No more As than Bs & non-standard quantifiers

0:00 ...unless the Cabbage Go works, which is fine. I'm going to do a Cabbage Go for you one day. Scott's finished with Cabbage Go for you. So that's... I suspect... But it's like the parameter is going to be quite far on the other side, but then you want the other... ...has a number of arguments and so on. So, these forms, the ones that are displayed, can be regarded as values... ...arguments, and the arguments themselves are... since the argument takes values, through values, the concept. So this form here, Frege argues, is best used as a second-level concept. So that's the beginning. And things stayed pretty much where they were for a long time until people started taking the subject of generalized quantifiers a little more seriously. It was a study by Moustowski in 1957, and continuously worked by Montague in 1974, and it spans a number of fields, right? Linguistics, philosophy, mathematical logic, and so on, with people in different fields focusing on different parts or aspects of the subject. So the linguists think of quantifiers, general quantifiers, a tool for semantics, logicians are more interested in questions about expressive power, axiomotricability, decidability, and so on. According to this view, the quantifier Q over some non-empty, say domain V, well, that's just a collection of subsets in terms of the domain. It's a subset of the power set of V. For instance, the universal quantifier is just the collection that contains only one member, the domain itself. The existential quantifier is the collection that contains all the non-empty subsets of D. Those are the two quantifiers we all know and love. But there's also the quantifier that exists exactly K, which is the collection of all the subsets of D that have exactly K members. And more recently, perhaps, even constant singular terms can be interpreted as generalized quantifiers.

2:30 So, for instance, the name John can be represented and identified with the collection of all subsets of data contained John as a member. Now, all the quantifiers that we've looked at so far are unary quantifiers, but in general, you can conceive of quantifiers as n-ary relations over the power set of d. This is sort of, again, sort of a crash course in generalized quantifiers. You always wanted to know about generalized quantifiers, but we're afraid to ask. So here's one. You can think of all as a binary quantifier. All A's and B's is the collection of all pairs of subsets of the domain, so instead, A is a subset of B. Some A's are B's It's a collection of all pairs that have non-empty intersection Most A's are B's You can think of them as a collection of all pairs A-B's Such that there are more A's that are B's Than there are A's that are not B's Here's a theorem of quantifier Twice as many A's as B's or C's That means that twice the quantifier as the collection of all triples from the domain, such that the A, there are quite as many A's that are C's as there are A's that are C's, and so on. So, you can have all sorts of kind of relations over the power set of D. Now, it just so What happens is just a logistic, I suppose, that some quantifiers, some binary quantifiers can be represented by means of a unary quantifier, which is applied to a Boolean combination of their arguments. So, for instance, all the A's and B's can be represented by the unary quantifier applied to this Boolean combination of their arguments, but this is just an accident. It's, to some extent, an unfortunate accident, because, for instance, when you teach introductory logic, then the students come away with the idea that this is the right way of thinking of the quantifier, whereas, in many ways, this is the correct way of thinking of the quantifier. And this is just an axiom. For instance, Mohs cannot be represented by a unary quantifier applied to a Boolean combination of Tsai-Dubas.

5:00 Okay. So now we're starting to look at a classification of these quantifiers. And so there are two variables here at play. One is the addity, which is the number of arguments that the quantifier takes. And one is, I don't know, well, for what's a better word, the addicity, right, in the sense monadic, so the adicity of the quantifier, which is the number of variables in the formula that the quantifier is applied to. Okay, so let's look at some examples. All A sub B is a binary monadic quantifier, all right, because it's binary because it It takes two arguments, but it's monality because the arguments it takes are subsets of D and not, so they're predicates, one-place predicates of a D instead of two-place or three-place predicates and so on. But for instance, the quantifier that contains as its only member the universal relation of D is a unary-diadic quantifier. So there are these two variables that we have here. So, one way to systematize this, when known from the literature, you can represent the quantifier by, you can assign a type, which is a 2.1 to a k, where k represents the arity and ni represents the adicity of the i-type, so the quantifier q belongs to this type, even if it is a subset of this, of the Cartesian project of the appropriate power set of the power set. Now, what about secondary quantifiers? Now, secondary quantifiers, those are collections of, or relations over, first-order participates. So, in the sense that there exists a theoretical piece of step to P, this type has this type, so it can be as defined as a subset, with a collection of all subsets of the second power set of P that are not empty. Now, in this talk I'm going to concentrate on first-order participates, but the point I'm trying to make is that the distinction between first order and higher order is a semantical distinction of the argumentation. So what makes a quantifier first order or second order is the type. So

7:30 most, first order, all, first order, and so on. All right, so let's go quickly through some properties of these quantifiers. Several important properties have been singled out. is conservativity that q holds of a b can only if q holds of a and a can set b okay so when this is all if you say all a's are b's that's the same as saying all a's are a's that are b's okay and the number of quantifiers have this property uh another property is like monotony if if q a b and B the subset of C, then QAC. So all has this property, right? Again, if all Bs are Cs, and all A's are Bs, then all A's are Cs, then most has that property as well. Right monotony. Left monotony goes away. Right anti-monotony. If QAB and all Cs are Bs, then QAC. So, for instance, if no anti-property, right, if no A's are Bs and all C's are Bs, then no A's are Cs. left anti-monophony but one that we're going to be particularly interested in is permutation invariants if pi is a permutation of D then QAB holds if and only if Q holds of the pointwise image of B on the pi so permutation, the D is the quantifiers have to do with the the cardinality of the arguments, of some relationship between the cardinalities of the arguments. And therefore, however you permute the domain, it shouldn't really make a difference. Of course, this notion of permutation in Bion's history goes back to Tarski and his characterization of logical constants and so on and so forth. We're going to get back to this as well. You can always get new quantifiers from old ones, and there are several constructions that have been singled out for this. One is iteration. So we're already seeing that a proper name, such as John, can be viewed as generalized quantifiers. Now, the slightly

10:00 more general point of view, still, is to view them as projections, that is, as I keep using So if R is a 3 plus 1 place relation, then John applied to that relation gives you the projection of R, the set of all couples that are related to John. And this subsumes the monadic case, which we saw at the beginning, by interpreting true to a fault as zero-placed relations, okay? And things work out very nicely, so for instance, if you want to say, compute the semantic value of John, Kiss, and Mary, that's just the relation here. The quantifier Mary applies to the relation to Kiss, applied to, sorry, the relation... The quantifier Mary applies to the relation to Kiss, and then the quantifier John applies to the result of that. And this is true if and only if John, Kiss, and Mary. as one would expect. All right, so, again, it's not in more general view. You can define the settable zero-place relations over the domain. There's two of them. False. And rn plus 1 is the settable of n plus 1-place relations over the domain. And then you can look at the collection R is the reducing quantifiers of degree k with a set of quantifiers q, such that if r is an n plus k relation over d, then qr is an n plus relation over d. and quantifiers in these classes are closed on the composition because if you want is not a reduced quantifier of degree n and q2 is of degree k then the composition is of degree n plus k and so yeah these are associated so you can write down q1 q2 through q n without without ambiguity okay again I'm going all this kind of stuff first of all the definability so a binary quantifier let's say such as qab is sort of the final of the domain even only if the following is true so you take

12:30 the language this is relative to the language you take a language you expand it by adding to one place predicate symbols p and q and then you want to be able to find the formula fee such that of because this displayed thing makes some sense, even if D AB satisfies phi, you can only have Q AB. So there is a, you can only have Q AB. So, for instance, at least two is definable, of course, by the standard formula. But there are lots of quantifiers that are not definable even on finite domains. So let's say that a quantifier Q AB is proportional over some You can only, if there are numbers M and N, such that the quantifier holds between A and B, you can only if the proportion of A's or B to the A's is at least M over N. So, for instance, most is proportional, at least half is proportional, but more than 10% is a proportional quantifier, and so on and so forth. And so we note here that proportional quantifiers are not just sort of definable. So far, so good? Questions? All right, now we start to get into the slightly more interesting stuff. There are two related quantifiers that deal with cardinality restrictions. been singled out and I think those are the ones. That's the label that I'm going to be exploring today. One is the Hertie quantifier that holds between A and B if only there are as many A's if there are B's, if the cardinal to A equals the cardinal to B. And the other one is the Ressier quantifier, R-A-B, that holds if only if there are more A's than B's. So, Ratio introduced the Higgs quantifier in 1962 and Hertie in 1965, and I discovered after working on this for a while that they had already been extensively studied. That's how you know you had a good idea, and somebody else had it before you.

15:00 So using the axiom of choice, you can define Hertie quantifier from Ratio quantifier, but not by sursa, so we can define that there are so A's and B's as the canality of A, and B's are greater than the canality of A, and the canality of A is greater than the canality of B. I don't know why I should set it up this way and have to use choice when there is another way to do it. Now, 90-25 is conservative. The following both fail, of course. It's not true that if there are as many A's as B's, then there are as many A's as there are A's that are B's. Of course, that's false, so it's not a conservative. What's the point of a conservative? I forgot to say that. It's one of those generalizations that sometimes are somewhat surprising. It appears that just about every atomic quantifier in all the natural languages known to man are conservative. So all quantifiers that can be expressed by one word, or most, are conservative, with one exception, which is only. only A's are B's, that's not a conservative quantifier, because it's not true that only A's are B's, it's true if only A's are B's, it's not true that that holds if only A's that are A's only A's are A's that are B's. Okay, so that failed. And people have gone at length to explain why only A's are a quantifier. But anyway, I'm not going to be concerned about that. Is any a quantifier? Yeah. Then, of course, any is ambiguous, of course, right? It depends on the appropriate disambiguation. Now, conservativity fails for the ratio quantifier as well, but you can see it's not true that there are more A's than B's, it's only if there are more A's than A's than B's. one direction holds out of one thing but both quantifiers are obviously permutation in an area now what I'm going to do here is I'm going to take this notion of permutation in an area as a hallmark of logicality I'm going to follow in Tarski's footsteps

17:30 I don't have a picture of the book to plot this notion of permutation in an area can be taken is a first approximation as a sign that the notion in question has logical character. Now, another point I'm going to make is that these quantifiers have typed 1, 1, and therefore they are first order quantifiers. Although you might think that some of the notions involved are intrinsically higher order, that's not true if you take the view that the order of a quantifier is given by density-semantical characteristics. So these are the first two. They are first-order quantifiers. Okay. Now, I'm not going to talk about either one of them, but I'm going to concentrate on the one that's closely related to them, and that I'm calling the Frege quantifier f. Because famously, Frege argued for, it was the first one, I think, to argue for the logical character of the Kalanity notions. So let's see. Let's see what happens if we take this idea seriously, that Kalanity notions have logical character. And we want to look at this question by having these Kalanity notions in the context of the barest of the logical, of the barest logical paradoxes. So nothing, as little around it as possible so that we can really see it at all. So what we do is we consider, first of all, a language with formulas built up from individual predicament and function councils in the usual way by means of the Boolean connectives and picture-favorite set of Boolean connections, and the periodic graphifier, fx. And this slide explains that if phi and c are formulas and x is a variable, then fx, p, c is also a formula. The intuitive idea is that there are no more phi than c. So there is, I'm gonna, it's sort of the reverse of the ratio quantifier, there is an injection of the . So F is a monadic, binary, first order quantifier, right, it's monadic because it takes formulas with one, it binds one viable, it's binary because it takes two formulas, and it's first order because the argument.

20:00 So, syntactically, so it's like, oh, all is a piece, okay? And the good interpretation, as I said, is that there is an injection of the p's into the p's. Just like we did for the ratio quantifier, we can abbreviate... fx phi c and fx phi phi by this, which is the healthy quantifier, okay? And because of the Shurer-Burst anterior, we have the axiom choice, of course. This gives the meaning of i, x, phi, c, which is true if only if the cardinality of the object that satisfies phi is the same as the cardinality of the object that satisfies c. Okay. Here's the standard semantics for the quantifier. Okay, so what's the model? The model is an unentered domain which provides an interpretation of a neurological constants in a different way. are mapped onto a subset of the nth power of d, and so on, okay, so nothing fancy like that. But we have to say how to satisfy the quantifier. So given a formula phi, the variables x, and an assignment to the variables, a function of x assigning objects from d to the variables of L, then we want to define when it's true that x satisfies phi in m, and all the causes Except that when you get to the quantifier case, that S satisfies Fx in C in M, even only if there exists a one-to-one function between these two sets, right? This is the set of objects. This is the extension of phi according to S. And these are the objects that fall under the extension, right? So, and this is the extension of C according to S. precisely when there is an injection of the feed into the ppses, which is exactly what you could expect. The notation is a little cumbersome, so I think sometimes I'd like to use this, right? So, you give an object a and x, you define in the way what it means to shift the assignment to the variables.

22:30 So, this function here is just like x, except that it's shifted to assign these objects, a1 to a k, to the object 1 to a k, respectively, And then you can define the extension explicitly, right, so this is the set of all tuples of objects, A bar, such that the shift of assignment satisfies phi M. And if you have the notation at your disposal, then the satisfaction clause for the quantifier becomes that S satisfies Fx, phi C, if only there exists a one-to-one function from here to here. All right, let's pass with it. Okay, so what can we do with this? That's all we have. We have the connectives, we have credit accounts, and so on and so forth. The connectives and the peri quantifiers. Then the first thing to notice is that the standard first order quantifiers are expressible. So, everything is phi if and only if there is an injection of the complement of phi into the empty set, that is, if the empty set. If the complement of phi is empty, then the form is full. Once you've got the universal modifier, the other one cannot be far behind, there exists xvx if and only if there is no injection of phi into the empty set. I'm going to help myself. Although, officially, we only have the f quantifier, so I'm going to help myself with these abbreviations. And then we can get some more interesting stuff going, right? Now, contrary to the case of first-order logic, you can have an axiom of infinity in the two identity fragments of the language, right? Which is basically the first-order logic. That's easy to see. It says that the universe is dedicated to infinite, That there is an injection, that there is an object Y, such that there is an injection of the universe, of the object satisfies X equals X, into the object satisfies X, not equal Y. So, there is an injection of the universe into itself, so the universe is the thing. And since this is in the pure identity fragment, this is true in all and only the infinite models. And the phonetic negation is through all and only the finite models. Okay, so we already have that this language is more expressive than the first of the logic, and compactness

25:00 fails. All right, so let's abbreviate. Now we have a way of saying that the given phi is finite, okay? So we can abbreviate fin x phi phi x, all right? So that's a statement that, somewhat informally, the set of x is such that phi x is dedicating finite, that says that no matter how you pick y, there's not going to be an injection of the phi's into the phi's different from y. Okay, so the extension of phi's dedicating finite. And then you can use this machinery to characterize the standard model of the natural numbers. By the way, this is where I, as far as I got before I started all this stuff. So, you can write out the sentence. So, you look at the language, which contains one binary relation symbol, less than, and you can write a sentence, phi, which is true, even on the if the other type of that ordering is less than or equal to omega. So, the sentence says that less than is a strict transitive linear order. It has an initial element. There exists x for y, for all y other than x, x is less than y. And the set of the predecessors of every x is finite. And then if you combine this sentence and you can join it with the axiom of infinity that we saw on the previous slide, then what you get is something which is true precisely if less than is accountably infinite linear order. So we're done, essentially. Now you can join that with a set of argumental axioms, such as, you know, the axiom for plus times pa without reduction and then you can characterize that model up to isomorphism so the language is indeed very very expressive okay as for the consequence we have that the language that the set of these validities is not recursive x-axis because you can characterize however surprise if you instead look at the head identifier okay so the language without the you can only assert the identity of the cardinalities, that is a tie-over.

27:30 And I haven't looked after two points, so I have no idea how that goes, but I was surprised. OK. But it's the f-modifiers that we're interested in. um and so the aluminum column theorem for for both for both in fact failed failed family okay let's look at some further properties now it is well known that uh over the structure and less than addition is not definable in first order logic however it does have to be definable using the f quantifiers. In fact, what you need is the healthy quantifier, right, because if a plus b equals c, if and only if b equals c minus a, if and only if there are as many things below b as there are between a and c. and I am supposed to drive the phone home, you can just test that that's true, and in fact, you can go even further, this is, again, things that I don't know, you can find two sentences, P1 and P2, in that language, so it's that P1 is true, to all and only the successor cardinalities, and if you choose to all and only the limit cardinalities. Okay, so that's how badly the organized column feels. We also have a figure of the VAT problem here, right? Since the set of natural numbers is categorically indefinable in this language, you can define a good number of finite sequences and finite sets, and using that good number you can define satisfaction implicitly. Okay, so you can add a predicate such that in every model the predicate is, in an axiom, such that in every model that predicate is an extension of the satisfaction. Okay. But since my class is theorem, of course, an instruction is not explicitly definable, then the best definability theorem fails. So, this is how bad things are, right? It's very, well, I don't know, bad, or good, right? It's very, very expressive, extremely powerful, extremely powerful language, okay? Even with the, you know, with the barest operations that we have here, right? Connectors, predicates, and this one quantifier, right? We have quantifier.

30:00 That's why what I wanted to do next was to try to see if you can sort of train the quantifier by giving it a standard semantics, the general semantics, and see if we could get something more manageable. Now, the general semantics is just what you would expect, if you know general semantics for a hierarchical style. Now, a general model, again, provides an empty domain, V, interpretation from the neurological constants, and moreover, instead of looking at the collection of all the injections between substances and the domain, now you're looking at A collecting all the injections between substances and the domain. So, we have a collection of f, curly f, of one-to-one functions, f from a to b, where a and b are subsets. So, you know, these models come equipped with a collection of functions, which is then u in the satisfaction cluster of quantified. So we're going to say that the sign of x satisfies x to the C in M if and only if there exists a function in M, which injects x to the C's. This is not the original move, but it's the one that we need to do. In practice, we want this collection of injections to satisfy certain closure conditions. We're not just letting it be anything at all. We don't want it to be empty before, okay? Okay, so, for instance, we want these two things to be true, that for each subset A to B, we want the identity map on A to B, including the identity map on the empty center, so the empty map. And moreover, we want to be able to extend, to get finite extensions of these injections. So if f, little f is in the collection, f is made to b, and you pick elements x not in a and y not in b, then you can find the g, the collection, which injects any union x into b or y. Okay, so we're going to be able to extend these maps one at a time.

32:30 And we're going to build that right into the definition of general models. Okay, so our general model had this collection earlier that satisfies these conditions. Now that we have it, we're going to start looking for a system of axons. It is a candidate for such a system of axons. I say candidate because I can also tell the So we want to put in all the propositional analogies. Fair enough. Then we want to have something which gives us universal distanciation. And so we want to say that if there is an injection of phi into the empty set, then for any term t, which is phi for x in phi, then not phi t. So if there is an injection of phi to the empty set, then And therefore, no matter which p you pick, that's going to be, f is going to be false. Okay? That's essentially the universal instance here. We want transitivity of f, if there is an injection of p to psi and psi to theta, then there is an injection of p to theta. You could have built that into the semantics, couldn't you? It's quite getting closer on the composition. What if I ask what time would I be finished? What time would I be finished? Yes, I could have. And then we have this clause that essentially says that if A is subset of B, then of course And finally, this axiom that corresponds to the fact that these injections can be extended, so if you can inject the phi's into the phi's, and x is on a phi and y is on a phi, then Then you can inject, this is the phi union X into C union Y.

35:00 I don't think I have it on the slide. In fact, we also have it on the slide. Rules, we have most ponens and generalization. So generalization, again, tells you that if C is an eigenvariable, and you're able to put it down to C, then you can infer that there is an injection of C. That universal generalization. And then, of course, the task is standard science and completeness. Now, science is fairly straightforward. it. Of course, all of these axioms are valid in all the standard models, but that's not what we're interested in. If you have all these, all the injections, then you have all valid. We're looking at the general models. So the one that, and I'm just trying to signal out when you need the closer conditions. So the axiom that if you raise a sensation doesn't The transitive of f is likewise valid. The inclusion axiom belongs, holds as long as the identity map on each A is in the collection of injections, which we assume it was. Then you can view the identity on A as a function from A to B, of course. And then the other axon, of course, just tells you the extendability on the finite unions of those injections. So that's not a problem. And you can also show that the rules to preserve a linear model, of course, problem, of course, and then use the existence of the empty map for the soundness of universal generalization. Okay? So, the actions appear to be complete for the classical norms and the models satisfying the social conditions, and when they appear, there is because I haven't had the time to work out this case. All right, so it looks like that we have now something and more manageable, right? The interpretation of the fragment modifiers that allows us,

37:30 that this seems more tame than standard interpretation. So let's see what we can do with it, okay? And one thing that I thought was interesting to try to do was to do some arithmetic. Now, of course, to the extent that you have for all and there is in there, you can just take the piano-ledicking axioms and be done with it, okay? So that was too easy. what I would like to do is do something else I would like to try to see if we can find a first order theory which formalizes arithmetic in the Frege-Russell tradition that is, I want to think of numbers as either equivalence classes of equiluminous concepts or something which is related to those equivalence classes so I want to think of numbers in that And this, again, is something that the motivation to do this is that usually people think, well, if you want to do Spreger-Russell-like, then you have to go higher order, because you're probably not going to be able to talk about it in the classes. And I think this is going to tend to show that this is not quite true. Okay? So here's the language. The language contains non-logical constants. Okay? Zero, less than. A one-place predicament, n. I'm going to head myself. Let me get you. there okay so zero is a logical is an individual constant and it's a binary relation and is a one-place predicament single the entire section is at the national numbers and then the what is going to allow us to do some play some tricks is this operator here no which takes a term and a formula and returns the formula right so I'm going to show you what it new XP expresses the fact that X number is the P, the P's, okay? So, the fact that there are X objects satisfying, satisfying. So, you think of X as a number. What number? The numbers of the P's, right? Now, I sort of lately, only lately realized that perhaps, In this case, there would be no problem using an operator instead, right?

40:00 So the fact that I use this operator allows me to say which fees have numbers and which Okay? Whereas if I use an operator, then I'm assuming that this is total, so I don't want to go there. I'd rather use special axioms, the existential axioms, specify which formulas are related Okay. So that's the language. And I want to put in there some actions. And they're broadly divided into two categories. One is the definitional and uniqueness actions. I think those as not doing much of the work. And then the second class of actions that, intuitively, those are the ones that do a lot of the work which is needed in order to keep some arithmetic OK, so let's look at the less problematic ones first. This is just an abbreviation, a definition, OK? It's a definition. It says that the numbers of C succeed the number of P's, right, if and only if there is an object in C, such that there are as many P's as there are C's other than A. of success. Then, something which allows me to define the ordering, and I put that in scare quotes because it's not really a definition, it's more like a conditional definition that Hannes mentioned earlier today, it tells me that if x numbers the p's and x numbers and y numbers of the C's, the dangers of color paste, then x is less than y if there is an injection of the P's into the C's. Okay? So that tells me how the ordering relation at the first order level is related to the variation of the carinology of the formulas at the higher level. But again, this is all first order. higher over there. Hume's principle, right? And again, the problem is to kind of paste. So if x numbers are t's and y numbers are c's,

42:30 and there are as many p's as that are c's, then those x is the y. Hume's principle. If phi and c's are equinumerous, then the numbers are the same, which is the... Hume's principle is usually the bi-conditional, but this is the... we lost and this is the part that does the work for your principles but i think of this as imposing some uniqueness condition it doesn't pop up the size of the domain at all yeah okay which is what i think of in the real work so you say it doesn't pop it doesn't increase the size no it only forces you to identify certain things right so far right it doesn't there is nothing that tells you that No, right? It's all conditional. If x number is the p, then... Oh, okay. All right. We don't know that anything is numbered yet. No, not yet. Okay. Um... Excuse me? Nothing. That's why you put Hume's principle in scare quotes. Why do you put Hume's principle in scare quotes? I don't know. Maybe that's the reason. Maybe that's the reason. And again, the definition of that is not a real definition. for a reason, right? Because it just tells you that something is a number, if it's 0, or else there exists a number y such that x number is equal to y. Okay? So, what I really want to put there is, right, something is a number if it's equal to 0, or the successor is a smaller number, but I don't have a way of saying that yet. So, I'm just going to then x number is the princesses of y. Can I just write to recursion action? It is kind of recursion action, right? Except that I have no way of enforcing the recursion here. It could be also, there's just a sentence that I put down there for people to contemplate, he doesn't force any closure on any, right? There's no way for me of saying that any is this small as such. See, is zero part of the Uh, probably not crucial, right? You could sort of define it. It's the unique, given uh, your principle, that's the unique number of the empty set.

45:00 And now we get to the axioms that actually do some work. Uh, this one that tells you that zero numbers, the empty set. No problem. That's why, given the other axioms, this is a unique object in numbers, like you said, so then you can introduce, you could use a definition of extension instead. So far, we've got finite models of this. Yes. This, of course, is the domain to be interested in. This is the one that does the work. This is the existence of success, right? It tells you that if x numbers are p's, then there is an object in numbers that are a sense of x. That pops up the size of the room for the domain. At least in the intended things that work out in the room. But interestingly, because you need also a principle which, in certain regions of the literature, is called counting, for instance, I don't know how many of you have perused in your literature on finding foundations, but there is a maximum of counting there that says that the number of the predecessors of n is n plus 1. And when added to the foundations, in fact, it produces a noticeable increase in consistency strength. But we need this here, let me face that, I think that, I don't know whether we need this here, right? In the best possible world, this would be a theorem, but as we know, this is not the best possible world. So, anyway, that's why I called it the principle, is there an action, because I'm still hoping that it might turn out to be good. In any case, it says, as I said, that if x numbers and p is then, the probability of, the predecessors of x is a success of the probability of p. That's the actual count. It says that there are n plus 1 things that precede n. And the principle of induction.

47:30 So I look for a way of expressing some kind of induction. And one way to do that is to say that the set of predecessors of any of these functions. A lot of these notations can be simplified if nu was an separation instead of a relation between formulas and terms, in which case the treatments of success would be built right into the notation. And I think that's a little bit cheating, right, because you don't want such a good thing to be swept under a rug like that. I think I'd rather stick with this, the way of putting things. And then, as I said, this principle of induction, because the seven parties, induction is meant to enforce in any formulation of some kind of minimality, and so one way of doing that is by saying that the seven processes of any x is finite. Okay, and again, this is all proof, but certainly we can represent a number of facts about the the target result is to say that, well, if you expand the language in a proper way with cross-site types and so on, you ought, at least, to be able to interpret Q, if not perhaps I just don't know which way it is going to be. That's a set of actions there, so therefore, again, people to contemplate in this way. This is more like a browser, but I think it's already something to have a set of axioms. OK. What else is there? Aha! OK. Now I'm going to play a slightly different game. OK, so we started out with the... how much time we got? Well, in six minutes you talk for one hour, but we know we have flexible. Okay. Alright, I think I'm going to finish talking about this and leave the rest for any questions that might be. Okay, so here's what we did. We started out with the standard interpretation of the Freud quantifier, right, and then in the usual way, that was the first order quantifier, right? But then we gave it a general semantics,

50:00 a non-standard semantics. So we discovered, by doing this exercise, that it is possible to give non-standard interpretation of first order quantifiers, which is something that we didn't quite know before, right? Because before, the only non-standard or general interpretations that we had were for second order quantifiers, or higher order quantifiers. OK, so let's take our old friend here, the existential quantifier, and see if we can play the same game. So it just notes out that the existential quantifier can also be given a non-standard or general interpretation. So what's the standard interpretation of the existential quantifier? The existential quantifier, as we saw at the beginning, is just a collection of all the non-NFT subsides of the domain. To give it a non-standard interpretation, we just identify it as, we require that each one of them provides an A non-empty, we have it, A, no, a collection of the non-empty substance of the non-empty substance. So instead of looking at all the non-empty substance, we're going to sum them. And we call the collection curly E, and then, of course, we change the satisfaction relation, sorry, the satisfaction definition to say that S satisfies that X is X phi, so the extension of phi according to X, to S, belongs to the collection. One thing to notice is that as soon as you do this, well, this quantifier is not permutation environment. Because you can take a permutation that takes a subset of E into, sorry, that takes a subset which is a member of E into a subset which is not a member of E. So it's not a permutation environment. And so, well, so maybe this is not such a good thing to have. I mean, I don't know. But some people might be led to question the logical character of this quantifier still the question arises like what is the logic of this quantifier well again somewhat surprising the answer to that I did find the answer a little surprising okay so what what logic of the generalized existential so the standard semantic sorry sorry the one way of

52:30 given the general semantics of extension modifier is to have this collection, this e, which is the subset of the power of 7e minus 3md. Now, instead of doing that, we can equally well use an inner domain and outer domain semantics. So we have an outer domain and a subset of which is identified as an inner domain, and then we can formulate the semantics by saying that s that is y is x, if there is a d in the inner domain, that the shifted assignment satisfies phi. These two concepts, the way of doing the semantics are equivalent, I would go back and forth between So, if you have a model of the second kind, you can pick curly E to be the settable subsets of D that have 90 per section with D prime to the union of D. And conversely, if you have this D prime, you can define, sorry, if you have E, you can define D prime to be just a union, okay? And the straightforward induction tells you that the induction of formulas, that is the first course of satisfaction. And so the set of validities, under the two ways of doing semantics, are the same. Not equivalent of the cursive, you know, some motion, right? They're the same. Aha! But the logic of this second semantics is well-known, right? This is a free logic, which was axiomatized by John Lambert, one time ago, by dropping the axioms of universal instantiation and replacing it with S, right? It says that if E is an existent, then you can instantiate the universal fortifiers. So what is the logic of the generalized existential fortifier? It's free logic. Okay? It's free logic. Aha! But Joe is not going to like this, because the non-standard semantics is not the realization And the need, of course, is the outer domain in the domain, because you can have a presentation that moves objects and not together.

55:00 Now, I think that this is a slightly more serious problem, right, because the—let me give you just a 30-second history of this—the inner-domain algorithm semantics for freilogy was initially proposed by Church in a review of some of the early papers in the field, by saying, oh, you can easily keep the semantics for this, for free logic, by doing this. I think it was a way of realizing the whole thing. This was taken seriously by different logicians, right, saying, ha-ha, yes, the semantics, complete and serious, and so on, and so forth, right. But still, the fact remained that the inner domain, outer domain semantics for free logic was considered to be somewhat artificial. Why artificial? Well, because if you take seriously the idea that the objects in here are non-existent, right? Well, the point about non-existent objects is that there shouldn't be any, right? Whereas here in the other domain, there are lots of them, okay? So the phylogicians were never happy with this, and so the, you know, I'm assuming that they might not be totally satisfied with, uh, uh, uh, nobody going to impress with the charge of the non-logical character of the existential quantifier of free logic, okay? But now, look. Now we can get to the same point by coming from a different direction. Now we have a well-motivated way of getting to the same logic by doing something which is a standard operation of relaxing certain constraints on the distribution of quantifontics to get to the same usage. Now, this is not an artificial move. It's a well-motivated way of getting to the same semantics. And so this charge that the extension quantifier in large degree is not really a logical cost. Now I think it's a charge that runs the risk of sticking a little bit more than it did before. I think that Saul, I just want to end with the moral of the story. Oh yeah, I know that Hannes isn't like all this. That's why I went back and threw him on out. but I had to leave this one in. OK, so what's the moral of the story? The moral of the story is, ta-da! There is more to preserve antipirates than just what there is. OK, and if we come away from this talk, which is that talk, I think I'll be happy. That's just a list of references that I can make available.

57:30 I think that's all for now. I just wanted to ask for a guess, you said that if you set up arithmetic with the F-quantifier, you expect that you will get Robinson arithmetic. Why not something stronger? Yeah, I have induction, in fact, I'm holding a hope and I'm going to get that, right? So you don't have a good reason for saying that? It's a funny way of expressing induction, so I'm not sure it's going to do all the work that we want to do, you know, with nothing like time, and I haven't had time to find it. Michael? This is just a clarification. I lost track when we were talking about the axioms where it actually used f. So basically, I just wanted to... The axioms for arithmetic? Yeah, I just lost track of where we actually used the quantifier for most of the axioms. Was there a use of I at some point was that? Okay, so, first we have these axioms. Okay. Oh, right. We got it. We got it. It's kind of a weird place. We got it in the first and the second. Mm-hmm. Oh, and... And here's principle. Okay. Here's principle. And also, here, let's see... I just noticed that many of the axes are actually axon schemes, rather than axons. I just want to point out that they are axon schemes. Yeah, I should have said that. Thank you. Yes, there are some schemes. Why? Well, because what you really want to do here is, yeah, but I don't want to go second. I mean, you have no comprehension principle. Presumably you could replace all these. Yes. I think if I had to do it, I would just have a second other comprehension principle and then. Replace replace everything with x, a couple of x. And all the fees by .

1:00:00 So wouldn't there be a... So suppose we had smaller equal given or something like that, right? We had, I'm sorry, what? Wouldn't there be a way to get arithmetic not using schemes? you're really aching to go second order. No, no, no, without second order, but you could use very special phi's, right? Oh, yes, yes. I think that's right. In the end, all the phi's that you're going to need are of the form, right? Because these are the ones that... Precisely, yeah. That was my point. Yes, right. And so maybe it would be circular, but then you could simply stipulate that you have smaller or equal, or something. Right, yeah. No, I think what needs to be done is, we use these axioms, and we go through these axioms and derive arithmetic from it, and then go back and inspect the proof and see exactly which phi's were involved. And, um, and... But the fact that you can write down this axiom of infinity in one, in this, in the standard semantics, right? So Jay would suggest that maybe there could be something done using No, this semantics doesn't characterize the standard structure, the standard model, of course, because it's a general semantics. And then there would be some analogy, rather, I think, to ACA0 or something. Well, it would be exciting to see what precisely it is. Yeah, this is a work in progress. The language you use to state the semantics is, of course, I guess, the usual first-order language, because if you give the interpretation of the language in which the semantics is stated, it's more the first order, right, in fact, right, but it's a first-order language with that theory oh well yeah yeah and you should want to know all and there is so probably you explain that the interpretation of the existential quantifier the set of all x x is a subset of d and x is not empty now how is not empty explained probably by the reason why such that why i don't want to go that way i want to say that the language which the semantics is given is a

1:02:30 It's not the accented mathematical English. Which the mouse is the same thing, actually. Yeah, so I think it's not bad, it's OK. It's just to the model. If the model is there is more to first-order logic than first-order languages, then you should quantify it. So in some sense, it's not more to the meta-language you do semantics than the usual quality. I use semantics in any way I like, frankly. But you have this existential quantifier there in the same particular language. And if you explain things like X is not empty, how do you explain it? Is it primitive? I could ask, what do you mean by X is not empty? X is different from the empty set. And I hope you would explain it using the usual quantifiers and not take it as primitive. Yeah, of course, yeah. I would use it using the language of mathematics, which is the language which is appropriate for the enterprise. So I think the meta-language, which you use to formulate the model theorem, uses the usual first order quantifiers, not the new modes. It's not circular what it does. No, no, no, okay, yeah. Yeah, but also set theory. Right, right, of course, but that's not the new quantifiers. No, no, it's a strange sculpture in the sense that you were using. Sure, sure. One of the interests in looking at... No, I just wanted to get something somewhat weaker than very, very strong set theories, which would still give some of those properties. Sure, but what I wanted to ask the second part is a question about Friege or a... So when Frege says, explains to existential quantifiers by saying a concept is non-empty, or something falls under the concept, it is not clear for me that he, in the end, will explain being non-empty by a quantifier in our sense. So maybe being non-empty is really some higher-order predicate. Do you have an idea of how he wants to understand it? No, I don't know how he wants to understand it. I mean, except for the very first thing I said, right? I think that there's a few places in the Gungesite where he talks.

1:05:00 He gives this argument, right? Section 21 is not one of them, right? But, of course, being on empty is, for Frege, it's a second level concept, so it's a high order. But you might have the idea that if you ask him what does not empty mean, that he explains it using a quantifier. Not in a higher order predicate sense, but in a quantified sense. And I don't know passages in his work which speak, be away with respect to this question. Well, don't you just have to go from his definition of what is the German letter and the concavity? If you know what that means, and you know what not means, then you want to know what. So he defines the existential, quantified by not for all, not. It's not like Frank gives a formal semantics of the universities. I'm not sure about everything that's going on, but you said that you could use to characterize the natural language. Under the standard interpretation, yes. Under the standard interpretation. I just wondered how that related to the kind of arguments that the Jewish period has given I think there's a misconception going around, which is that anything that goes beyond the reason for all is higher order. And that's not true, right? That wasn't the point of my talk in some ways. There's a lot of interesting things that go beyond the standard existential universe quantifiers without going higher up than first order. To be fair to Stuart, in his book, he has a whole load of stuff about intermediate things between first order and second order. Intermediate levels. But I think the proposal here is that first order is really, well, the one the entire range over, right? It's a semantic of property, right? It's not like there's anything intermediate Yeah, right, between first and second order, right? Look at what the higher, you tell me what the type of quantifier is, and I can tell just by looking at it. But then I think there's a problem, because what we call first order logic, we should call, according to you, standard first order logic, right? Or something else, yeah.

1:07:30 Further questions? Well, I'd like to add a remark to what you said at the end, concerning inner-order-domain thematic. In his book, A Logic Consequence, John Etchermandy criticized Tarski, because Tarski presents the notions of logical validity and so forth in terms of this old Zano-like style. so take it let us forget about arguments just just sentences so you you replace all the descriptive terms for the corresponding variables and then you universally quantify them and if that sentence comes out as true then the original sentence would be logically true something like that and you also have to do that with the extension on the wire according to Trotsky because at least if you've got this idea that you want to fix the domain of quantification and but But then what John Edgerman criticized was there is an additional constraint, because, for example, whatever is denoted by a singular term should be a member of whatever is the extension of the quantifier, so whatever is the domain corresponding to the quantifier. They have to fit together, so to speak. And that's not natural if you have this idea, so that's a kind of critique that he said. And then I asked him about that, because he could now think, well, let's fight the bullet and say, okay, let's look at the logic where you replace the descriptive terms for the variables and take the universal closure of that without this additional constraint, so what will happen then? Obviously, then you need a kind of free logic. A singular term might know something that is not within the domain of quantification. And it's easy to show that that gives you semantics, and that's the amount of domain semantics. So the logic that comes out of that idea of the Bortz-Oak-Bowen-Bortzano idea, and later the positive free logic that you get corresponding to the inner-order domain semantics. But then, as you said at the end, one would just have to say that the existential quantifier is not a logical sign. But still, you get a logic. I mean, you get a logic according to the Volcano style. That's a quite good theory. It depends. If you think this notion of proposition invariance is the hallmark of logicality, then Yeah, there's an issue there. Now I'm myself torn between this, right, because, well, I also gave the semantics of positive theology, which is not in another domain, I think, and so I have a vested interest in promoting the cause of positive theology, but I think there's an issue here.

1:10:00 in this problem this is just a lot off the top of my head when we were talking about this issue about what makes first order or not i mean there is a there is a different kind of model theoretic characterization of historical logic floating around from from lindstrich's theory oh yes um and this is really a very off the offhand of thoughts i'm not sure where this is going but i mean in fairness to somebody like shapiro all these logics of the kind of you're looking at theoretically much at all right for historiologic standard first yeah very confusing yeah yeah but i think alba was right about the terminology You know, the little theorem applies to the laws, you know, what's the next problem is, but irregularity, compactness, and so on. Yeah, and I was wondering about the way to draw the kind of object. Yeah. Well, I just have an extra comment on that. It's that usually when the argument goes about, say, most, sorry, more A's. Mm-hmm. There are more A's than B's. of the quantifier is that although you might think of it in that field discuss this last chapter of his books, he wants to use a quantifier like this as primitive, you might think that what makes statements using those expressions true are just the A's and the B's, but when you try and define them using the techniques that we have, then we have to refer to functions, let's say there is a function from the set phase, so it's Only when we have that kind of project of saying, how do we explicitly define them, then things that can kind of get bumped up in a set-favorite or get bumped up into quantifying over functions. Whereas intuitively, you might want to say that the things that Fure's are actually referring to are the first-order domain, so they are first-order quantifiers. And in fact, if you were to go with general models, then assuming that we have the completeness proof and so on, You might be happy with just doing the document, right? Because you don't have to talk about injections of functions at all, or you have a calculus, right, in which you grow consequences about this quantifier, so as with the F quantifier,

1:12:30 where we have a semi-semantic level at all. No. We've still got time. No, but if there are one more thing to say, if you want. Yeah, yeah, that's a good idea. All right, so there's been a lot of talk about the recent work of Green and White about the logical epistemological studies of cognitive notions. And these guys have focused on Hume's principle, right? by specifying identity conditions that will make of terms in the screen. So, you will say, you know, the number of F equals the number of G, even if there is a one-to-one correspondence of the F onto the G, and where this part on the right-hand side is the second statement, right, that expresses the existence of such a correspondence, okay? And that's taken his principle to have some sort of privileged epistemological state. First it's analytic, then it's privileged epistemological truth, then it becomes challenged, it becomes analytic, then it becomes challenged, it becomes explicative, then it becomes challenged, it becomes constitutive. But in some, it has some sort of privileged epistemological state. And the point is that the operator's number of, right, maps concepts into the objects in such a way that the humanus concepts are mapped into the same object, okay? And the right-hand side is taken to be logically innocent, right?

1:15:00 So this part says, well, this is just pure logic, right? Say a lot of logic, I mean, it's pure logic. And the fourth, if that's purely logical, then it's logically innocent, then so it's the other side as well. and the equivalence between the two, but, you know, some of the motivation for this exercise is that if there's a notion of equivalence to be so logically innocent, right, then it seems interesting to see what happens when we take that decision as we did by positing this quantifier, right, and see what happens, right, but they take that quantifier only, put it in the barestoological practice and see what happens. Well, a lot happens, right, because what you get has a lot of expressive powers, right, you can categorize it from the, you get axiomal infinities and so on and so forth, right, you can categorize it as times and models and actual numbers, right, so it seems to me that we have, yet again a stronger case in favor of Bullock's objection to the program. Bullock said, well, this notion of carnality or equinomenosities is just too strong for it to be logical. And, in fact, this is perhaps yet another instance that when you take it and you drop it in, a logic, you get something which is really, really very, very expressive. And so perhaps, if there was such a thing as a philosophical fallout, this would be it. So perhaps, here again, another argument to question some of the claims of the neologistist a program in this sense, right? Because if you want to say, cardinality is a logical notion, well, let's see what happens when you take it seriously. Too much happens, right, when you do less, and so perhaps that should lead us to question whether really cardinality has this logical character, in spite of the fact, of course, that you have permutation in practice. So it's a somewhat different take on permutation, on logicality than pure I'd just like to make a historical point that Fraga and Hale and Wright, of all three,

1:17:30 misunderstood what Hume said in the treatment. And interestingly enough, the notion of number that Hume uses is the notion of number that Aristotle and Hume used. He's talking about equality of numbers. It's the same notion of number that you find in the Dixieland jazz song, I want to be in that number when the saints go marching in the night. What you want to be there is in the set of the saints. So that's the notion of number, Hume is using it. So what's not innocent in this is to think that there has to be some hard object, which is some entity, which is the number of x's, the number of phi's, and the number of psi's, and so on. That's how you get, that's how what seemingly innocent remarks of Q get blown up into an axiom of infinity. Yeah, I agree. I would also say that in order to get enough arithmetic going, at least we probably will not agree with this, but enough arithmetic going to get Q, for instance, instance, you do need some injection of the concepts into the objects using Prager's terminology. Okay? Now, you cannot have an injection which is total and, right, which is total, because otherwise you get Russell's Paradox or Cantor's, you know, we run a file of Cantor's theorem, right? But you need some way of assigning objects to the concepts in such a way that you get this feedback loop point, which is the one which is in the successor that I had, which is also the same as Frege uses in the basic principles. That may be true, but don't voice that off on you. No, no, no, no, this is just, it's just that the label is, I think, it's stopped now and there's nothing we can do about it. Charles Parsons tried for years to get us the psychiatrist principle and nobody listened to it. Can I add something? In it, there was this best trip for Dummert, where Boulos published the paper, also always used that time. Dummert, I think, also objected very often to call it a few different things. And then Boulos added this footnote that, well, since this is the best trip for Sir Michael Dummert, we will call it HP.

1:20:00 that was all thank you very much I'm going to just make a short announcement tonight we have to leave by the main entrance we also have to come back in by the main entrance tomorrow morning because it will be the only one open at the weekend so we start again tomorrow at 9 of course if you arrive extremely early before either John or myself, who won't be able to get in, so... Is the back door going to be alarmed? Please do not use any of the other exits tonight, or tomorrow. I expect you can jump out of the window. No, despite all these alarms, I wouldn't... But please just use the front door from now on. So, at 8pm tonight, we are meeting for dinner in the Sands restaurant, which is on Queens Road. I think it's number 96, Queen's Road.