Theory & Object (Session 5)

Hello and welcome to the fifth session of Theory and Object. I'm going to pass the mic to the course instructor right now. Thank you very much everyone. So we are going to work on the work of SNID. As a similar, we are going to look at the projects of SNIDification. what FireAband calls a scintification of scientific theories, which essentially means investigating the logical structure of scientific theories and see according to this logical

structure how the evolution of science can be reinvestigated, can be re-seen both in terms of the normal course of science and also a Kohnian paradigm of scientific revolution with theory dislodgment. So because last week you know we didn't have course If you have something on the topic that you are vague about or you need to kind of have a more better grasp, please open.

Otherwise I will just go right into the topic. I wanted to ask two quick questions, if I may. Sure. I have, the first one is pretty basic and I think I just skipped over it for some reason, but in the first chapter, when he refers to, he calls it informal set theory, does Stagmuller refer to like a Kantorian naive set theory or does he refer to… Hilbertian, informal Hilbertian set theory, yes.

Okay, okay, yeah. And the second question that I wanted to ask is more of a topic-related one. So I tried to find for the last couple of weeks or even months as some sort of coherent summary of what Burbaki's project with his new kind of formulation of mathematics was like, but I couldn't find anything actually coherent. And since this book kind of really aims at such a method, could you possibly maybe elaborate Burbaki's work generally? Okay, the second question, I will try to work through it both in this session, but more specifically as soon as I start off about Carnap's logical structure of wool.

I will get into that. So the first question, so are you asking that what is exactly informal Hilbertian theory and why is it called informal? I mean I sort of understand that he basically tries to stay away from the formalized axiomatic aspect of Zermelo Frankel's theory. No, not really. Not really. That's not the case. You see, informal, actually they don't, actually a Stegmuller is not informal, what you might call, he doesn't propose, he just basically shows that there are these, what you might call the hierarchies of axiomatized systems.

The axiomatization of which belong to a different class of axiomatization. Like for example Euclidean axioms are simply intuitive axioms. They are under defined most of the times which means that they are undefined. In a sense that like what we have a point or we have a line. A line is not really well defined in basically Euclidean system. It's just an intuitive data that you work with. And then for example, depending on how you are approaching it, you can say well that a point, another elementary axiom of Euclid, is made by the intersection of lines.

But then what would be lines? We don't know. It's just that we have a geometrical intuition of them. So then we have different classes and like almost toward the top, we have informal and formal Hilbertian axiomatics. The difference between the two is not really that it does not correspond to the formal axiom of Zemmerlofranco. The reason that it is called informal, it is essentially the axiomatized system that we developed out of the informal Hilbertian axiomatics is what you might call to be a a system that can be couched in terms of set theoretic formulas or sets and all their properties.

But nevertheless, the sets that we are making are elaborated within a natural language. Hence, it's not that it is naive set theory. it is still set theory but a set theory whose axiomatization and relations between the sets which are obtained are not elaborated in set theory itself but in natural language. So that's the point. Whereas the formal Hilbertian axiomatics we do basically what you might called to be couch our system in terms of set theoretic relations and properties.

And even when we are trying to elaborate the relations between sets or the logical structure, it is not being done inside natural language but only set theory itself. So this is basically the difference. So he's using formal as in proper formalist mathematical sense? Yes, absolutely. One that cannot be done in natural language. Yes. Perfect then. Thank you for clearing that up. Any more question? Anything? Yeah, I was just hoping when we go into the text that you can elaborate more what exactly Carnap's approach and Sup's approach or the statement view and non-statement view.

How, just sort of give us a breakdown of what they are. Yes, yes, yes. I will go. This session, first of all, I mean, you remember that when I was trying to give an introduction to Stegmuller, essentially I saw it as the extension of Carnap's view, the structuralist view. And I highly recommend reading of Bao's introduction, logical structural world. But particularly there is this rather very interesting and curious section in Carnap's work which sheds light on the motivations behind this structuralist formal

approach that extends from car nap to a sneet and a stegular. It is section 66 of Baal. Also read the prior and the succeeding sections, before and after sections, because precisely then you see that what they are trying to do is essentially what you might call to be within general historical philosophical context. Essentially their project is in response to Kant and Horserl.

The idea of transcendental deduction, how can we make objective claims about the world, What do we call objectivity? How does objectivity has anything to do with intersubjectivity? And then how intersubjectivity can be really understood at different levels by way of transcendental logic and formal logic. And just those of you who weren't in the Kant session, transcendental logic is what you might call to be, so we have two different kinds of logical rules, logical laws.

Transcendental logic is essentially what you might call to be about, it's about the application, correct application of logical rules to our sensory experience. Okay? Whereas formal logic is about the kind of rules that is being obtained without any reference to experience whatsoever. Now this is actually a very interesting thing. You see, you think that, for example, formal logic essentially means that it is non-intersubjective,

precisely because we think of intersubjectivity in a continental sense. Intersubjectivity is about our various streams of experience which are diverging and hence They need to correspond with one another so we can have something like correct application of rules that can be applied to our individual experience as well as commensurating these conflicting different experiences that we have of, for example, one and the same object. Now the thing is that Carnap's belief is that intersubjectivity, so as a little bit of also, think that intersubjectivity require exactly like logic to be basically

distinguished there are different classes of intersubjectivity one an intersubjectivity that we might call to be experiential intersubjectivity or personal intersubjectivity like the one that content of philosophers usually we talk about but also there is a logical intersubjectivity imagine that everything that happens in the world of logic as divorced or disconnected from the world of experience also has its own intersubjectivity that for example like in today's interaction is paradigm of logic like Jean-Yves Girard and so on so forth, we see that the meaning of logical

connections are being given by how logical processes interact. That's also, this interaction is also an intersubjectivity. Okay. So, let me start with a little bit of SNEED, which I told you that, my apologies, Essentially, Joseph Snedt's The Logical Structure of Mathematical Physics was one of the most

important books written in philosophy of science in the 20th century. And precisely because of the complexity of this work, it remains obscure to, you know, non-expert philosophers and particularly philosophers of science. Nevertheless, this is precisely the work through which Estegmuller tried to investigate, criticize and somehow confirm the view, the Kuhnian view of scientific progress.

So essentially, Sneed begins his project with two things, two main topics. One is logical structure and the other is axiomatization. The notion of logical structure of a scientific theory is actually a notion that provides a logical reconstruction of an existing scientific theory.

Dar esnith introduces at least three methods for this kind of logical reconstruction of all scientific theories. So as to study coherently and clearly without any kind of vagueness the courts of scientific theories, how they are being constituted, how they are being shaped, but also how they stand in relation to one another whether one theory dislodges another theory whether a theory moves forward and essentially makes you know still contribution to the course of science

The thing is that SNP thinks that the nature of how we logically reconstruct scientific theory is not by any means a clear-cut issue. It's not even intelligible as such without adequate constraints and adequate qualifications. First we should know that the most common accepted view of scientific theory is something like

that scientific theories are sets of statements, some of which are empirically true or false. This is what you might call to be the most bare minimum and widely popular claim about scientific theories, about the nature of scientific theories. This is of course a general claim about the sort of thing scientific theories are. It is of course a very plausible claim. Even those who believe it to be false or misleading do not deny this. Presumably what one finds in the textbooks and journals pertinent to a particular scientific discipline

are statements. The sort of thing one may properly claim to be true or false. At least some of these statements are claimed to be true or false because they are or are not supported by experiential data or observational data. And presumably also it is at least in part scientific theories that are being expounded in these textbooks and journals. Even though this claim, this popular claim is perhaps more widely accepted than most philosophical thesis about the nature of scientific theory, its acceptance is by no means universal.

There are those who claim to see beyond its superficial and vulgar plausibility and propound and more sophisticated view of scientific theories. For the moment, we can stop thinking about these other alternatives to this popular view. Even if A is usually regarded as plausible, even the proponents of more sophisticated accounts of scientific theories, it is not usually regarded as interesting, even by those who believe it to be true. It does not really say anything very explicit about what a scientific

theory is like. What is needed to characterize more explicit scientific theory? That's the real question for as needed. Philosophers who accept this claim generally believe that sets of statements which could constitute a scientific theory may be distinguished from those which could not by their logical structure. A statement may stand in various sorts of relations with one another, which we call logical relations. So when we are talking about logical relations, essentially we are talking about various NRE, you know, NRE relations

between statements, scientific statements, statements in a theoretical sense. And one thing before I move forward, I remember that I promised to you, give a very, very, before moving forward, a very brief definition of what we call theory. So obviously we are not talking about cultural studies or such cargo cult philosophy stuff. are talking about what actually theory is so theory formally as speaking is a is a triple

is a tuple is a well-ordered set consisting of three elements L s and you so what are these L is a language by that we essentially do not mean sorry we do not exclusively mean natural language, language in general. And specifically within the domain of scientific theories we mean formal languages, not natural language. Language as basically

a vehicle or a framework by which you can arrive at syntactical semantic, basically by which you can investigate syntactical semantic properties of the very expressions that are being made in that language and essentially in order for something to be language it should at the very least I'm saying that at the very least should have a symbol design the symbols are meaningless they absolutely have no semantic features okay you can think about sounds you

can think about text inscriptions they are basically just inscriptions inscriptions can be attained by any means possible. So you have symbol design. But symbol design, you require that every symbol to stand in a syntactical relation with another symbol. And hence the The concatenation of these symbols or syntaxes also stand in relation with other expressions made out of symbols or inscriptions. This is what you call syntactic. So this is what we call a language.

I mean at the very least, at the very least, we are not essentially talking about the idea of semantic at this point. that this is the most rudimentary characterization for something for framework to be called a language so language L is language S is a structure so what is exactly a structure A structure is precisely and in the most broad sense is itself a set of NRE relations between the stuff

that can be addressed, that can be indexed, that can be captured by your language. So a structure in this sense is not something in the universe. A structure is essentially what you might call to be a subclass or the subset of the domain of what Plato calls a logoi, language and logic. What is you? The third element of the theory. You is what you might call to be the universe of

discourse. Those of you who know universe, the term universe of discourse invented by the morgan the logician essentially universe of discourse what you might call to be a domain in which you explicitly you ex this is really important explicitly a study a particular set of properties a set of relations and a set of entities.

So this is what essentially, if—so in order for a theory then to be systematic, It should, so for, let me, sorry, I flash forward a little bit. So every theory should have these three elements. Without these three elements, there is no such a thing as a theory. All so-called theories implicitly use these three elements, even if they deny it. However, for a theory to be systematic, like philosophical theories or scientific theories,

the requirement for it to be systematic is that they should be capable of explicitly express and elaborate the relations between the relations within L, the relations in S, the relations in U individually, but also all these three elements, language, structure, and the universe of discourse explicitly. So it's essentially what you might call to be systematization, what we call systematization

or systematic theory is the explicitations of how these things hang together. Any questions? Sorry, if there is anything vague or… This third-fold distinction comes from Pantel, correct? It's actually coming from Frigge. Oh, originally from Frigge. Yes, it comes from Frigge, then Carnap, then Stegmuller, and then Pantel. Okay, okay. Yes.

So I just wanted, precisely because we have thrown around the word theory, the word structure, the word systematicity, these things I understand that in general philosophy are never paid attention to and there is quite a good amount of vagueness around them. So I just wanted to make this clear, what we are exactly talking about, even though it is you know abbreviated so well okay before I just I have I one minute I'm I'm coming back. I'm coming back.

Thank you. Thank you.

I actually forgot to tell you one more thing. I'm sorry. Justin, which one are you talking about? What author? uh artem had asked me or had asked about burbaki and i just um put out a link of something that i had found when i was interested in background on him it's not it's not it's not it yeah burbaki is a collective yeah uh it was actually an interesting article for me i did i didn't go so close i don't want to sidetrack it but it it's that that chapter of that book um was sort of suggesting that

there's like these um smuggled in aspects in borbaki's claims of like creating this pure structural aspect without philosophical baggage but there's still baggage being brought in which i thought was like an interesting question that kind of like carries into the background of reading stegmuller and in terms of what other assumptions are being smuggled in inside of the framing of their formalism but i don't want to sidetrack the this conversation here but just very quickly i mean you know the the essentially the the the the intention or the mission of the burwaki group or working collective was that um a structuralist view of the mathematical

universe in the sense that what you might call to be a meta theory or a theory of how this universe grows and how things are standing in relation to one another how mathematical objects being produced so on so forth they They thought that there is possibility of a program by which we can in fact give a view of this mathematical universe, of mathematical universe as such. And that's what you might call to be the structuralist view.

What is, how mathematical structures are being generated? What is mathematical structure? Can we look as if we were outside of mathematics, but in a meta-theoretic kind of way, look into mathematics and see what is exactly happening? So the reason that Stegmuller compares it with Borbacke, precisely because he does the same thing with science. Stegmuller's program is essentially what you might call to be an equivalent, a scientific theoretical or logical equivalent of Borbaki's program in mathematics.

One thing that I forgot about talking about this triple of LSU as the three well-ordered elements of any theory and then systematic theory is that where does axiomatization come to the picture here? You see, the universe of discourse essentially is about a specific kind of data. Are we talking about poetry? Are we talking about scientific discourse? Are we talking about philosophical discourse? So on and so forth.

So the data is what you might call to be the designation of the kind of stuff that this domain of discourse is capable of articulating and elaborating and works with. Now axiomatization only comes in the domain of systematic theory. Why? Precisely because when you are talking about a set of data, for example as pertaining to some domain of discourse, then you need to have a corresponding structure and a

and corresponding language to make the relation between such data as explicit. In the realm of language, that's basically what formal languages do, axiomatize languages. In the realm of a structure, we are not talking about axiomatic systems, like, for example, Hilbert, Euclidean, Carnapia, and so on and so forth. We are talking about axiomatized systems. essentially we are talking about the kind of a structural relations by virtue of

being axiomatized in a specific form of language their relations to one another their in our relations to one another can be made explicit and also can be constructed in logical terms, in logical mathematical terms. So axiomatization essentially a prerequisite for having systematicity in theories particularly theories pertaining to a special sciences

So when we talk about axiomatization, what exactly would be the axioms themselves? Axioms are purely linguistic. You might say, you see, the vision of language that I've mentioned doesn't need to be natural language in fact for a special sciences we are talking about formal languages well I just mean if you are formal languages is like a calculus okay a calculus a calculus in which according to for example your linguistic framework your formal linguistic frameworks there

There are certain elementary rules which should be obtained between symbols or inscriptions which encode some observation or encode some data, so on and so forth. These are what you might call to be axioms. They have nothing to do with the world. They are precisely formal. They are only happening within the domain of the language as a calculus, as a general calculus. So the axioms, the syntactical rules themselves, or are they themselves formulas or propositions? Could you please elaborate on your question a little bit? I guess I'm thinking if you have sort of a mathematical theory, you would have a set of axioms,

and the set of axioms would themselves be formulas. Yes, yeah, yeah, yeah. And they don't need to be always formulas. they can just be, I mean not formulas as an equation, but if you mean formulas as set relations, yes, absolutely. So I guess I'm wondering in axiomatization of a scientific theory, what are the propositions or the statements, what kind of character do they have that are the axioms themselves? It depends on what kind of axiomatic systems we are talking about. Like for example, formal Gilbertian, everything should correspond to the kind of axiomatics

that we deal with in non-naive set theory. memberships you know coherency forcing so on so forth and so for a scientific theory what when we've axiomatized like Newton's physics what exactly are these are the propositions like what do they say

I'm going to actually, that's exactly what I'm going to do today. I'm going to give two examples. Classical collision particle mechanics and mostly in fact emphasizing classical collision particle mechanics and also the second law of Newton to show that what exactly once we axiomatize them, axiomatize such systems how the what you might call to be the scientific axioms can be made explicit by way of set theoretic axioms we will go we will go to this very very shortly and it won't be very pleasant

so So, I said that what is needed to characterize more explicit scientific theories? Philosophers who accept that claim, that popular claim that I mentioned, generally believe that sets of statements which could constitute a scientific theory may be distinguished from those which could not by their logical structure. The statements may ascend in various sources of NRE relations with one another, which we call logical relations. Examples of logical relations are is entailed by, is consistent with, is confirmed by. The first two, for example, are deductive logical relations. The third is an

inductive logical relation. The logical structure of a set of statements is roughly speaking, the logical relations, both inductive and deductive, holding among members of the set, Thus it is alleged that certain logical relations must hold among members of a set of statements if this set is to be a scientific theory. So now you see why we are trying to axiomatize. Precisely because, you know, kind of like coming back to idea of poker that's distinguishing

scientific theories from non-scientific theories but also distinguish different scientific theories from other different scientific theories. So, a great part of the activity of philosophers of science has been directed toward describing the sort of logical relations that must hold among members of a set of statements if it is to be a scientific theory. The aim is to provide a stronger, necessary condition in a Kantian sense for a scientific theory than the popular claim we introduced.

Clearly, since it is a necessary condition that is sought, a working assumption of this endeavor is that all scientific theories have the same logical structures. In fact, they cannot have any other logical structures because if they have, then they wouldn't be scientific. That is, it is assumed that there are some logical relations among the members which are common to all sets of assessments which are scientific theories. The task is then to characterize or describe this common logical structure.

So, in association to the very, what you might call to be task of philosophy of science, that I just described, there is also a second task, a second enterprise, which is that of clarifying or elucidating the logical structure of a particular scientific theory and not just general scientific theories, scientific theories in general.

This enterprise has also been the concerns of philosophy of science and of philosophically oriented scientists as well it is related to the first enterprise which i just mentioned characterizing the common logical structure of all scientific theories in at least two ways first successful completion of the first task would provide a pattern or prototype that might be helpful in dealing with a particular theory the hedging is very important here it could turn out that whatever we could say in general about the logical structure of scientific theories was so general as to be of no help at all when we came to trying to clarify the structure of some

particular theory. Second, successful completion of the second task for some particular theory would provide an example from which to build or to check a general account of the logical structure of scientific theories. A clear acknowledgement or appreciation of the second way or the second task these enterprises might be unrelated, sorry, a clear appreciation of the second way these

enterprises might be related is important to understanding the significance of the results to be presented in this presentation. And we go over it and how does it work exactly? How can we move from the general to the particular? And how they can be correlated in fact. So, since the notion of clarifying the logical structure of a scientific theory is exactly

the topic for today's session, it is important to understand it very clearly. According to a SNEED conception of this enterprise is roughly this. are presented with an existing scientific theories as it is expounded in textbooks and canonical technical literature and perhaps the unrecorded colloquies of scientists working with the theory we have reasonably clear intuitions about what the empirical claims of the theory are and what the logical relations among them are. Here reasonably clear means that in most specific cases we can confidently claim, for example, that such and such is or is not

an empirical claim of the theory. Intuitive means that we do not in fact appeal to explicit criteria in justifying such claims. With this as our starting point, we can produce some comprehensive and perspicuous form for exhibiting the claim of this theory and their logical relations. Let us call this a logical reconstruction of the theory and the activity of attempting to produce it. a logical reconstruction. So, when we are talking about

a signaler's and a SNEED's attempt and logical reconstruction, by that we meant exactly what I just said. Of course, we should demand that the logical reconstruction be in some sense compatible with our intuitive ideas about the structure of the theory. Beyond this, it should provide a tractable and systematic way of codifying these intuitions, which could be appealed to as justification for specific claims about the theory itself. We should provide a means of answering questions about the theory on which our intuition seems

to be hazy or conflicting. It may be that in the process of attempting to provide such a logical reconstruction, we come to believe that some of our intuitive conceptions about the claims of the theory, whether you are a scientist or a philosopher, are confused or even incompatible. We might be forced to make a choice to preserve some intuitions at the price of giving up others. In this sense, the enterprise of logical reconstruction is a normative enterprise. But in overall outlook, it is descriptive. You are simply explicitly making a description of both general structure of scientific theories

and the structure of a particular scientific theory. You presume the practicing scientist's conception of what he is doing to be roughly correct until proven otherwise. Thus the task of providing the logical reconstruction is one of codifying and systematizing an existing scientific theory, applying no external standards of judgment beyond simple clarity and logical consistency. In particular, we do not conceive the enterprise to be that of providing an epistemological

critique of the concepts employed in the existing theory from the viewpoint of some epistemological credo. For example, operationalism, logical empiricism, so on and so forth. If a scientist claims that he observes certain things in his laboratory, according to this view, we are committed to at least attempt to deal justly with this claim in our logical reconstruction rather than denounce it as conceptually confused from the viewpoint of some external criterion, external to science.

So in this sense, I think the task of logical reconstruction as initiated by Sneed and Stegmuller is essentially what you might call to be a scientific counterpart to what Brandoom calls explicitation, the task of explicitation. It's not that we are going to talk about, for example, metaphysical assumptions or epistemological methods that a scientist does. what by virtue of this logical reconstruction by virtue of making explicit what is going on

when that scientist observes and works within the framework of a theory what does actually this scientist presupposes and what follows logically speaking in an explicit manner from its implicit commitments to a certain framework of epistemology or metaphysics. It is evident that the aim of logical reconstruction is to provide what might be called a static account of a scientific theory. At the very least, it aims only to provide a clear and accurate picture of a particular

theory at a particular time, an account of what the theory claims at this time about the way the world is and how these claims are logically hanging together. Well one might contend scientific theories are the sort of things which change with time, certainly conceived as a body of empirical claims, quantum mechanics now encompass a larger body of such claims than it did, for example, decades ago. For example, since that time we have incorporated into theory a body of claims about electrical conductivity in metals. In the light of this fact about scientific theories, it might be held that our notion

of logical reconstruction is too restrictive to be, in fact, interesting. The really interesting questions about the scientific theory are dynamic ones. Questions about how theories change, grow, come to be accepted and rejected. It might even be hoped that if we could answer these questions for some particular theories, we might generalize the results and arrive at what might be called a general theory of scientific method or scientific methodology. For those philosophers of science who see their task to be one of providing such a general theory of scientific methodology, the enterprise of logical reconstruction might appear to be so modest in its scope as to be uninteresting.

though the enterprise admittedly modest in its scope we can say that the results of the subsequent investigation into the means of providing a static account of a scientific theory do in fact shed some light on dynamic questions of the set theories the implication of these results is something that we are going to talk about. Now if we are interested in clarifying the logical structure of some scientific theory in producing a logical reconstruction of the theory, how should we proceed?

Well, this brings us to another second widely popular accepted claim. And what is this claim? The logical relations among the statements of a scientific theory may be exhibited by an axiomatic system. This claim can be most plausibly maintained for deductive logical relations. However, it is reasonably clear that any account of inductive logical relations will presuppose

an account of deductive relations. Thus, the subsequent discussion may be regarded as, strictly speaking, confined to deductive relations. But it should be remembered that this is a necessarily a purlude or introduction to any serious discussion of the inductive relations, not by any means exhaustive view of inductive relations. We shall not, however, completely avoid talk about inductive relations, such as confirmation. But such talk will be on a naive, intuitive level, almost in a kind of early Humeian way.

No explicit account of these inductive relations and their connections with deductive relations will be given. These difficulties aside, it is clear that the second popular view is not much help as an answer to our question until we understand what it means, what is meant, what we mean by an axiomatic system. So in discussion of logical structure of scientific theories, the term axiomatic system has been used to denote different things, fundamentally actually different things. Among these are, one, a certain kind of set of statements. Two, an axiomatized deductive theory in some form of language.

Three, a definition of a set theoretic predicate. The traditional meaning of an axiomatic system as being pronounced in that second popular claim is actually number one, namely a certain kind of set of statements. The kind of set of statements which is an axiomatic system is characterized in the following way. There is some finite subset, the so-called axioms, of which all other statements in the set are logical consequences. The relation of being a logical consequence is an informal one,

applicable to statements in ordinary discourse. The paradigm of this sort of axiomatic system, for example, a good example of it is Euclid's axiomatization of geometry. For this sort of axiomatic system, the relation between the axiomatic system and the scientific theory, whose logical structure it exhibits, is obvious. The two sets of statements are usually held to be coextensive. In order to understand the second meaning of axiomatic system, one must first understand what a formal language is and how it may be related to sets of statements.

A formal language, for example, the first order predicate calculus, is a set of symbols called sentences. They are purely syntactic. For most interesting examples of formal language, this set of symbols is defined recursively by listing a finite set of basic symbols and stating rules for constructing sentences from these basic symbols. Defining the set of sentences in this way is often regarded as analogous to stipulating the rules of grammar or syntax for so-called natural languages, like for example, English,

Spanish, so on and so forth. And Artem, in reply to your question, informal, Albertian axiomatic is more in tune to this second definition of an axiomatic system. The sentences of formal language may be related to a statement of ordinary discourse by providing an interpretation of the formal language. Formally, an interpretation of a formal language is a function which maps a mathematical function, which maps the set of sentences of the formal language onto a set of two objects.

Tf. For the first order predicate calculus without identity and operation symbols, this function is specified in two steps. First, certain basic symbols, the predicate symbols are assigned to subset of a set D, the domain of the interpretation, and its powers, Dn, D index n, or subscript n. Intuitively, this corresponds to an assignment of meanings, namely references, to the non-logical symbols, quote predicates. Next, rules are provided for employing this assignment to determine whether T or F is to be assigned to any given sentence in the formal language.

intuitively, in a very intuitive manner, this corresponds to an assignment of meaning to the logical symbols, sentential connectives and quantifiers. If we assign meanings to the logical symbols in such a way that they correspond roughly to the meanings of logical symbols in a natural language, then we may regard the interpretation as establishing a correspondence between sentences in the formal language and statements. The corresponding statement is true or false depending upon whether the sentence of the formal language is assigned T or F by the interpretation. For example, let us fix the assignment of meanings to the logical symbols of the formal language

to correspond to the meanings of logical symbols in a natural language. Then each different assignment of meanings to the non-logical example predicates symbols determines a different function from sentences of the formal language into a set of statements. Let us say that each interpretation of the non-logical symbols determines a function from the set of sentences of the formal language into a set of statements. It is important to note that the entities which an interpretation assign to the non-logical symbols of the formal language are set theoretic entities. In particular, they are subsets of the domain of the interpretation and its powers. Okay, maybe I should...

It's 1.17. Maybe we should hold on. Things are getting a little bit too dense, and it's going to get even denser. Let's have some brain-dressed resting session. You said this with a smile on your face, though. Sorry? I said you said this with a smile on your face. Yeah. Have you watched Infinity Wars when Thanos says that balancing universe is not about fun but it puts one or two smiles on my face that I can say about this formalism. Okay let's have a break, let's just a little bit think about this stuff and

then coming back and continue. Alright, how about seven minutes? Is that okay? Sure, sure. Excellent. I like pacing and rocking to try to maintain my heartbeat and oxygen rhythm so I can focus at such a blade of an hour. Okay, see you. Thank you.

Thank you. Thank you.

Thank you. I don't know exactly.

I don't know where we were planning on going with this. They made this sort of difference between, say, a static approach to theories and a dynamic approach to theories. Now, does this sort of kind of have to do with, say, the difference between axiomatic approaches to scientific theories and maybe more like interactive approaches to theories? Like, say, like sort of more like Hilbert systems versus like sort of like Gensen or structural proof systems? You see, those are essentially, Gensen is not really working on axiomatics.

Right. He talks about a system for manipulating, you know, or for understanding how, you know, the relations between antecedents and consequence, premises and conclusions, hold, and build the system for observing how the mechanisms work. But it is not essentially about axiomatization. Hilbert is about axiomatization. So as, for example, Carnap. So as these are essentially that you might say that you can in fact give a, again, a view of all these things or an interactive view of all these axiomatic systems but these

are I think two different stories all together. Yeah sure sure. So I mean what I was kind of asking is like if the move from nautic to dynamic has something to do with the difference between axiomatic versus structural systems, but that might be sort of imbued in my own sort of my assumptions. Can you repeat it one more time? I mean, I was basically setting up kind of like an analogy of axiomatic systems, or static scientific systems and dynamic scientific systems to axiomatic logics versus structural logics. But that might be like just completely

incorrect. No, the thing is that essentially a segmuller tries to show that even if we begin with an aesthetic view, essentially a static view is what you might call to be dynamic in disguise. The dynamic of such theories in disguise, precisely because once we arrive at a generalized logical reconstruction and then we are capable of seeing particular theoretical logical structures or a logical reconstruction of particular theories then within with in comparison to that general

view of logical reconstruction then we can in fact understand the dynamics better so you have so you have this sub-strate which is what called to be this uh the general logical reconstruction of scientific theories and then you have these foreground you know uh particular reconstructions, logical reconstructions of particular scientific theories. When we see these as two kinds of views and two forms of logical instruction, then we can in fact deduce or analyze the kind of dynamic relations that these particular theories

stand in relation to one another. So we wouldn't preserve axiomatics to the end, so to speak, of where this is headed? No, no, no, no. Essentially it's what you might call to be... it allows us to arrive as a view of scientific theories which even though is mainly a static but in so far as it gives us a generalized form and a particular forms of a static theories then we can indeed like a almost like a kind of uh zalamaya sense we can indeed by virtue of comparing or contrasting

this general and particular global and local form, see how these local or particular scientific theories stand in relation with one another. Okay, so that makes some sense. It's also like, I mean, this might be unintelligible to most people, but for example, you can have axioms that aren't just propositions, you know connectives which are able to do like implication or something yes oh so i mean maybe then it makes sense maybe maybe i might risk overextending here but like extend the analogy of you know uh static theories being um uh uh uh dynamic theories in disguise is that like

axiomatic theories or axiomatic logics are interactive logics in disguise Mm-hmm. So I had a question about the universe of discourse. So I guess when you're using an interpretation in the model theoretic sense, what makes this informal is that the universe of discourse is not itself mathematically constructible. so it's not like the set of all natural numbers which we could construct formally but it's something that's taken from from the world is that right not from the world from the natural language actually

so you see it's a well-ordered set theory is a well-ordered said what is the world where you're set is that there are relations between the members of your between L, S and U, language, structure and universe of discourse. The way that your universe of discourse is not for example formal is precisely because of how it stands in relation with language but also the question of structure. Depending that if your language was formal but also your structural relations were also couched in terms of this formal language then your universe of discourse could be formal or informal.

So can you give an example of like an informal predicate that would be taken from natural language in this context? Well, early theories of theorem, early in philosophy of science I will I will I have to get some of the references but if you see for example early in philosophy of science when they are talking about thermodynamics and by that I mean thermal thermodynamics rather than statistical it is absolutely usually couched in terms of natural language predicates. But let's even go outside of

this whole idea of science. You can say that ordinary theories of the kind that we usually talk in philosophy are essentially informal. This doesn't mean that we cannot formalize them, but this formalization is no longer axiomatic or axiomatized. So we're basically using like a translation key in this sense? The translations, as I mentioned, are more like a function, interpretation function.

Perhaps maybe we continue with the material and we can attend in the seminar. May I ask for something? Absolutely, absolutely. like, could you, if you will continue in the same kind of vein, would you slow down perhaps a little? Because I guess I assume you are reading from somewhere. Yes, I have some notes. I can actually, one of the things that I can do, I can, actually I will try because my notes are a little bit, you know, disorganized. That's why I don't feel embarrassed to show it to you.

But from next session, I think it's precisely because, you know, there is huge amount of details and technical details. I can actually read over it while you can, I share my screen so you can actually read the text as well. Yeah, that I think would be a good solution to it. Yeah, that's useful. Like visual and auditory stimulation just sort of is better at provoking more attention and thought, especially at 1.30 o'clock in the morning. Yes, and not to mention that we are talking about some esoteric philosophy of science and stuff. Anyway, so…

I mean, I can go a lot around these topics, but I think that I should actually let you to simmer your brain for a while on these questions and explicit connections so that we can actually talk about them without me overwhelming you with the idea of this kind of the relation between logical reconstruction and axiomatization.

But for now, I would say all I can say as a kind of introductory level is that these definitions of axiomatized systems roughly correspond to the hierarchy of axiomatization introduced by Wolfgang Sigmüller in his later works.

So we have at least five different meanings of the concept of axiomatization of a theory. The method of axiomatization then used is the informal set theoretic axiomatization via the introduction of a set theoretic predicate. This method is illustrated with the help of miniature theory which also appears in what I'm going to talk about, the classical collision particle paradigm. Roughly speaking, what are these, what you might call to be five meanings of axiomatization

of a theory? The first two meanings have a common formal feature. They regard an axiomatic system as a class of statements which are logical consequences of a finite subclass of this class. For example, if we call Sigma, symbol Sigma, a Euclidean axiomatic system, if and only if Sigma is a class of statements and there is a finite subclass, Delta of Sigma, capital Delta of Sigma,

whose elements are self-evident and thus true, such that each statement of the difference class, the complement relation, sigma complement relation, delta, is a logical consequence of delta. The elements of delta are the axioms of the system sigma. In so far as right now all we are concerned with is drawing a comparison with the Hilbertian notion of an axiom system which is what you might call to be the canonical definition, we could just as well have spoken of the concept of an axiomatic system in the Aristotelian

sense. Historically, though various differentiation would have to be made, that for the most part remain neglected in philosophical discussions of this topic. For example, existential quantification plays a different role for Euclid than it does for Aristotle. Furthermore, axioms in the strict sense, namely relation sentences, would have to be distinguished from postulates, namely construction sentences, and underlying which presented considerable difficulties even in ancient times. Now, modern axiomatics, the ones that we are interested in, was completely and

and systematically exhibited for the first time in 1819-1990 in Hilbert's work on the foundations of geometry. Hilbert's efforts were bent toward freeing geometry from recourse to uncertain intuitions. You see, the whole point of axiomatic, as I mentioned to you, is about explicitation and logical reconstruction. Okay? So hence intuition has undefined, namely under-defining, undefined terms, postulate and construction sentences.

By virtue of that, it cannot really give us a great medium for making explicit what is actually going on in a certain theory. One could therefore also regards such axiomatics as abstract axiomatics in contrast to Euclidean's. For example, graphic, you know, diagrammatic axiomatics. Like, you know, that, I mean, I'm sure you're familiar with Euclidean system that is essentially a diagrammatic form of reasoning. All of our axioms are presented in diagrammatic configurations.

You know, drawings and such. According to Hilbert, the axioms of geometry are merely assumptions about the mutual relations obtaining between the elements of three classes of things. The elements of these three classes are indeed called points, straight lines and planes, in accordance with pre-systematic intuitive ideas as represented by Euclid. And similarly, the expressions of lies between, coincides with, and is congruent with, suggestive of intuitive spatial relations are used for three fundamental relations.

But the question as to what sorts of things and which relations are involved here is explicitly left open. The usual intuitive ideas should not be associated with the fundamental notions in a Euclidean sense. Now let me read this quote from Seigmuller because in Hilbert's view the axioms are free of intuitive spatial components there can be besides the normal model based on the original intuitive interpretation. There's still other models for them, some intuitive, some not. It is then entirely possible that a non-normal model be found within the same regions of the intuition which originally yielded the notions for the intuitively

perspicuous model of the axiom system. An example of this within the framework of Euclidean geometry would be the so-called sphere bundle. Here the axiomatic concept, point, is to be understood as an arbitrary space with the exception of a single particular point. Line in the axiomatic sense may be any circle that passes through the point in a question, and by plane in the axiomatic sense is to be understood a spatial sphere touching the point in question. The geometric relations mentioned above are accordingly relations between these new geometric elements.

despite its geometric perspicuity, this model for Euclidean geometry has naturally little, if anything, to do with the model which furnished the intuitive basis for Euclid's own axiomatization of geometry. That this and other models are admitted within the framework of Hilbertian axiomatics illustrates how the transition from Euclidean to modern axiomatics increased the latitude of interpretations. Instead of admittingly only a single normal interpretation, infinitely many models which make the axiom true are now in principle possible.

So what is really the consequence of this statement? And can anyone tell me about this? The move from the intuitive axioms that Euclid talks about, the formal and non-formal Gilbertian axiomatics, what is really the consequence of such move, which I would call the revolutionary move provided by the formalist approach? It seems to me that maybe it would make more explicit the types of rules which are employed in the construction of new propositions. Yes, that's one.

What I wouldn't call this fundamentally revolutionary, what you might call to be a byproduct of something more fundamental. It seems like to me it's allowing special cases that can in some ways like be a probe head to keep moving the whole net forward. It's like that's the moving... Not the special cases, general cases. But you are absolutely correct. it allows us precisely because our intuitive axioms are rooted in our intuitions sensory intuitions they have they are essentially restricted

especially what you might call to be special cases okay with the advent of formal axiomatics now we are capable of talking about a domain of systematicity in which such a special cases can actually be taught from a more general perspective. And hence we can we can arrive at new possibilities of new intuitions about the space, time, so on and so forth. So you see formalism here is a force of revolution. And of course, what I would call breaking from the Arsotelian-Kantian prison of intuitive

way of looking at the structure of the world. I have a question. Absolutely. So just to clarify certain things, because this is new material for me. When we talk about relations, when you announce this second popular claim that is done when you try to answer the question, how does logical reconstruction of theory proceed? You were talking about axiomatic relations, and now you were talking about axioms as basically statements that captured relations between different things.

So it occurs to me that it's just like, is it getting mixed in my head? Yes, yes. When we talk about axioms and we talk about relations, we're talking about relations between things that axioms kind of talk about or also relations between axioms or maybe there are some other relationships. Okay, I think to best to, and you're right, I actually was not careful and hence it led to a confusion. So imagine that we are in fact talking about two different things, but they are in correspondence. One, an axiomatic system, namely a system that is fundamentally based on a class of axioms.

Gilbertian, Euclidean, Carnapian, so on and so forth. Okay? Axiomatic system. And then an axiomatized system. Okay. Axiomatized system tries to capture the relations between structures or relations between structures, namely all it wants to talk about is the notion of a structure within a specific domain of discourse, between the kind of data that you are working with, right? But such, the reason it is called axiomatized is precisely even though the priority under structural relations or structural entities, nevertheless, such structural relations are

codified by axiomatic system. Now so this is this axiomatized system. The axiomatic system, what you might call to be, we are not really talking about the the general notion of a structure or even a universe of discourse is what you might call to be a purely mathematical logical domain where axioms can have relations between one another. And of course, such relations are a structure but purely mathematical, exclusive to the mathematical domain that those classes of axioms you use enable you to articulate

and elaborate now such relations are essentially what you might call to be quidifying relations codifying relations they are purely abstract they are purely logical and mathematical. They are not about all universes of discourse. They are specifically about the universe of discourse which we call mathematics or logic, the relation between those axioms. There are structures but there are not any sort of structures. There are very specific kind of a structure. They are codifying structures. When we try to axiomatize a system essentially what we are doing is that we use the axiomatic system the relation between axioms

in order to codify our general notion of a structure or data and how's that another question maybe you were talking previously you were talking about inter subjectivity logical inter subjectivity and I was I'm not familiar with this concept very much but is it something to do with this relation making relation building it's in different yes you see since so you see that there is a progress in classical logic and attack on psychology them

since friege that's two since friege but also after russell so it's essentially uh you might say that logic now uh if you are going to give a very generous classical logic is neither the laws of thought are not the laws of the world namely there is no metaphysical correlation between laws of thought and laws of the universe okay one two the idea that logical laws of thought or logical rules have nothing to do with our psychological way of experiencing the world

and in fact they should be subtracted from them so this is what you might call to be the first germ of logical intersubjectivity now why is it I am calling it intersubjectivity because as As logic matured in 20th century, they understood that the stuff that we call logical connectives or logical relations, the meaning of such connectives like conjunction, disjunction, so on and so forth, you know, complement relation.

It's not as if they are given in advance to us in their totality. Their meaning is undecidable. But how can we actually see their meaning, decide and determine their meaning? This led to a new revolution in logic, at least from the late 60s onwards. So Karnam had already thought about it even though not coherently. This is a paradigm that is usually called in today's contemporary parlance and lexicon the interactionist view of logic.

What does this mean? Is that essentially in order for us to study logical behaviors and logical connectives, We should see them in terms of some more fundamental logical or computational processes. This is what you call, so the interactionist view of logic is, Jean-Yves Girard calls it the move toward the foundations of logic. What does this mean? You see, imagine that we are essentially working with symbols, right? With syntax. but not any kind of maximalist view of syntax, but just minimal view of symbol design, just

meaningless inscriptions. So the point is that this can be interpreted computation as a form of interaction, namely intersubjectivity. Imagine, so in Church-Turing paradigm of computation, the abstract machine, the Turing machine is a syntax manipulator. It takes the classical version, it takes a set of discrete inputs from the environment me for example and my computer, right? It takes a set of discrete inputs from the environment.

It goes through a set of transitions. This state transition is what is called computation. It's basically syntax manipulation, okay? Zeros and ones or whatever. You don't need to essentially think about just digital computation, but nevertheless. Then during this is a state of computation or syntax manipulation, the machine does not admit, does not receive or accept any more input from the environment until and unless it yields an output. Only then it can accept a new set of input from the environment. So this is what you might

call to be essentially Turing's machine is a deductive system, is a deductive system, is a logical system. But do you see here that there is something missing? There is no such a thing as a machine that can be said to be waiting for the environment to finish its job and then interact again with the environment. This led to a generalization of the concept of computation and hence of logic.

That computation ultimately is what you might call to be interaction, confrontation of axiomatic actions between a machine and the environment, the system and its environment. In the sense that the machine just does not wait, does not take for granted that it should do its job and then take something from the environment. It's what you might call to be the interaction between these two, between the machine and the environment, is in real time. It's truly concurrent. And by the way, the machine can make, you can think about how the machine computes and how the environment reacts to the system as essentially two game boards,

two different game boards. In this game boards I make some moves. Environment also makes its own moves. But Church-Turing paradigm of computation, the classical Church-Turing paradigm of computation, which is essentially encapsulation of classical logic, is what you might call to be a game, a special case of computation for logical behaviors in which it is already assumed that our moves are in coordination with one another. Synchronicity and sequentiality of moves, of logical moves,

the syntax manipulator and the environment that gives these symbols to the system are not essentially in all cases are coordinated, synchronous or sequential. Like imagine like me and my computer are working, you know, I am playing the chess so I basically I open different apps and I do some other stuff while I'm playing the chess and basically chess might actually halt, I mean pause for 10 minutes and then make a move. while during that I make some other moves and calculation in my own domain. So this is what you might call to be the vision of computation and the ultimate proto-foundation

of logic, that interaction, a truly, truly interactive way of confrontation of actions by the environment and the system ultimately decide the meaning of logical behaviors, the meaning of logical connectors. Really one of the things that I really really fundamentally suggest is a work by... Sorry, I have a very short memory.

John Baptist Jonet J-O-I-N-E-T. The name of the essay is Proofs, Reasonings and the Metamorphosis of Logic. You can find it online. And it basically tries to show, in a very rudimentary and intuitive

way it tries to articulate that essentially logic is an object of computation, but not all sorts of computation, a generalized computation. What is generalized computation? Interaction par excellence. The truly concurrent confrontation of actions by a system and the environment. no presupposition about how these two things interact. Can I ask a question about that? Yes. I'm not exactly sure how the picturing of logic as interaction between these two domains,

I'm not exactly sure why it's not collapsing modes of thought into the way the world is again and it's and therefore it's would be equally susceptible to psychologism because you see it only becomes susceptible to psychologies and other kinds of unpleasant stuff like vitalism and psychism stuff if you take logic and computation as a physical model of the world but if you only see it as a logical as a logical basically the proto foundations of logic but in in this connection with the world

then it is not susceptible to such uh i would say uh you know consequences precisely all we are going to talk about is that the view of a syntax essentially what is logic logic is essentially a fundamental view of syntax by which you determine the meanings of logical syntax connectives behavior so on so forth with no reference with no representational reference to the outside world because if you do that you're essentially in the uh rossalian trap hole laws of thought are correlated with the laws of the world right this is this is a view of logic

that has been fundamentally challenged and is no longer tenable. So we are not going through that. All we are saying that computation is essentially what you might call to be a protologic in which we can have a more fundamental view of how the meaning of syntactical relations are being determined in the domain of logic. So, just to be clear, I think my understanding of psychologism is that thought is a byproduct of psychological circumstance. How did you arrive at that point?

Well, that's what the psychologistic claim is, right? Yes, psychologistic claim. But it's another way to formulate that psychologistic problem would be to say that the inverse is equally plausible, that the world just is logic. Yes, the world is constituted by logic. Right. But what you're suggesting, I feel like, has to open up three different types of interactions. One, a logical domain, another, a psychological domain,

which might be like a subject, and then the domain of the world. Is that correct? Can you repeat the three elements? A logical domain? Yes. a subjective or psychologistic domain which is like the conditions of a subject observer and then a the domain of the world right yes and essentially what Carnap tries to say is that since at least the time of Frigge and Sneed Carnap and Stegmler are essentially the children of frigate the ultimate assault on psychologism of the subject that science has nothing to do

with the subjective experience of a particular observer And you see that the move to formalism is essentially not only in terms of, it's not only meant to explicitize relations, the structural relations, so on and so forth, But also it has, I think, a fundamental rational philosophical import. As Carnap said, that we can only see the world objectively if we have something called structural

entities. These structural entities are not given in advance to us. They are the products of logic and our labor of theorization about the world. Because the material of our incompatible streams of experiences and not only me and Theo have different incompatible experiences but also I myself have different incompatible experiences of one and the same object. So you see that logical intersubjectivity, this view of the interactionist view of logic is essentially trying to show that logic is what enables us to commensurate

or coordinate at least, not commensurate probably. Synchronize and coordinate are different diverging streams of experience of the world so as so as to arrive at veridical namely objective claims objective claims do not mean that they are essentially true all they mean for them to be objective is that they mean it means that they are very means that they can be assessed they can be assessed they can be revised and corrected

without an initial reference to particular observing subjects of course Now you see that this whole idea of philosophy of science that we are talking about, this is essentially a very, very corrosive asset in a Donetian sense against many of this stuff that are being taught in content of philosophy about, you know, I have my own lived experience of the world, you know. But what is this lived experience? A lived experience that cannot be coordinated with another lived experience by way of a logical infrastructure. It's just the logic of illusions.

I just want to push back on that because it seems the whole notion of logical intersubjectivity, the way that you're saying it, imagining logic as interaction, it needs to posit that type of what whatever you're saying is the kind of hyper personal so isolated yes and what how do you think that they does it essentially what is really the ultimate basically the task or or the the grand view of logic it gives us a a structure. What is a structure? A structure is something that is invariant across the

board. Precisely because it's invariant across the board, we can codify our diverging experiences with it. And hence we can see them in correlation with one another, in relation with one another. Hence, we can assess them through such a relation. Can I push back more? You can always push back more. I wouldn't expect less than a skeptic. Well, the interaction requires dynamism, but you're saying the objectivity of logical structure requires stasis so and

those are start are not mutually exclusive absolutely they are not mutually exclusive in the sense that the dynamicity is a dynamicity of processes of logic and computation logic computational processes that in In interaction with one another, they decide upon the meaning or a specific structure of one single symbol or a set of expression of symbols. Context sensitivity and that's what you get essentially a Brandomian view that ultimate pragmatism, a Brandomian analytic pragmatism that essentially what we call meaning is a

stabilization of this dynamic so we can so so the dynamics dynamics they can actually grow it doesn't mean that there is these are fundamentally firmly stabilized but precisely it's a kind of a stability that allow for more qualitative form of dynamic logical computations to grow as they interact and hence new fundamental kinds of a structure can be logical structure can be derived. I mean, I'm super, I'm incredibly sympathetic to pragmatism, but the whole notion of growth conceptually to me has very little relationship to stasis

and to objectivity in the way that we're talking about it. You see, no, no, objectivity is not a stasis essentially. We were talking about a static view of scientific theories. Right. That's a different topic. But what you might call to be objectivity is essentially a stabilized structure. a structure that what you might call to be is stabilized by virtue that there is a coordination, there is some sort of synchronization of, for example, the meaning of a particular syntactic expression or syntactic symbol that holds, that plays the same role for these computational

processes for the environment as it also plays the same for the system. Nothing else, nothing new. But is there such a thing as stable growth? It's not a stable growth, I said a stable structure. Stable structure is what you might call to be essentially a computational problem. How can we decide that this symbol plays in our board games the same roles? Well, of course, we have to interact to see how this relation with the symbols change as we move forward. And at some point, we might agree that, for example, in the game of chess that we are

playing, pawn has the same functional role. this becomes a rule of course in another board game namely another interactive scenario the pawn might be a goal the game goal uh basically piece you see it's a functional role that the syntax plays for both of us like a game of chess the pawn Imagine that we started to invent the game of chess, so we had to interact and we played with all these things and then we decided that this such and such symbol of such and

such relations with other symbols played the same functional role for me as it plays for you. I can only move it one or two cells. I can only do it in this diagonal way. And that's what you might call to be the meaning of a symbol that undergirds the idea of objectivity. And it's essentially objectivity, I would say, at its very base, a logical computational problem. I know what kind of want to move forward, but I have maybe a few things to say that... push back against the pushing back

or something like that. But, I mean, the three things that you mentioned, you know, is world, subjectivity, and its logic. You know, and I mean, I think, you know, one important thing would probably be that the way that the world is given is according to the logical structure by which it is given. You don't have unmediated access to the world. Yes, yes. But then it's also that the constitution of the subject, first of all, the perception of the world, the perception of the object, is part and parcel of the subject,

but also the subject's interaction with the world. is a big part of the subject, and both of those things are articulated by logical structure. Yes, absolutely. One other thing would be, you know, I mean, Reza was talking a lot about sort of like the, say, invariances of, say, pieces in a chessboard, but more than just that, you know, I think Reds would probably agree. Just to take an example, in game semantics, which is very interactionist paradigm of logic, there's a huge emphasis on strategies and the interaction of strategies within a proof game.

The proofs, the programs themselves are what's interacting. You can say that a certain proof is able to successfully interact with all these other proofs. Yes, and proof essentially, you should also tell them that proof is essentially the core of meaning. Yeah, the core of meaning and the core of dynamism. Yes, yes. It can successfully interact across, like a winning strategy would be a strategy that can successfully interact over a whole range of opposing strategies given by the environment. Yes, yes. And so you have dynamism here while also possessing an invariance of strategy.

Yes, a stabilization. Yeah, I mean, this isn't to say that, you know, like your strategies won't change, that you're not revising your concepts. You know, the structure of how you're interacting with the world doesn't change. but the point is though is that you can have stabilized strategies, invariant strategies which are productive of the dynamism in interacting with the world yes, however and I want to get back to the topic, just as you with me, if I don't get back to the topic in regard with just what tears wrote at the sidebar well we

can't say that the world is logically given that psychology yes of course the whole point is that the givenness of the world or even war as such essentially Ultimately, what we are trying to decide is that what a structure actually is, logical structure is, where does it come from, and according to what criteria it can codify our experiences, our observable experiences of the world. We are absolutely making no reference in the first instance to the given world. In fact, our given world is that of logic, of thought and structure.

So, I might be able to find a way to tie this into the text with the question that I had, if that's alright. We can both, yes, absolutely. So Stegmuller differentiates between Carnap's approach and Supes' approach. Carnap's he calls the formal language approach. Supes he calls the informal language or the informal set theory approach. Yes. and at one point he says it's on page uh six at the bottom but it's uh let me just 11 of the pdf

he says if an advocate of the statement view which is carnapp's view starts with an assumption like this, quote, suppose L is a first order language in which the physical theory T is axiomatized, end quote. And closer inspection reveals that T is a complex theory whose mathematical part makes use of tensor analysis, partial differential equations or even just theory of matrices then we may inquire again where on earth do we find the formalized theories containing these branches of mathematics nobody knows they simply do not exist just as most of the material in the burbaki volumes do not exist rephrase i don't yeah i don't

know burbaki well enough but he's pointing to this problem where the what the axiomatic system has trouble even, I guess, a crude way of what it would be, what, making the axioms axiomatic? Yes, you see, no, actually I would say that this is, to be honest with you, I think that if you wait for a little bit of intelligence and spirit, it's the last chapter to the last one, I think it's the seventh chapter, where I actually take side with karma. You see, the thing is that a signaler tries to talk about is that when we are talking

about these kinds of axiomatic systems or axiomatized systems, essentially we think that we can couch everything in terms of a syntactic language or calculus. But nevertheless, we do not know or it is not basically obvious to us that how much of this stuff that we call axioms are actually not really based syntactical entities, but actually are semantic interpretations and thus not axiomatic.

And this view coincides with later work of Putnam that he tries to talk about that essentially the labor of theory is the labor of semantics rather than syntax or formalism, pure formalism. Essentially you can't even talk about theories without some kind of semantic space by which you can articulate these kinds of syntactic axiomatic relations. Otherwise, so many things that you take as your axioms might actually be axioms. They might be just semantic byproducts in disguise. However, this is all true and good. This is all true and good. Nothing against

But what I would say in response to Stegmuller and Pottnay is that if we don't take the vision of the logical syntax, a la Carnap, seriously, essentially we are in what you might call to be that very intuitive, naive Kantian view of logic and the world. because why because first of all two things happen we either see an inflated cement syntax and of course the secular things that syntax is always just basically about pure formalism

the idea of inflated syntax that they critique leads to a deflate sorry a deflationary syntax view of the formal language that they criticize a la Carnap leads to an inflated semantic. An inflated semantic, deflated semantics leads to an inflated syntax. What does this exactly mean? It means that essentially if we don't take the idea of Carnapian calculus or formal language seriously in which semantic space imminently unfolds from the interaction of syntactic

processes within the domain of logic and language, then essentially whatever we talk about semantic interpretation of those axioms might be actually byproducts of our psychologistic view of language and that's why Carnap like Frigge and I really took me a long time to arrive at this conclusion I think that Frigge and Carnap did fundamental thing precisely because they see that okay if you are going to talk about logic as what constitutes this structure and hence constitutes of our

claims of objectivity about the world, then if we simply resort to the features of natural language, like the priority of semantic over syntax, then we are essentially in the bounds of a representational intuitive language. The whole point of Carnap, get the fucking shit, divorced, namely language, from all such parochial concerns. And then we can in fact coherently talk about what we are actually talking about. Whether in terms of syntax or semantics. We will get into this heavily logical syntax of language when we are talking about Carnap.

Does this require treating natural languages as if they're formal languages, like in a Chomskyian way? So we're always kind of… No, not essentially, no. It's the idea that what you might call to be natural languages from a Carnapian sense or even today's theoretical computational sense are a special case says of more general languages not one just universal language but more general languages exactly like mathematics like set theory can in fact be seen in terms of more generalized categorical structure within the domain of mathematics.

But so a natural language would still be purely syntactical and algorithmic before it's... It's actually more semantic than syntactic, more semantic. So it's not Chomsky in that sense? No, no, no. I mean Chomsky essentially is basically the Chomsky hierarchy of formal grammar is essentially about syntax. has nothing to do with semantics. Chomsky, in fact, for a very, very long time didn't want to talk about semantic, precisely because he thought that there is an incommensurability or difficulty,

very fundamental difficulty, in order to map his hierarchy of formal languages, which are basically computable and algorithmically elaboratable to what you might call to be semantic complexity of natural language, how we use concepts. Okay, let me start now with these all introductions, biological reconstruction, let me start with the what you might call to be the an introductory account of the most fundamental concepts that

a stegmuller tries to work with and then once I introduce these hopefully next session we'll have a very brief talk about how within this S-need S-segmuller paradigm we can in fact formalize particular scientific theory and understand their dynamics. And as I mentioned, our example is about classical collision particle mechanics.

So building on Sneed's work on logical reconstruction of scientific theories, a signaler develops a certain family of new concepts that can in fact be elaborated within this logical reconstruction explicitly. So according to Stegmuller, theories, by that he means scientific theories, are reconstructed

as concepts or properties and philosophical ideas are defined accordingly. Assume a theory has domains d1, d2 to dk and non-theoretical functions like f1 to fn. also theoretical functions capital F N plus 1 to FS also axioms A1 to AR and of course correspondingly theoretical axioms capital A r plus 1 to a v a specifying

properties of the function such as the domain co-domain differentiability and so on the a adding further information and possibly relating relations between the functions essentially what and that's that's essentially what you might called to be one of the tasks of a scientific theory. So in every scientific theory we have some non-theoretical elements and some theoretical elements. I mentioned very briefly in the previous sessions that for example one of our non-theoretical entities for example in classical collision particle mechanics is the position of a particle. The position of a particle doesn't need

a theoretical framework. It's just observational or even a statistical. But then we also have theoretical functions and theoretical axioms. The tasks of these theoretical axioms and functions is that they elaborate further information, they articulate further information about how these non-theoretical functions or axioms stand to one another but also within the theoretical framework.

In such a way then we may define a predicate T, a standing capital T, a standing for theory, by stipulating X is a T, a theory, if and only if X is a triple, a well-ordered two-pole set consisting of capital D, a small f and capital F, namely the domains of theory, its non-theoretical functions and theoretical functions.

In addition, the T, namely theory, requires two more elements, a, a small a, and capital A. Axiomatized non-theoretical statements and axiomatized theoretical statements. The non-axiomatic ones correspond with the small a and the axiomatized theoretical ones correspond with capital A. Applying the predicate to the concrete situation, for example, S

index i concrete situation which can be denoted by a singular description in which the small f non-theoretical functions are replaced by concrete constants F . FCI, concrete situation, is a particular situation containing Jupiter, for example, in our theories of heavenly bodies. F, the position, small f, the position, as I said, one of the most canonical non-theority

functions is the position of your observable. a small f position then FIG are positions of the first and the second Jth ingredient Jth Jth ingredient of the situation including the position of Jupiter then we obtain CI concrete situation I is a T a theory which is said to describe the Ith application of theory of T a measurement of a concrete function phi

iL that appears in the i-th application of T is said to be dependent on T if and only if there exists an X that belongs to the set or is a member of DI. And what was D? D was the domain of our theory and D I simply shows that the domain of applications of that theory. Such as in every available exposition of T, the description of the measurement that leads to a value phi i l of x involves a cr a c index r concrete situation cr such that cr is a

t the abstract function phi position of phi i g is the position of jupiter in the i-th application of the theory is said to be t-theoretical if and only if the measurements of phi ij depend on t our theory for every application of t non-t theoretical otherwise now these models of t which what are these models of our theory t they are essentially classes of models, theoretical models, are entities that satisfy our theory T, the prerequisites

of our theory T and the kind of logical relations that hold in such a theory. So, this is how the, these are the basic concepts of a Stegmuller formalization. models of t essentially a class of m index t or m subscript sorry superscript t are entities that satisfy our theory t then we also have possible models which estegmuller denotes as m subscript t sorry superscript t and subscript p m index theory

uh and subscript p satisfy t prime where t prime is defined as t but without the a a partial possible model which a Stegmuller calls M index T subscript pp superscript T, satisfy T zigen T prime prime, where T prime prime is defined as T but without the A. The F and those A, small a, that specify properties of the F alone, capital F, the theoretical function.

So, thus given T as defined in this formula, we can envision any model, any scientific model in terms of its I-th applications, different applications. and hence the corresponding partial possible models are associated with such applications of a set theory. Let me get a little bit less torgid here and say that essentially what a signaler calls a partial possible model of a theory

theory can be regarded as the facts of this theory, which he calls T-facts. The factual content of a theory might now be explained by statements of the form that I mentioned the bar. Were it not the case that the correctness of such statements can often be ascertained only with the help of other statements of the same kind. A statement, for example, concerning an a small a belongs to the partial possible model MTPP, on the other hand, can be ascertained

independently of the theoretical framework of T. To relate them to T, we proceed from the bottom up by selecting all those possible partial models that can be supplemented by further functions so that they become models of T. Again, coming back to a little bit elaborating and articulating on this whole idea of partial possible models, which essentially what you might call to be models that allow us to extrapolate the dynamic of a theory from its static logical view.

So what are these things? What are these partial possible models? According to Seymour, there are observable facts or physical systems, and he calls empirical in this sense. All investigations dealing with non-theoretical magnitudes, it is important to see how these notions of observability and empiricity differ from the epistemological notions that have so far dominated investigations in the philosophy of science. epistemological notions aroused in connection with problems of meaning and confirmation. One postulated statements that were intrinsically meaningful and conclusively verifiable and tried

to explain the meaning and the empirical support of other statements in their terms. Estatements of the first kind were called observation statements. Estatements of the second kind are called theoretical statements. The distinction was retained by thinkers who did not accept intrinsically meaningful and conclusively verifiable statements, but who still thought that some statements being further removed from directly, though not conclusively, verifiable statements were more doubtful than others. The new version of the dichotomy that arose in this way

contained two elements. One, the logical fact that the examination of some statements of theory involves other statements of the same theory, while the examination of other statements does not. And two, the psychological fact that some statements are packed with perception while others are not. The elements are independent. Perception-packed statements of a theory may involve other statements of the same theory. For example, in case of psychoanalysis. While statements only loosely connected with perception may be tested without

analysis while a statement only loosely connected with perception may be tested sorry without reference to the theory example the position of one component of a stereoscopic double a star that is being examined as an instance of Newton's theory of gravitation now a stagmuller resolutely separates the two elements concentrates on the first and defines observable fact and empirical accordingly in this it seems to be in agreement with scientific practice for perception plays a negligible role in a mathem in mathematical physics

So, that's what a segmuller means by factual instead of observable or empirical in the naive Humian sense. Instead of observable facts, accordingly, a segmuller suggests that we are dealing with objects while in fact we are dealing with objects that are described in a certain way i.e using non-theoretical functions again fact or t-fact as a singular calls it seems to be a

A more appropriate term for a fact is always object in a certain situation. Of course, even now, it is easy to fall back into naive realism when dealing with such entities and to give in to an intuition that might be described as follows. The elements of MPP, possible partial model, are the things that are lying about in the world out there. We must always keep in mind that key facts are relative to theories. Asserting their existence makes sense only after a theory with clear and unambiguous methods of measurement has been specified.

This is important to understand, you know, what exactly a singular calls fact as differentiated from simple observable, so on and so forth. Hey Reza, real quick, where in the text is this discussion? Is that from chapter three? I can't remember exactly. I think it's actually later on that he… let me just get it.

Sorry, one second. I don't want to sidestep the discussion so I can... No, no, no, don't worry, don't worry.

So okay, here, I got it. Let me just go. It's actually section eight, Kuhn interpretation withdrawal of objections against Kuhn. Thanks. It's around page 61, something like that. So just to give this idea, so basically very, very, you know, brief introduction to understanding a model-theoretic view of logical reconstruction of scientific theories.

So as we can go on and actually investigate examples of how this logical instruction for case of you know some more classic theory particular theories works. Let's continue and then we can we can you know stop it today. So sorry one second I lost my line. Sorry, okay.

So theories, when we are without going into much formal details and I will again come back I just wanted so I think the best thing is that I will try to share my notes let me just go through them a little bit adding the symbols because it takes time I will share it so you can see it on the classroom so you get less confused about what are these formal relations so you can think about them. But just to conclude, having this view of this model theoretic approach to logical reconstruction of scientific theories, we can think about theories can also be

defined in an objective way, independently of linguistic expressions. If we start with a pair of a tube pole, a weld order set, SI, consisting of a mathematical structure, S, and a set of intended applications, capital I. The structure S consists of a core, you remember when we talking about a theoretical core or core of the theory when by that's a singular means this core contains the mathematical structure of the theory in the proper sense as defined by m by the model

the class of its models a function r capital r related to distinctions between theoretical terms and non-theoretical terms. And what he calls constraints, capital C. And these are constraints that what you might call to be essentially restrict the domain of application to the kind of logical core that you in fact have, the logical theoretical core that you have. So essentially, what we call the structure of a theory consists of a core. A core contains the mathematical structure of the theory as defined by the model, the

class of its model, a function R which is related to the distinction between theoretical terms and non-theoretical terms like position of a particle whereas as in contrast to the mass of a particle the mass of a particle is a theoretical function whereas the position is actually a non-theoretical one and also some constraints with regard to how dynamic or application of the theory is constrained by this logical core, by this theoretical core. And various expansions of the core, what Seymour calls expanded core.

And what are these expanded cores or the expansions of core? they are essentially they essentially introduce applications and the laws and special constraints that are valid in them remember that we talked about for Newtonian physics the second law is the core the second law is the core and in fact we are going to look at precisely how it can be formalized in this is neat a statement or so as we can make explicit the relations between all these

theoretical not theoretical a statement view not a statement view constraint so on and so forth in that logical structure. So the second law is the core in Neutronian physics and other laws are the expanded core. They are derived from the core but also they show, they exhibit or display the range of applications that are already implicit within the core itself in the second law here as i don't mean to cut you short but mo is telling me that there are some students who are in

adam burgh's class which is starting yes okay okay yes yes okay uh let's let's just let's just stop here is way too much details but believe me a signaler is the signal around Steve is the most difficult part we get into again some breezy landscape but not until next two or three sessions so hammer it down yes yes I will okay as Svitlana, would you continue? For next class, we keep reading Signular. Yes, and I actually share the notations and the symbols and the formalization

so you can go over them. Because I realize that it's very hard to concentrate about such relations if you don't have the exact notations, the exact symbols and formulas. Great. Thanks. all right thanks everyone i'm gonna end the broadcast now