 My pleasure to introduce Sara Alhá from the University of Mujo. She's working on topics that are very relevant in her institute work project. So for instance, the topic that she's going to talk today, she's working on uniqueness of logical connectives and bilateralism of other things. And today, she's going to talk about how to secure uniqueness of logical connectives in multilateralism, multilateralism. So, thank you very much. You have to tell me where I should stand. You're good, you're good, yeah, you're fine. You can walk around. Yes, okay. Thank you very much. Yeah, to your group for inviting me here. I was very glad, this invitation, especially because you're still doing this in person. It's nice. Yeah, as you can see, this will be a topic of my talk. I will also explain this a bit more in detail, but just in general, so background question here is about the uniqueness of connectives, which is the question whether a connective is characterized by the rules governing its use in a way that there's at most one connective than playing this specific inferential role. I will explain this in more detail later. And the problem I want to talk about here is that in a bilateral proof system, the usual notion that's usually considered for uniqueness does not suffice. So, this is the plan for talk. I want to present and present you a bilateral sequence calculus system for bi-intrusionistic logic to end and then discuss some general settings in which uniqueness can fail, then show the specific problem of bilateralism and how they relate to the uniqueness-related problems that have been detected so far. And I want to propose a solution on how we can still guarantee uniqueness in bilateralism, show that these considerations could be a guidance on questions about representations of proof systems, there might be this or that way, and you could argue that this way is a better way of representing a proof system than another. And really in the end, just hint at possible extensions of these results to multilateral proof systems. It sounds like a lot, but I was told that I have an hour or so, but it's also not so. We'll see. Okay. Background, bilateralism and proof-theatrics semantics. Now I'm kind of glad that I have these slides because at first when I prepared the talk, I thought it was really only people who worked with bilateralism, and now I was told, no, the audience is a bit more, like, broader, so this may be good. So these are some definitions of bilateralism in the literature. It all started, you know, started, but this specific term started with Ian Rumpfitz, yes and no paper. They say that affirmative answers, however, enjoy no priority over negative ones. The acts of answering a propositional question affirmatively and of answering it negatively, the acts of accepting its content and of rejecting it are conceived beyond our force. I shall call this view of how sentences might have specified the bilateral conception in contrasting to the unilateral conception. And for example, in a paper by Mr. Francis says, bilateralism is an approach to meaning taking denial as a primitive attitude on parable assertion. Or then this comes from a bit different around bilateralism when Dave Ripley saying that bilateralism, a particular form of inferentialism defended in Rumpfitz and Rastall is the view that which inferences are valid is served to be explained in terms of conditions for assertion and denial. Or bilateralism holds that we must consider conditions governing the speech acts of assertion and denial. For bilateralism to genuinely be by, it must hold that denial conditions cannot themselves be understood as deriving only from assertion conditions. Our needs curb us, bilateralism which proposes a wholesale revision of what it is that is assumed and manipulated by rules of inferences and deductions, rules applied to speech acts, assertions and denigres rather than propositions. If you have an idea what is conceived of by bilateralism and what most of these definitions have in common. So there's a reference to speech acts of assertion and denial and it's emphasized that these notions have kind of symmetry, they're both primitive, they're not reusable to each other, they're all in power. And another point that is often mentioned in this context is that in bilateralism denial are considered as conceptually prior to negation. So denial of formula is not interpreted in terms of, or is not interpreted as the assertion of the negation of this formula, but rather the other way around. Okay. And Ripley then distinguishes two kinds of bilateralism, I find this quite helpful because there is definitely a difference in the literature and he really made this distinction clear for the first time I think to say okay there's, on the one hand there's this what he calls warrant-based bilateralism which takes relevant conditions to be the ones under which propositions can be warrantedly asserted or denied. And this then motivated this, what Ramford did, a proof system of natural reduction with assigned formulas for assertion in 49. And Ramford these papers were by price and spiling rejection. Okay, and the other form would be the coherent space bilateralism which takes the relevant conditions to be conditions under which collections of propositions can be coherently asserted or denied together. And this motivated a specific kind of reading of the secret calculus. And I'm actually not really sure I know that you work with bilateralism I'm not really sure in which direction your bilateralism will go. Maybe later. So this reading of the secret calculus then would be to read to say the left part cannot be asserted while the right part of the secret sign is denied together. So this is why these collections are together then. Okay. And now I told you about these conceptions of bilateralism just to tell you that our notion and we will say our maybe the both of notion because it's it's done with my PhD supervisor Henrik Manzig I would say have working in the same direction so that's our so we do not consider bilateralism to be about speech acts because the context of bilateralism is of course proof theoretic semantics so saying that the meaning of the connective is given by the rules they use so we're considering this in this proof context not to be about speech acts per se but more to be about other notions which should be considered on a par like proof and refutation refutability or verification falsification and if you do this then you can see proof theoretic semantics as a study of proofs but also of something that Henrik calls dual proofs it's meant by this but the thing is then you get a duality between different inferential relationships so the thought is to implement bilateralism deeper than on the level of formulas so you don't have side formulas but to have bilateralism on the derivability relations in a proof system and then you get two notions of logical consequence a usual consequence relation capturing notion of verification from premises to conclusion and then do a counterpart capturing notion of falsification from premises to conclusion this is a device in one thing's natural reduction system for a bi-intrusion to stick logic to it and this natural reduction system comprises introduction and elimination with both for proofs and for dual proofs show you here an example so the upper line is I guess familiar for everyone it's just the usual natural reduction used for conjunction and here we have single lines for proofs and the other part these would be the dual proof rules for conjunction and we use double lines for these and as you can see these look very much like the ones that you know for disjunction so this is basically the way this proof system was captured by taking the dual connective of conjunction with disjunction and just replace single by double lines and then you get the dual proof for conjunction and you read this just as saying from a refutation of A you get a refutation of A and B or same for B of course and the other thing is that you have two parts of assumptions namely assumptions take to be true which is gamma here and delta is supposed to be the counter assumptions take to be false and single square brackets for discharge and double square brackets for discharge of counter assumptions the proof rules was distributed to you so if you want to take a closer look at this I tried to come up with an equivalent sequence parallel system for this natural reduction system and this can be shown to be cut free so for the natural reduction packages I think Van Zing showed normal form theorem so cut elimination here okay this would be a slightly stronger result even and here we have two consequence relations denoted by this sequence of signs with plus and minus and yeah sequence of this form where you also have the two parts of the assumptions on the left side of the sequence and then a sequence that is a sign with plus or minus and these would be for example the rules for conjunction again the upper line is basically what you are familiar with and the lower line would be the dual proof rules in sequence calculus the thing is now here within the right introduction rules we have to distinguish between different derivability relations of verification and falsification within the left introduction rules this is not necessary but we have to distinguish here whether a parameter is introduced into the assumptions or to the counter assumptions that's why we said these superscripts a and c yes the following structure rules can show to be admissible in the sequence calculus of course here we need then two parts we need two weaklings we need two contractions, we need two cuts because for example you can weaken into the assumptions and into the counter assumptions and the same for contraction and so on and these are the dual proof rules in the sequence calculus we have also two two reflexive rules because we can say from the assumption we can verify from the assumption that p we can verify that p from the counter assumption p we can falsify that p so basically we just double the mod of everything and these would be the rules for this junction and as you can also see here the lower line are the dual proof rules and they are again basically the dual so like the mirror of the conjunction of the positive conjunction rules so here you say if you have a falsified that a falsified that b then you can falsify that a or b is the case and here we have the implication rules now a natural deduction system and this one thing I told you for how how do you come up with the dual proof rules for for the conjunction where you take the ones from this structure and you do dualize them but for implication we usually don't have a dual connected here we do I will show you in a minute but still okay what does it mean to falsify it so to introduce the refutation of a implies b what would you say you have a proof of a and refutation of b and if you have a refutation of a implies b then you can prove from the a or you can refute b these are the rules in secret calculus they are also just not the same very interesting now a quick detail about co-implication I just told you here we have a dual connective that's why it's a bi-intuitionistic logic we call this collective co-implication in the literature it also goes by names like subtract and the other ones I forgot we will think of it later and sometimes also different signs I use so it's a bit of a mess in the literature but it is okay it's also a dual connective to implication and the thought is that this is desirable to have in a bilateral system why I will show you what the rules are so that you know what's about so again these are just the rules for implication and then you can see how they just lead to the dualized rules so to if you want to introduce we really did the other way around so b coincides with a you have to have a proof of a refutation of b so basically what you had to refute the implication and from b from a proof of e plus a you can get a proof of a or also a refutation of b and these are the rules for single calculus but I think data that would be more interesting things to say so why is it why is it desirable in a bilateral system to have this because both you can say that both implication and implication can be understood to express then certain concepts of entailment in the object language and if we take falsification just as important verification then this might be desirable because in our normal case we have that we have this implication to capture this entailment relationship we have a derivation of a and plus b if and only if there is a derivation from a to b and here we have the same thing we have a dual derivation of b coincides a if and only if there is a dual derivation so falsifying the analysis needs falsifying the conclusion from b to a so and if we say we want to take this falsification from parenthesis to conclusion seriously and it makes sense to also have this in the object language okay that was about this bilateral system we want this bi-intuitionistic logic I didn't say this before but you hear bi-intuitionistic logic maybe you think rather of the more well known bi-intuitionistic logic the Haydn-Brahmer logic they also have this complication but it may be important to mention that you have the same sign there but the interpretation is different so in Haydn-Brahmer logic complication is interpreted as preserving non-truth from conclusion to premises in a valid influence here complication is said to preserve falsity not non-truth, not falsity from premises to conclusion in a dual derivation just to mention this now about this thing of uniqueness so this was I call it here the overall condition because it was introduced by Werner in his famous response to the right as tongue problem so I don't know how much everyone knows about tongue so the question was okay what requirements must rules for connectives meet in order to be meaning conferred and kind of solution because Pryor's tongue problem was meant to say okay I can also give you a nonsensical connective and just say well these are the rules and this is the meaning or what but he wanted to make the point you cannot just give any rules there have to be some requirements for the rules and okay Werner then said yes I will give you some requirements we have to demand the utility of extensions that if we extend our language with certain connectives then that should basically not mess up the system it should not trivialize the system and so on this was the answer to the existence issue of connectives and then he says basically like in one paragraph he says okay there is an existence issue so is there such a connective then we can ask okay and is it uniquely characterized by the rules and yeah this is then the question when is a logical connective uniquely detailed by its rules and in general okay if the rules that defined at most one inferential role for this connective to play two connectives characterized by the same set of rules but otherwise playing different roles in inference this should not happen and this should definitely not happen we consider something like proof theory takes one place maybe that and the question is now okay how can we check this okay we say we build a copycat connective of that connective defined by exactly the same rules and the connective is uniquely determined if they play exactly the same role both in premises and in conclusion and how can we test this then usually inter durability is then taken as the so far said with this sanctions for a proof system we said this is then one fine criterion must be able to show that we can derive one from the other and the other basically if that is the case then it is uniquely determined and we see here for conjunction for example okay this is very unproblematic we can show if we consider this conjunction prime to be defined by the same production elimination rules then we can just get this inter durability no problem and about the example was this clock and blink we said okay we can consider two connectives which are both specified by just one rule we can derive from B A clock B and from B we can also derive A link B and given only these rules for clock we cannot show that clock is unique because I mean there is there is no way to show this inter durability right it can go from B to A clock B then you stop you have no no elimination rule or something like this so you cannot get rid of it you cannot get to A and B so the same for click so this is possible and thus balances they could in theory stand for different connectives and this was the beginning of this and then there was not that much literature on uniqueness I would say but especially large number stones one of the people who worked a lot on this and we all know when he writes something he writes very extensively about this so all newspapers are 80 pages or something like this or the 2000 years of course this book on connectives but he gives very interesting examples so because he doesn't give only these like made up like prok and pink examples but from actual logics which we use otherwise he shows that there can be a failure of uniqueness and what he emphasizes is this that it's really system dependent so one of these examples is we have box in system K or negation in FDE he says ok one does or does not uniquely characterize a connected set of rules of course and collection of rules of the system are in turn should be seen as conditions on consequence relation and then he shows that the failure of uniqueness can occur due to for example just different ways of formulating the proof system or this would be for my part important due to non-conferentiality or impure rules let's say a minute or what this is about but just about this first thing he shows that whether we have rules or axioms so zero premise rules plays a role or the number of formulas on the right side of the sequence side for example can make a difference he uses as an example for example this junction is not uniquely characterized in the classical intuitionistic rules but they are formulated as zero premise rules or negation and minimal logic cannot be uniquely characterized by set formula rules where there is exactly one formula on the right side of the sequence side but it is uniquely characterized we say there is at most one formula on the right just to give you an example there is really weird stuff going on but yeah well congerentiality is not it is not connected to uniqueness so we say that a consequence relation is congerential if for all formulas whenever A and B are interderivative then we say that got equivalent whenever they are interderivative interderivative they are also when they are integrated into more complex formulas these formulas then are also interderivative then if that is the case then we can say that A and B are synonymous okay we say that the logic is congerential if the defined consequence relation if there is a congerential and then and then we have to distinguish between uniqueness to the equivalence and uniqueness to the synonymity the stronger notion would be then the interderivability of all compound formulas containing the connective and so in congerential logic uniqueness to the equivalence or uniqueness to the synonymy they are coincide in non-congerential logic this makes a difference whether we demand this or that and just quickly about impure rules rules are said to be impure or sometimes we say we say they are inseparable another term for this when more than one connective occurs within the rules and usually also from proof-theorical semantics point of view this is something that might be considered problematic because we want to say that the rules define the connective and more than one of course then there will be some circularity probably ok so and then if this is the case then we can only say that the rules characterize the connective in terms of this other connective which also occurs and an example for this would be Nelson's constructive logic with strong negation and form which is a parapresistence extension of the E basically and I chose this because there is a strong relation between N4 and this logic 2 and I told you before then if we add strong negation to 2 and this could be read as a toggle which basically does directly between from proofs to view proofs and the other way around so and in usual unilateral the rules for strong negation are impure and also Dr. Pravitz gives us a natural reduction book he gives the rules for N4 and they are impure so the question here that could only be asked is then is this strong negation is it uniquely characterized by its rules in terms of and then conduction and implication because these also occur in the rules for strong negation and 4 is also non-concerned because for equivalent replacability in all contexts provable equivalence is not enough what is traditionally needed to secure this is an equivalent between the strongly negated formulas so here is an example these two formulas are equivalent in this system of N4 but if you put a strong negation sign before they are not yes so in this case for N4 we have non-concerned logic and like I told you before in non-concerned logic we have to demand then uniqueness to within synonymy to get real uniqueness but uniqueness would have to be tied to strong negation here because the inter-derivability between strongly negated formulas would need to be the additional requirement but strong negation can only be defined in terms of other alternatives because of the impure rules so this leads to some points I have a question for clarification the first point I don't get the sentence uniqueness to within synonymy must be demanded is it a technical term within synonymy? you mean the subliminal synonymy is the technical term okay what does it mean? what does it mean? exactly yes yes yes it means to within equivalence and it means to within synonymy and this would be the inter-derivability of all components it's great no if you have any questions no problem and now for the problems for units in a bilateral system I want to show now what those are and that they are quite similar to the ones I just mentioned but that I think here we can give a more evident solution so what causes trouble in a logic like 2-int is that we have these two sets of rules and these two consequences so we could think that we have the proof rules and we have the dual proof rules and we have a relation for approvability and for dual provability so if we think about what Hammerstone said about collection of rules should be read as politicians and consul inspirations we could think that proof rules generate consequence conservation of provability and the dual proof rules one for dual provability the thing is now that the connectors of 2-int are already uniquely characterized on such a way for both consequence relations individually by only a part of that set of rules so let me show you what I mean if we take this so this test for uniqueness whether it's to with an equivalence or synonyms is now irrelevant there's always some kind of inter-graphability and we can show now that we get this if we take conjunction prime again to be defined by the same proof rules as conjunction in 2-int then we can show I showed here for the sequence calculus that we can get then we can get this inter-derivability between these formulas containing these different conjunctions because they are defined by the same set of rules it's no problem basically likewise we can show if we now take conjunction double prime to be defined by the same dual proof rules not only the dual proof rules as conjunction we can show now that we can get this inter-derivability with respect to this dual probability relation again that's the problem but the thing is there's no possibility to determine by this characterization that there's only one connective of conjunction because of course we do not get these following sequence so we cannot get from we cannot get from the so the conjunction double prime is characterized by the dual proof rules so we cannot get this for the positive inferential relation and we cannot get this for conjunction prime of its defined by the proof rules for the negative consequence relation so the question would be can we then know that there's only one conjunction with a unique meaning since there's no inter-derivability possible might it be the case that we have to say there's a conjunction for proving and a conjunction for refuting and this is something that we wouldn't want right because we want to say conjunction has one meaning and not if we prove that it has a different meaning than if we refute this this would be bad however way out is if we take a look at our rules again so the proof rules for implication or the proof rules for co-implication then we see that here we have a mixture of the consequence relations within one rule so here we have double lines and single lines in secret calculus this is expressed by this mixture of plus and minus or in the left rules of a mixture of so we get something from the assumptions and from counter assumptions into counter assumptions or into the assumptions so what we can see is that the consequence relations are actually intertwined in the characterization of the connectives so the rules for implication and for co-implication they need both inter-derivability relations in a single rule so that would mean that it is not correct to think in general of the proof rules as generating the consequence relation and the dual proof rules to generate the dual consequence relation but actually yeah they work together they must be there are intertwined in the characterization of the connectives and so we are not allowed to use these different duplications of a connective to want to show its uniqueness so we are not allowed to show the proof rules and for the dual proof rules but when we duplicate this we need to use the same duplications for both proof rules and for the dual proof rules and this way it is insured that we are not talking about different connectors in different contexts of proving or refuting so what I would then as my suggestion to modify uniqueness criteria in a bilateral setting is to just basically just modify it and say it has to be sure that they are not only inter-derivability that they are usually inter-derivable but they are also usually inter-derivable which means in this context just inter-derivable with respect to this dual derivability relation so formally expressed for the case of two end we can just say that this must be the case this inter-derivability with respect to the plus relation and also with respect to the minus relation this is the case then we have uniqueness secured the advantage of this modified definition is that because two end is just like in four it is also non-conferential so we would also get the same problem here basically the lot of all formulas which are equivalent with respect to our usual relation are also equivalent with respect to this dual inter-derivability relation for example these formulas which is basically equivalent example to one end and four just differently written down but if we have now equivalence of formulas both with respect to the one end the dual consequence relation then we also have synonymy guaranteed and this is one thing in this paper so if we have inter-derivability both consequence relations then we immediately get synonymy so with this definition that I just gave not only uniqueness to the equivalence guarantee but also synonymy and that is what we want in non-conferential system how am I doing on time yeah now that was three steps of my plan so that was the modified definition of uniqueness now I am going to this part where I want to say something about what this lead us towards going for this or for that kind of proof system what I want to compare first I told you in the beginning about these different kinds of bilateralism and that Ramford for example has a system with syn formulas now I yeah and we have these syn consequence relations in Ramford system it's always I'm always struggling I must be honest but many people here have a way better understanding of this but there's a bit of an unclear status which also I mean I'm not the only one thinks that but it has been criticized with some people this unclear status of what is exactly this plus and minus these would be the rules that Ramford gives so basically this means we assert A and we assert B then we can assert A and B and where he motivates it is that you can give a plus if you would answer yes to so you ask A then you say yes then you ask B you say yes and you can also say yes to A and B and no for the minuses but okay the plus and the minus it can't be really part of the formula because like connectives I'm a part of complex formulas because it is not possible to iterate the plus and minuses because it is also not possible to iterate or embed a speech act so yes it makes sense that you are not allowed to do this but okay it's not part of the formulas it's that it must always act on premises or conclusions of rules and thereby a speech act is formed if I get it correctly if you ask yourself okay where can I put this plus or minus then it must always be when you are allowed to put it in front of premises or conclusions of these rules and thereby you don't have formula but then you have speech acts but on the other hand rules in this system how would you define a rule you would have to define it in some way that it acts not on formulas but it acts on speech acts as premises and as conclusions and this for me it sounds a bit circular but this is not my strongest argument because I'm really not sure if I'm just not understanding it correctly but that is always something that bothers me a bit okay another point that has been made by Restor and by Kürbis is also about the actually status of plus and minus making and distracting assumptions is still part of the system and you can say that making an assumption or distracting assumption is itself also a speech act but speech act cannot be embedded as we just said and as Raphael also said so what exactly would it mean to write something like this because this is something like making assertion of A or distracting assertion of A okay so there I have to say that the speech acts are embedded maybe and the problem with defining uniqueness I think is now part of this it's not part of the formula so if you really want to give a precise formal definition of uniqueness we would have to remind something like interactively of formulas with copycat connectors with otherwise same components but the plus and minus are not components of the formulas so I have problems to see how we could really give a precise formal definition of uniqueness in such a system if we cannot form copycat formulas with otherwise same components so this would be something for me to say the sign of consequence relations are better than the sign of formulas another thing now is to say I am still talking about what might be a better or worse representation of the proof system there are different ways to represent the proof system and I told you before there is this the usual unilateral version of N4 has impure rules but there is also a bilateral sequence calculus for N4 given in the paper by Nore-Nido-Permida and one thing where they call it SN4 for sub formula calculus like that and there we have a bilateral consequence relation where we have sequence with two sided assumptions and conclusions of this form this would basically then stand for what I had as plus and this here so there you have two sided conclusions too you can do it I mean there I did that now we talked about it now we are both thinking that this is a better way than this because this basically just hides the fact that you have different derivability relations but it's basically just another point to write down and the relation to GN4, GN4 is the usual calculus given for N4 is that in GN4 you need so you have these zero premise rules and these so for the usual one and the same immediately for strong negation so a unilateral proof system a la bravitz in natural reduction would look like this so here you see you have individual rules because here we chose the one for implication two so for strong negation implication and so on and in a la bravitz proof system in the sub formula calculus it looks like this and as you can see I mean you don't have to see the details or whatever but you can basically see that the negation doesn't occur here so the thing is now like I said before in a unilateral calculus for N4 to get uniqueness to the synonymy we would need to use strong negation in our definition for secure in synonymy but strong negation is only definable in terms of other connectives so as I said before this is circular whereas in a bilateral calculus we could easily manage to give definition of uniqueness like the way I did before for SC2 and we could easily transfer this to this calculus and there I get uniqueness to the synonymy without any circular conditions so I think this here this would be a point for me where you could say that if you want uniqueness and there are reasons to want uniqueness then there might be a better presentation for a proof system where you get uniqueness whereas otherwise if you formulate it like this then you don't get it and yeah just only two slides this was something that I am advancing and I it's not working for us anymore because we handed in the manuscript no more two weeks ago or something like this but it's still very new but we thought about extensions to IT lateralism there is something like this around in the literature not a lot and coming from a very different round I would say so from not that much proof theoretic but more of it I would say more of it but still if you consider something like okay we want these speech acts to be on the path then some people are going okay but there are not only speech acts of assertion and denial but we could also consider something like being indifferent or being in doubt about something so so at least different I would call these rather different attitudes on speech acts but okay like I said we are not really about speech acts at all so but there might be reasons where you would want to say okay you could even go for more than bilateralism but for multilateralism and yeah for this we have expanded in four by two unary connectives m and m m a means that it is meaningful that a and n a it is unsensible that a and in fashion of this sub-formular cultures in four we devised tetralateral sequence calculus s n 4 m n with four different sequence signs so just like just like transthyme minus is supposed to capture preservation of falsity instead of having to incorporate these impure rules with negation which you would otherwise have to do here you can say okay this transthyme m and transthyme m capture preservation of meaningful or respectively nonsensicality from consensus to conclusion and the motivation to go for these different sequence signs then is that just like strong negation these connectives are also confidentiality breaking so if you have them in their language then you do not have this interoperability of all compound formulas with respect to your usual consequence relation so if you don't have this then you cannot say that the rules of your proof system are meaning-giving so at the same time to make proof theoretics and antics however if we then use these multiple sequence signs this allows us then to define confidentiality just as a purely structural property without any appeal to any connectives at all and thereby you can avoid the characterization of the connectives uniqueness in terms of other connectives so in the same fashion as I proposed this for bilateralness settings you can secure uniqueness in such a multilateral setting if you just demand interoperability with respect to all the consequence relations that are present in the system and yes this can get a bit messy I mean that's why I didn't show you only these slides and not all the rules or something for this for this calculus because of course you saw in this other calculus you need double demo now for a tetromaterial you can imagine now rules you need but there might be reasons to say that something like this might be philosophically interesting to put up with these proof systems ok in conclusion from point of view of proof theoretics one thing's uniqueness is definitely a feature that is desirable and as I showed you in the beginning there are several features in logic systems like representation and non-conferentiality and so on that may cause problems for unique characterization of connectives and in a multilateral setting with consequence relation for probability for dual probability interoperability we must demand interoperability with respect to both consequence relations the thing now is ok these multiple consequence relations these were in the beginning also what actually caused the problems for uniqueness so in the beginning you have these two sets of rules and dual proof rules and two consequence relations and that's why you need it seems to get messed up but the thing is yes this causes the problem but so do many other features in other logical systems what I want to emphasize is that they cause the problem but on the other hand they also offer a very nice and elegant solution to just modify your uniqueness criteria there with less than one hour for the discussions ok thanks a lot thanks a lot here so I just want to go to the matter about how to be a bilateralist and especially I would like to press a little bit on difference between signing formulae and signing consequence relations because so from purely like syntactical point of view so if I take your calculus where you have assumptions and counter assumptions and you have positive derivability and negative derivability or if you want to call that and suppose I take all the rules of the calculus and in front of every assumptions and put a plus and next in front of every counter assumptions I put a minus and then I have a single conclusion so each time I have a conclusion that comes from a derivability put a plus and this time I put a conclusion if a conclusion is compared by a negative derivability or a refutability I put a minus so I have a calculus with sign formulae and it seems like it's just a notational variant so I I thought hard actually I saw this talk already I think it gave something similar already online and I saw it so I was thinking hard about it because we have similar discussions within the group biopharmic formulae biopteralist theory of logical ground that we are interested in and it comes away from a point where I'm really puzzled about what exactly the difference is so maybe the difference is purely conceptual whether you want to bring in speech acts even if you use pluses and minuses if you sign formulas I understand why speech acts look bad because it looks too linguistic but you don't have to interpret them necessarily as speech acts you can interpret them as attitudes which is less linguistic and more related to like how to relate to to to formulas and when you prove them or when you refute them it's kind of an attitude to take towards them depending on I don't see a super big difference maybe I'll stop this rant so my question is so if I press this is just a notational difference maybe there's something I miss in the behavior of the two proof system that introduces a different maker different makers somewhere yes I think I know I discussed this with with different people maybe also with people in Berlin but yeah so several people have pointed this out because absolutely in one way you could say okay you can easily transfer it into the system just like you said but I would say so for me two things are so on one hand it's if you want to stress that it's so on one hand it's maybe really this conceptual thing that you want to stress that it's about the derivability relations rather than something that happens just on the level of the formulas but there it always must be something that happens between the formulas I don't know if it's I mean on this on this account it wouldn't make you couldn't have just a plus A right on my account now because there's always this relation between different rules so this would be one thing and to stress this I think it makes more sense to then sign the consequences secondly about this about this unclear status which I criticized in Rumpitz system I'm really always always a bit puzzled about this that you can only have plus A but A cannot stand again for plus or minus A and so on and also this making the certain assumptions I found this criticism quite convincing and then thirdly about this uniqueness definition that it's about derivability and that makes it way easier to have different derivability relations to give a definition of uniqueness then to have these differently signed formulas and come up with this and incorporate the connectives within this like I said I'm not sure how to be precise definition then in the signed formula system but I might be wrong about this but the first thing I think that is great Is the question related to this everything is related but no so regarding this issue of defining uniqueness so it turns out that I read the last paper last week several times a day working on these things the last paper very fresh in my memory and so the way actually the criteria is not exactly the same that he gives so it's not an entire derivability it's just a special case what he has is two conditions so that for example he has Plink and Plonk and so you have a sequence with Plonk as a premise you should be able to derive the same sequence with Plink instead as a premise and similarly with the conclusion so of course the hard part is to prove the identity you have Plink and Plonk and you need to prove that Plink and Plonk and that's what it's hard to do but the general definition of the criteria is a bit more general I mean I had this from Hamburg not from from me and I think since it's a side issue but the system's dependencies are already present in the last paper oh yeah but I think so if you have this more general definition in mind I don't see why there will be a limitation you just want that for example if you have A Plonk B plus in premise position in a sequence you need to derive exactly the same sequence with A Plink B plus similarly with there's minus and the same in the consequence position and I guess that's so and if you secure that you will have Plink less so would that be a solution to the difficulty that you put in? I don't know I mean I'm just thinking about that it's not just a matter of iterating I'm not really sure but what is the more general can I use a platform here? I don't know if we have one there I don't think I don't have any she put some in this box I mean do you mean that I mean he talks about and basically this comes down to showing the interoperability right because he assumes of course some structural processes of the but I think the interoperability just a special case of the uniqueness condition it's just I guess in most in the context of the disability it's interested in it's equivalent but not in general I mean it's easy to if you have like a logic which is not reflexive why not? Yeah so so what is key to the criteria is the fact that whenever you have the formula in one place in a given sequence and you replace it with this cookie cut you preserve the reliability of the sequence and so if you have only one notion of reliability but several signs to put into formula you would just need to secure the fact that a signed formula when you replace only the when you substitute it with the cookie cut is all so it's hard to explain the word but if you have a sequence and among the the parameters you have a signed formula with a connected and then take exactly the same sequence with the same and before that the same sign would just replace going to the cookie cut and it should also be derivable and you do that with all positions and with all signs it seems like this will give you a method to define like this this definition of uniqueness will not have the drawbacks of pure interpretability in the context of the the sign formula cultural yes I'm also not really sure about this thing it just seemed to me I mean I tried to work out a precise definition of this and it just seemed to me problematic that then do it for example for any compound formula because of this thing that plus and minus is never a part of it but then you have to somehow construct compound formulas where this connective is still in but the plus and minus then doesn't really apply to the formula where the connective is in you know what I mean okay I cannot explain so it's not important that I do yeah it seems to me a bit problematic to have these the copy cut to do this copy cutting with respect to then compound formulas where the connective in question is just embedded and the plus and the minus is not it's not applying to this connective right because it's it only can apply to the outmost formula basically and this seemed to me a little bit problematic if I get it right it's like if you have the plus and minus but I mean I may completely misunderstand a bit just to check whether the plus and minus they kind of form proof theoretically together with the connective you're introducing a specific connective like long plus long minus and this one connective the one that is signed so to say cannot be inside the formula so inside the formula there's a pure plonk or pure connective and not the signed one on the inside while proof theoretically you're always dealing with signed ones on the when you introduce them you don't introduce plonk but you introduce plonk plus or plonk minus and on the inside formula you don't have plonk plus or plonk minus you just have plonk I don't know whether that's the idea you're expressing I think I mean it's also it's just more a problem in my head more than idea but somehow I saw that as problematic when you really want to give a very general definition capturing all these cases I'm sorry I need anything more but that's a little bit helpful we'll have two more questions I don't know if it was a follow up or a question our first ever question coming on YouTube who says thank you for the great talk and also just wanted to ask a verificatory question so why is it that you see the impure rules as undesirable I mean okay this is this is of course only if you if you assume something like proof theoretic semantics and you say that the meaning of the connectives is given is characterized by the rules of influence that we have and then you have impure rules where not only this connective occurs but some other connective occurs then the characterization in some way hinges on this other connective I mean this seems undesirable that you want to give the meaning of connectives in terms of rules but there are also other connectives and the meaning of these connectives must also be given in some way of by their rules I mean this is general in general I would say seen in proof theoretic semantics as undesirable in some way you want to have these pure definitions of the connectives and not in terms of other connectives that you might get into some kind of holism which people don't want or not everyone wants I hope this helps you there is a follow up but thanks Peter? Thanks a lot for this great talk which was very helpful for me so I wanted to ask first other questions but a very general question about the whole concept of uniqueness and it's kind of related to Pierre's question so I've always been puzzled and this is nothing specific to your talk but by this interderivability concept as a way to formalize uniqueness I mean this is almost like insane because the informal way to express it is like having the same roles the copycat and the other the original connectives should play the same roles in both premise and conclusion but this doesn't mean that they should be interderivable for me because I mean they are can I say this was an important part of my question interderivability says something about the relation between the two how they work together well that's not issue their role inside the system should be the same but this has nothing about how they work together and certainly not on different sides of the turnstile so if you have interderivability one is on the one side of the turnstile and the other on the other sides and they may only do work on one side and they do the same work on that one side we have only the introduction rules for disjunction for example that's going to be of course not as you mentioned not a unique connective you introduce but on the other hand it's correct it uniquely defines the role it plays namely only as a connective that is that functions on the left side you haven't given anything about how it performs on the right side so it doesn't perform at all you can never conclude it you can just eliminate it but uniquely you have given the rules for elimination it's not that there is some freedom allowed on the right hand side because there is nothing you can do on the right hand side so it's unfair to require for a new uniqueness something about stuff happening on the right side even for connectives that have no rules where you have said nothing about what you do on the right side so if you think of premise behavior of premise usage as something that is fundamentally different from an differentialist point of view then conclusion behavior that it's for me but this is probably a very specific way to look at logic that I like that it is really weird to express uniqueness in terms of something that requires usage as premise and as a conclusion for always, for any connective even if you don't have any rules for conclusion use so that bothers me a bit it seems like if you only it should be separated the premise usage and the conclusion use as you are talking about uniqueness if you have saved like the premise sorry I'm coming I just at one little thing if you have premise if you just have the introduction for disjunction rules you you have uniquely defines its left hand this is premise usage and you have said nothing at all about it it's conclusion usage but there is no conclusion usage no problems but I know my question is not very precise but I don't know how do you get this intuition that you cannot just combine them into a sort of inter-derivability where at the beginning of your question I was more with you I think because I also remember when I read about the first time and also in between again and again it looked like okay but is that really what now all the things you said I don't have to repeat them but now when you come to something like in the end I would disagree to say that if I understood you correctly that there can be a difference premise usage and conclusion usage because this sounds a bit like what I said in the talk about we don't want that conjunction has a different meaning in proofs versus in refutation we also don't want that conjunction has a different meaning when we use it at premises when we use it in the conclusion right? I mean this seems at least to me very very not what we would want because there is one meaning and you can use it as a premise, you can use it as a conclusion the meaning doesn't change and the other thing would be it would be weird to have a connective that can only be used in one or the other why? I mean I don't know I mean we're talking at a very abstract level to logic so of course you have to specify both sides it doesn't make sense to say that well these are the introduction rules and now I have a connective that is uniquely defined you should say these are the introduction rules and for example there are no elimination rules which is also a specification of the elimination rules right? meaning uniquely defines a connective or well there might be other criteria for it but if you have specified the rules exactly for premise and conclusion I don't see like from a sort of very general perspective why you should dismiss all logics where this doesn't completely correspond the premise use and the conclusion use like if you have cut free systems or something then they can go apart premise use and conclusion use I mean non transitive systems I mean where cut is not even admissible or non reflexive systems or something like that so you by such definition of uniqueness you sort of immediately exclude all these sort of approaches to logic it seems to me fits well in a framework where you have these structural rules for free maybe not brawler than that yeah I mean I'm also not definitely not in general against structural approaches or something like that but it's still like philosophically I would get bernab's argument to say they should behave in the same way in so in premises and in the conclusion so philosophically I must say this makes sense to me okay then I'm not very familiar for example with non reflexive systems so I don't know how convincing I will find them but I know that there are some but I don't really know I would personally say like in the premise the original one should work in exactly the same way in premises should have the same role in premises as the copy and in conclusions also it should have the same role as the and the same true conclusions in conclusions should have exactly the same role as the other one has in conclusions but not the one in premises should have the same role as the one as the other as a copy in conclusions that seems to be comparing apples and pears if that's an expression I think but that is also not what he says well, if you have inter-derivability you derive a premise from a conclusion so it's a premise use you compare it to conclusion use you plonk is on the left side and plonk is but to derive it is not to say that they are playing the same role you can derive one from the other yes because they should both play the same role both in the premises the same and both in the conclusion the same yes but that doesn't mean that the premise should be the same as the conclusion the premise of playing should be to give you the conclusion of plonk because you you know, derivability is always from premise to conclusion right so you take the original one on the left and the copy gets on the right if you have inter-derivability if you say they have to have the same in premises and the same conclusion then comes down to well, in structural systems that as soon as you but maybe there is literature maybe I do substructurally focus to probably in humbuston but of course the logics you talk about in that sense it's not a very criticism or something but I thank you criticism it's actually not a follow-up it's just me ok, ok those were like this this time ok and he said that he can't be here to discuss more I think I'm asking a kind of similar question but I'm asking it as a philosopher of biology who used to know a lot of logic 15 years ago when I was a graduate so maybe I am too class, I certainly haven't ever worked in logics as bizarre as these so perhaps I'm just too classically classical logic focus but one thing that would help me is so when you say uniqueness is about only having the maximum of one connective that plays the same kind of role in and proofs so imagine what would happen if you had two connectives that obeyed the same rules how should I understand the concept of a role that doesn't involve talking about the introduction of elimination rules because that's where I'm immediately like wait but you just told me the rules are the same so that's what it does so if the rules are what it does then it's the same connective so what is there to what does role start to mean how does role come apart from the introduction of elimination rules I guess this is also a variation to just question so the thing is I mean you assume now that we have introduction and elimination rules which we have in our normal and well behaved systems and so on and I think the problems they would definitely start like when you just have only an introduction rule only an elimination rule which is exactly what was the case with Planck so they only had a right if you want to know for a sequence type then you had only a right introduction rule and no left one so nothing that tells you at all how this formula behaves on the left side of the sequence sign so well I don't agree with that I think it does tell you something but namely that it does it should never cannot do anything just asking this is what you can do with it you can do nothing with it it's a very nice characterization of a product I guess there's a sense in which that was my intuition as well when confronted with the Planck and Plink example my initial response was just well if you tell me that the only thing I can do with Planck is I can go from B to A Plink B my gut reaction is just by the same connected cool yeah I mean that's okay this is maybe I didn't emphasize this enough that what balance actual augmentation is maybe you can it's fresher in your mind but he explicitly says and also the tongue issue he solves on the background of saying well we have these assumptions about the reliability namely that it's transitive and permutation and so on and so on so basically the usual structural against the rules are assumed and then against this background we cannot do this which you should be able to do so against this background it makes sense but as Peter said of course nowadays we are very much like well you don't have to assume that the consequence relation is transitive and so on this will of course create a problem so this is probably a beard question I need you to talk me through an example of a logic weird enough that this stuff starts to fail because I think it's my like I've never worked with one so like my brain is unable to see how you could start to drive a wedge between these notions so I need you to tell me over the beard you can tell me about a really bizarre logic then I'll start to get where you're coming from I think it's wrong I think it's wrong I think it's wrong I think it's wrong I think it's wrong I think it's wrong I think it's wrong I think it's wrong so the consequence very consequence relation and according to me the relevance is neither reflexive nor transitive so it's reflexive it's not more like more that's more contractual I have my own so I don't like they are normal so I'm trying to connect this notion of sequence that you give at the beginning of the presentation and try to connect them with other readings that I'm more familiar with and the first question that I have for you is how this I don't know if you have an answer how this could be applied if possible in a non-intuitionistic setting so with multiple conclusions for instance think about this so I don't I mean my memory was from the assumptions the final assumptions then if we have multiple and really so this first question as I said it's very general it's related to so when I was trying to see your definition here of sequence connecting to more simple ones for instance you cannot accept the premises and reject the conclusion at the same time my question was is maybe this setting in which we have in the premises of the inference gammas and deltas accepted and rejected and then with this accept as a rejection we either accept or reject some conclusion C whether if we accepted multiple conclusions we couldn't have this complicated system because we could already move the deltas into where the C is so we we would have for free something like simple you cannot accept gamma and reject delta at the same time in which in delta you have more than one form okay so if we have this if we read this so from the assumptions now yeah we start gamma and delta are assumptions yes we have gamma and delta which we accept gamma and reject delta and this leads us to accept C for instance okay we could move the delta maybe and say and then have a restalian reading more which we would say we cannot accept gamma and reject at the same time delta and C but I am not this is something this is a yeah I was thinking if I I think right now I have seen something like this right which looked a bit but I cannot really remember how exactly you were supposed to read so I am not really sure but that yes you can maybe someone else in this type now okay by by Greg it was at the last year's AL they gave a talk which was also really just about letterism and it was about also having assumptions and counter assumptions but yeah I don't really know how it would work because because I never worked in classical systems basically so I am okay but it was the question if there is something like I don't know if you had like an idea of how this would apply in the classical but anyway that is something you can just do put your assumptions then just in the conclusion because I mean it still strikes me as something you assume you take it to be true take it to be false from this you draw something isn't it something different then but yeah in my conclusions you would have to say okay and then you draw either a verification of this or a falsification there is a version of that talk on Greg's YouTube channel oh okay from last March last May something like that yeah I am not sure yeah I think I would want to know um yeah so I wanted so this was my last question was very general and a little bit more critical but sometimes I am very attracted to to this the uniqueness thing combined with bilateralism and the thing you proposed um and to me it seems like an argument in favor of an argument to convince a unilateralist that you have to go bilateralism or you don't you don't define connectives basically in the right way you don't give the full meaning of the connective if you don't also have the bilateral characterization like it seems that a conjunction that a unilateralist gives with this proof system it's just lacking it's ambiguous it can both you just give the proof rules for it and not the anti-proof rules and so you say how you deal with verification but you don't say how you deal with falsification of it and this is an act to be engaged in in falsifying and um it seems that the states in the world they can falsify formulas or the worlds can falsify whatever it's not something that people find philosophically problematic or something so it seems that a unilateralist should either like in the background tell an implicit story about how to deal with the anti-proof rules like implicitly that you can calculate them from the proof rules but if that is not present then what she has defined or a unilateralist is simply an ambiguous connective not a full characterization of conjunction for example what would you think of such an argument I mean I personally like it of course absolutely I was just thinking because I was thinking of something that Parvitz wrote he's like very just verificationist I think it might be about this but what I read was very much defense this only verificationism and I was just wondering if somebody who is for certain philosophical reasons I don't really know but who would be very convinced of that whether they would accept that they have to think about falsificationism at all because actually what I showed you with the viewer rules for implication that was for Parvitz a reason against falsificationism basically that you have these two different that you have to give a proof of A and a refutation of B and this gives you a refutation of A and plus B that was for him this what I these constructions that was for him saying this shouldn't be the case maybe then you could say why not press a maybe on this but this shouldn't be the case and that's why we shouldn't consider falsificationism at all and I mean ok and one things I would against this was just because we just consider this implication if we would just as naturally consider co-implication then you could say the same about the verification rules for co-implication basically but of course you can say you just consider co-implication if you take falsification seriously and so on so a little bit I think that's the way out saying well there's only verification we should never talk about falsification it's a ill-conceived concept so that's I guess naively I would say that's the way out but if you think if they take falsification seriously then then and you're not pressifying you only give the usual rules for co-implication for example then you didn't give rules for co-implication you didn't define and connect if uniquely you defined something as ambiguous yeah and I mean I think there are I would also say just philosophically there are reasons to press the people who are just considering verification rules or some other proof like you could say okay why is there something like the same things that were said about the social denial basically saying that denial is just as non-reductive as assertion I think there are reasons for that so yes and then you can say you have to go that's good that's good for all of the questions I think you have to be careful because it depends what you put under basic commitments of binataryism if the basic commitments of binataryism is having like denial as being primitive then the kind of argument gets sounds a bit like begging the question because you say hey you have very or partification should be taken seriously as if you mean as being primitive then you put in the premise kind of I mean philosophically at least so that's why I think this kind of I would be like a lot hinted on the details but that's just a basic remark the question I wanted to ask you is about non-territorialism so of course this is like a very interesting generalization so I'm curious to read the paper and why and like your examples with meaning and nonsense I think this is quite interesting but I was thinking of perhaps even like simpler ways to do ultritorialism by considering many biologics for example like the different attitudes you have so it's not like denial and rejection but one attitude corresponding to whatever true values you have in your logic another example that seems like interesting and it's related to the discussion I had earlier actually have a lunch about the attitudes you can have towards like believing God and if you play the square game and you distinguish those four attitudes like I believe God does not exist I mean there are four possibilities and I mean this could be perhaps another example I mean I would just so my question is why did you choose this particular example and how far do you think this ultritorialism could be implied I mean just a question for you then so because I mean when I am talking about bilateralism or multilateralism or something like this always in the context of proof theoretic semantics which I mean which is kind of justified because the origins of bilateralism were if you now say okay you have different truth values like how does it go yeah sure start with the semantical characterization of that's a fair point okay here's one thing so for example think about eddy eddy has you can do certain countries like you have countries you can can choose for eddy but it seems pure you have to have rules and have mitigations it seems like from the proof theoretical semantics it's bad if you look at the but of course you have many value matrices for eddy in which you don't have this problem because if you have more space if you can take more attitudes to propositions you can decide to take them as true or to take them as only true or only false or false and you know all the rules then you can say you can give rules if you have like four value sequence I don't know what that means so it's like this N side sequence exactly you can give those four and then you can give inference rules for I mean I suppose you can even this is only to bring the technical thing but seems to be able to to axiomatize to give a different rules that are pure for eddy for example and of course the technical realization would be motivated by the fact that we know that there is a matrix semantics that works but I think it can be philosophically justified also by I don't know different kinds of inferences or at least you could if you can make sense philosophically of those like four sided sequence it's purely inference terms then you would have inference like a co-share so to speak a way to look at eddy as a as a logic with like meaningful connections and not just like purely algebraic again I think this paper by from 2014 where there is definitely multilateralism in the title that's why I came across which is about multilateralism and I think if I understood it correctly this is exactly the point that he is making I think he is doing it for right you can give a three sided sequence for k3 this example and I mean this is fine for me and it's also fine for me to be honest to call this multilateralism for me it's a form of multilateralism but like a minor point that we also criticized in this paper I mentioned was like still if you come from proof theoretic semantics this paper by Hortland is quite like traditionally semantically motivated and like maybe if you want to have a stronger proof theoretical argument for this then but that was basically the motivation for this was really basically also because there you also have this congregationality breaking and that you can secure this with different forms of secret arrows and then also retain uniqueness and so on so that was the motivation to go for this and I'm not saying it's the only form of multilateralism I have a quick follow up so it seems to me that of course I mean it's possible that we came from a matrix point of view but I mean this shouldn't be an argument against it being proof theory how it's where it comes from or how it was originally conceived if you take seriously bilateralism and look at it as attitudes then having the attitudes of indeterminacy or something like that in the case of FD3 or case of FDE you could have the attitude of being in contradiction being conflicted about the proposition let's say or being in doubt about the problem to say that it didn't neither true value being positive about it and being negative about it something like that you could just generalize to the whole matrix of FDE the four-valued matrix and then why not go further I mean of course we use these matrices for traditional semantics but there is nothing that's just the inspiration you could see that as pure if you involve these attitudes in your proofs that becomes proof theory just like anything else except with multiple sides of signs you might have the true value as a sign to not just a plus and minus but true, false, both and neither a sign I mean it seems to me that there is nothing more theoretical about that it's just as proof theoretical and as inferentialist and so on but just a broader spectrum of ways of thinking about a proposition than just accepting or rejecting maybe I want to help you here so I think this is where the difference between signing the consequence and signing the formulas matters because so the kind of view seems like that you're proposing generalizes from I mean I can understand very well generalizing from the run fit style so it's kind of easy to see how it works if you add attitudes to propositions you could say well I can have that attitude of being like I want to say yes and I want to say no at the same time or I just want to say yes but it's much harder to apply those signs to at least for my intuition to apply those signs to consequentialism and your side of this debate is even though I was inconsistent but your view seems to be that criminally the sign belongs to the consequence so this game could have played as easy part I mean this would have been somehow also the attitudes that she mentioned I mean this is like this one form of bilateralism but as I said in the beginning this is not really what we are posing here and also maybe I had this my answer was maybe not really to you but to this Portland paper which I had in mind where he is really concerned about showing that then for these proofs in the category a problem doesn't apply or something like that and that's very, for me that's very coming from a model of theoretic point of view because why are you concerned with a category of the problem if not like you know what I mean maybe that was more of a response to this having in mind and a category of theoretic sounds very much like a unique use of facts now sure but from the model of theoretic it might be a good inspiration to have an altered characterization of uniqueness you could have a characterization that is proof theoretic of gothegorecity let's thank our speaker