior relating logic and grounding essentially what we have is at least to me I want to stress because I don't want to talk about other people saying this already is in this sense what we are here to me is really just the links which are exclusively due to the logical strategy I had some other questions I have some questions you didn't ask any of you can I have a couple of questions just a clarificatory one is about activity so activity was introduced at the very end when you want to eliminate the grounding so in the of the language and then you can eliminate it by activity however I was wondering whether this is already at the beginning because otherwise it's a bit strange so you have these rules you go from classic A B grounds A and B and we didn't say anything about whether A and B have to be true and then you introduce grounding and then we have magically A and B factivity yes so essentially actually I would say that this is one because this calculus is kind of has been developed by changing a bit the original one by Pozolesi and this is one of the main differences I would say so essentially what you have is what some people call potential explanations obviously you right because anyway you have formula so you don't you cannot pick true formula you have to suppose they are true this is quite obvious as a background but I would say that the proof theoretical behavior of the calculus works perfectly without too many strange things with respect to the traditional way proof theoretical rules work in the sense that so when you first of all in these rules both premises must be derived or assumed right so we don't discharge any impotences so what you have essentially is that you can apply a rule for grounding only if either you have proved that the formula is true or you have assumed that the formula is true or you have derived from assumptions that formula is true so essentially suppose you just apply one rule then what you have is something saying supposing that they is true and B is true they explain A and B and what you can do for example is for example when discharging comes into play is for example when you introduce the connective right you could say for example you apply the true then you derive that A comma B grounds A and B and then you can for example introduce an implication and say if A is true and B is true then A comma B grounds A and B so I would say that the behavior is exactly what you are expecting the fact that you can derive by this activity rule matches this perfectly because you can do it only if possibly you have open assumptions so you actually have assumed A and B to be true or if you have discharged them in some way it could be a logical theorem in that case you can even prove it right so yes I didn't mention but we thought about this and I think it should give does give what you would expect have a kind of follow up on that so in the rules for the factivity yes there was one that you can also get the condition from there that's really surprising to me like if you say that given the context C I can grounds A on gamma can we then conclude that the context is true or something like that? yes no because it's conditional like if the context is there yeah no no but how we use I mean that's my intuition and I think it kind of matches with this as well but our use of the context is really for example if we look at this junction is really saying A fully explains C considering that B is false so while you say that you actually are saying that B is false I mean the context intuition is just one way of reading it but when we use the context the context is really the actual context it's not like we're going to a context no no no it's just to say that the context is not part of the explanation it's just something we need to add to make the explanation an explanation in a sense sorry I moved from explanation to grounding some people might complain when I say explanation I mean grounding here so for some it's not the same thing actually but no, our context is really actual the actual is just the information from the rest of the truth yes yeah I will have another question we're really getting black switch so I don't know is that something no I was wondering if I don't know the literature on this so it seems very clear that in the case of introducing logical connectives like injunction is true that we go from something more simple to something more complex that there seems to be a disanalogy in the case of the quantifiers and this sense of through and through all doesn't seem at least as straightforward sometimes it seems more simple outside logic it's true that we have this more simple but not in the case of the universal but we want to have the same phenomenon that we have with the connectives in the case of the quantifiers but sometimes it's more simple in natural language I mean to say that something is true for every object and this explains why this particular object and this particular object can be more complex so this through and through all it seems to that there is a disanalogy between and whether this is something that is considered in the literature that there is this disanalogy no I mean I don't know you mean in general possibly the intuition of about really kind of parts whole in a sense relation that's probably closer to the full grounding approach for the universal quantifier at least and I agree but it doesn't I mean I don't think that's good you might have the intuition possibly metaphysically it works it's true that in our case the virtue of reading loses a bit to grasp for the quantifiers but I still don't think the other is better possibly there is really kind of diverging intuitions concerning quantifiers there is literature for example defending the or at least discussing the fact that often we would like to have general facts grounding specific facts that could be the case for example for what some generality is a big feature of explanations for example and the literature explanation in general generality is very important so you might have that it's true I think this is a hard issue to set because there are indeed conflicting intuitions but I would say that if we restrict ourselves to a logical sense of the thing it does make sense to say that the universal statement runs in the specific ones while in other contexts might for constructing ideological formulas yes but maybe not for the genuine meaning of for all of X yeah I don't know I really think that then it's an issue for which it makes sense to make clear what you're what do you mean by grounding in that context because it seems that there are really different intuitions all equally valuable because we have nothing so the discussion you just had is difficult because it's typical of people writing about groundings that the model you have in mind of explanation is closer to explanation and after that you extend out the grounding and you look for the same feature simplification but what you said is that the general case explained all the instance the law explained all the instance it's another way to understand explanation that is less interesting today so it's not just a question of intuition I think it's a question of what is the thing you want to model so my question when you said you want to read that as an epistemological U.N. system that will be your discussion and after that you went away from put more weight I would say just to be put more weight on the epistemic micro-products and I can understand that but you must have a notion in mind of what you want to map with formalism you don't do formalism just do formalism I'm sorry you want to you don't do formalism just do formalism so we have an idea to map some idea of grounding explanation that's why you switch to the other so it's not a question of intuition it's something in the world in the epistemic practice that you want to map what no not in the epistemic practice no I would just say that so grounding is supposed to have epistemic micro-products right so it's explanatory it's objective but explanatory and if you stress that and if you take for example grounding already just fixing that it's between truths it already contains quite clear choice right so in that sense I would say in this book but also the other thing I would say I think is about intuition in the sense that if you consider why people say that the general law explains the specific instances I think that's due to semantic issues I trust you no no no I mean it's not clear I'm not giving it just as a solution but um it's not about the form right in that case it's more about um yeah I know I would not say it's probably because I don't believe in the distinction between syntax and semantic so it's why I think I think every syntax it's just semantic described under the rug you're not just putting symbols in an order you start some you apply some based on a certain conception or a certain model in your mind what do you call intuition no I guess what I would like to distinguish is when because when you if you look at an explanation method now I'm just talking about the final goal this is sort of a dream but when you look at explanations in the sciences in formal context where you give meaning to the names and everything you do have sort of a syntactic simplification some steps which are purely due to the form of sentences just like in a normal proof even non-explanatory and some parts which are actually due to the meaning of terms right so you say that zero is an even number sorry zero is a natural number not because of the form of the words but because of the meaning of the terms but at the same time if you say that A implies B and A is true that step is due to the form of sentences it makes sense to try and and treat the two problems kind of maybe not separately but to first settle how you want to deal with the purely syntactical part and then see how whether you can generalize it to to a sentence because it's harder essentially right to treat this in an article so I would in that sense when you say the general sentence or the general formula explain the general formula it's a case of semantic in that sense and I'm not saying it's obvious how to combine it I just want to understand your basis yeah yeah but I would say and indeed I mean it seems hard because sometimes simpler things are more complex semantically and then you want to it's not easier but I think the divide in the problem could be so I wonder whether you have looked at applications of this for essence for sure essence nature actually I think it might be a good solution to a problem that Kit Fien talks about in Guide to Grounds where he tries to not reduce essence to ground but say to what extent we can learn something about essence of facts or essence of things via their grounds and he uses full ground the whole time and it doesn't work and he has to look for an ad hoc solution for it but actually for complete ground as you're doing it with the Epsilon operator the problems seem to go away now maybe you can quote so the nature of the facts it seems to lie in the nature you could think that it lies in the nature of the facts that it is grounded in such and such way if it is indeed grounded that says something about the nature the grounds it has then he says unfortunately this view will not quite do as it stands the fact that someone is a philosopher is grounded that Socrates is a philosopher and perhaps also that he exists and is a person but it does not lie in the nature of the fact that someone is a philosopher that the fact is so grounded given that Socrates is indeed a philosopher so it's not in the nature of someone being a philosopher that it is grounded by Socrates it could also be grounded in Plato it doesn't seem to be essential to the fact that someone is a philosopher that the particular example he uses is grounded and for complete ground especially with the essence of the Epsilon operator the whole problem goes away because you don't commit to the specifics yes I think this is worth looking at I didn't remember the particular it's in quite the ground section 11 essence and grounds I think in general I mean saying that just an example is a ground I don't think it's a problem it's a matter of essence yeah it's already a second question of someone being a philosopher that this is grounded by Socrates the grounding itself is not a problem for me at least no thank you very much for that yes it's also a non-technical question also not a logician I wanted to ask about this Epsilon operator so I think it's really cool it seems extremely useful for one thing I mean I'm thinking back to so I'm not a logician but I did use to teach introduction to logic and it seems to be a lovely way to capture a pattern of reasoning that I think we probably all know is really natural for our students and very hard to convince them not to do that is to say when you introduce a name that's very arbitrary that you get to keep it for a while and use it to do stuff it's a very natural way to think and we really have to like beat this out of our students that when you're trying to do existential introduction you don't get to keep your name around you have to be very careful with what you do with it I'm wondering I guess what I want to ask is it feels like too good of a trick so is there a catch? are the semantics for this operator challenging? is there something about this operator like it seems too good to be true to just be able to introduce this epsilon operator that solves lots of these really challenging problems I would say use the choice function essentially I don't know if that's problematic for you sure of course it is I mean to be it's expressive VTYs it's quite a lot of machinery because you could actually eliminate all of them you could have a logic without quantifiers actually even universal quantifiers just using the epsilon symbol it's really a way to do all the work that quantifiers do just by means, by terms sure it's very expressive but no I would say it's quite I mean the semantics can be defined it's used and it's very powerful yes we actually use just part of it also just by the way as a remark it wouldn't work very nicely for the universal quantifiers we could have technically but then when you use it because essentially the epsilon gives you an example and you can use it showing that there is no counter example essentially to characterize the universal quantifier but showing that there is no counter example is insert in a sense non-constructive so it's using essentially non-configuration or classical and for example it's quite common it's quite clear about the fact that explanations shouldn't be by absurdity so it felt quite wrong just to give a bit of context it is very expressive yes it's but no technically it's not as problematic as the infinite logic for example I don't know it's quite interesting possibly the intuition is a bit right but technically I think it's a very neat solution too neat very cool we have a bit more at the end so this is a bit further away from your talk but it's about the paradoxical thing you started from how should I say this more and more I get convinced that the only argument saying that grounds cannot be symmetric it's kind of a begging the question argument it's like a dogma of course karate well I think there is no good reason to impose this if you have I don't know whether you read this paper by Scott Dixon on the well-foundedness of grounding he gives like a couple of several different ways to characterize the well-foundedness of grounding in a mathematical sense which he says is too restrictive and I agree with that but the thing that seems to be quite right and I agree with that is that for full grounding at least it's sufficient that every fact has is fully grounded by fundamental facts or ungrounded facts so this allows infinite cycles sorry not cycles infinite chains as long as there is something below it that is more fundamental still so you can go 0-1-2 minus 3 minus foaming up for example you can have a cycle but you could also choose to go below well cycles that's my own thing but there is he also rejects and this rejects symmetry so he's not well founded so there is not well founded so there is not well founded in the technical sense of the word where we have infinite chains are allowed as long as there is still something firm below sort of like a limit yes and you can go on this close the set in a sense you can have infinitely elements but there is there is something on it that's a minimal requirement on grounding that there should for every fact be like a bottom even though that can be there is minimum but not necessarily discrete but once you have this requirement you don't need anymore for asymmetry because it's not a problem that you are at some point in a loop if there is a way into the loop that fixes your true values from problematically also the example you gave if you assume that there is some true proposition like 2 is equal to 2 or something then there is another proposition and then you get into the loop that this is self grounded sort of a truth teller situation where you don't know but if you fix it you are fine yes if one of the things in the loop is grounded itself the things in the serve also become all determined yes so I believe that these kind of cases are seen like paradoxes but if you buy that there can be self reference like if you which is usually assumed in theories of truth yes like you can do grippy analysis analysis for example and then you can determine you can determine perfectly true values for these seemingly paradoxical cases so you can then set up grounding rules that determine these true values even though obviously there is self reference so there are there are cycles so I think the argument against full grounding is perfectly acceptable that you gave if you also accept as a dogma anti-symmetry I would like a challenge yeah no no no anti-symmetry and then I was wondering whether your argument still holds for you if you think it's well I mean if you remove anti-symmetry clearly it doesn't I mean then it's alright I would say that if you have in mind grounding in a critical sense sure like truth determination in the sense of having that truth is determined but if you essentially want to capture a notion of clarification or right display reasons then the loop is a problem because then you you say well just because because A why because B why because A I mean well that seems to be a problem there is a way into there the reason gets A is a reason for B and B is a reason for A but C is also a reason for A and therefore also of B and this can then ground over stuff I mean it's circular but it's not viciously circular in the sense of having truth which is determined I see absolutely is not right but if you were about clarifying in a more epistemic sense I would say it is a problem because you just go I don't know so it clarifies an aspect but not everything but it only this or inter-mystic for the complete one you have this is not possible of course but for full grounding it seems that you can always have multiple ways in which is ground and then you can have an entrance that is not circular into a circle and then you just and that's not clarifying but I see that you're weird but I agree that essentially I'm just saying I want anti-symmetry it's not an argument in favor it's just essentially requiring it by much of it it's very interesting thanks for the a very short question at the very end the future work and what has been already completed I was wondering I'm very interested in this connection between grounding and the rules all my calculus I said complete it I'm wondering if it's published or if it's something that if it's something that you have already published or I yes so first part of the work yes it's essentially the normalization part essentially I just published five days ago or something you didn't really so essentially there I study I just remove the introduction rules and use grounding rules instead so grounding and elimination rules and I prove the normalization of the calculus and then in another paper because otherwise it was too long there is also the analysis of what that means with respect to classical logic and intuitionistic logic so what kind of try to see what's the real difference between logical rules and grounding rules that's not yet published but if you're interested I can also find the other okay okay four minutes we can also