Episode 194: Alfred Tarski on Truth (Part One)

0:00 Mark: You're listening to the Partially Examined Life, a podcast by some guys who at one point set on doing philosophy for a living, but then thought better of it. Our question for episode 194 is what is truth? We are focusing on the work of Alfred Tarski, reading his 1944 article, the Semantic Conception of Truth and the Foundations of Semantics, plus Tarski's Theory of Truth by Hartry Field from 1972 and Donald Davidson's essay the Folly of Trying to Define Truth from 1996. For links to these articles and other information, Please check out partiallyexaminedlife.com this is Mark Linsenmayer, materially adequate, yet formally deficient, in Madison, Wisconsin.

0:35 Seth: This is Seth Paskin, referring to Seth Paskin in Austin, Texas.

0:39 Wes: This is Wes Alwan, satisfying the sentential function quote. X is in Cambridge, unquote in Cambridge.

0:47 Dylan: This is Dylan Casey, bi conditionally in Madison, Wisconsin.

0:51 Seth: There is an X such that I

0:53 Dylan: was going to do air quotes, but then let's do so much better.

0:59 Mark: So we had all of truth to pick from, and this is what I picked.

1:04 Dylan: I knew by the time I got done with it why you picked it. Tarski seems to be the guy, right. Since the 1930s, he's like the stake in the ground as far as 20th century discussions seemed, at least in the secondary literature that Wes referred us to. Field, of course, is referring directly to him, but Davidson, Right. He talks about other people, but Tarski keeps coming up over and over and over again.

1:26 Wes: Yeah, Tarski comes up so much in analytic philosophy, and this idea that he's given a sort of metaphysically neutral definition of truth that kind of serves as a starting point for other debates. That's very common even in this book. I recommend all the time, Nietzsche on Truth and Philosophy, Maudemarie Clark uses the Tarskian definition of truth as sort of the neutral beginning point.

1:51 Mark: I think the example that you gave at the time is just whatever your definition of truth is, it has to fit the description that the sentence snow is white is true if and only if snow is white. Right. So that's the very, very short version that doesn't really capture his theory, but that is often taken as the starting point.

2:09 Wes: Yeah, that's not his definition, but it's what he calls his material criterion, the equivalence of the form T. Until we discuss this a little bit more, and it sounds crazy and trivial, and it's always sounded that way to me. I hadn't dived deep into any of this stuff, but I'm glad after doing these readings that I Feel justified in my idea that it's insane.

2:31 Dylan: Well, I liked how he started with Aristotle's definition and then proceeded to try to revise it. But the Aristotle one still seems to make a lot more sense. Tarski, at the very beginning, refers to Aristotle's definition, the classical Aristotelian concept of truth from Aristotle's metaphysics. To say of what is that it is not, or of what is not that it is is false. While to say of what is that it is or of what is not that it is not is true.

3:07 Mark: So that ties it straight. It sounds like to a metaphysical claim, right?

3:12 Dylan: Yes, because of all those is.

3:15 Mark: Yes, it sounds like then truth should have something to do with, like, you have to have a metaphysical theory before you know what truth is. So you might think that you're a realist, so that there really are things out in the world, and then a statement is true if it's an accurate description of the way things out in the world are. Or if you're an idealist, you think everything is just in your head. Well, I mean, what does that mean? It has to be something more complicated that the sentence is true if it corresponds with the ideas that are in my head. But the ideas then have to be described. You know, idealism always has this problem of how do you account for objectivity? Truth is sort of indelibly tied to some form of objectivity. And so whatever metaphysics it is, as long as it has the capacity to talk about objectivity, then you can talk about something being true or not true.

4:00 Dylan: Yeah. So who knew Bill Clinton was the great critic of Aristotle when he asked what it means? It depends on what is is.

4:09 Wes: Right. And, Mark, when you say objectivity, we just mean mind independence, right?

4:14 Mark: Not even necessarily. Because the way I was just describing it is if you're a idealist and you think truth is in the head,

4:21 Wes: independence from our particular minds, from any individual person's mind. So the way Berkeley grounds his idealism, for instance, it's ultimately God's mind is the sort of, instead of the material world, we are measuring things against what's in God's mind. And instead of being affected by material objects to have certain perceptions, God is just putting those ideas in your mind.

4:46 Dylan: And that's sort of the most extreme version, at least as far as I can tell. There's a kind of gradation. Right. Going to idealism in that direction versus realism.

4:56 Wes: Yeah. So the most famous mark, as you mentioned, the one that we've dealt with a lot in our analytic philosophy podcast, is this correspondence theory of truth, where a true proposition or belief or sentence in some way corresponds to a fact or state of affairs in the world, and what corresponds means can be more complicated than you might caricature. It can be about a certain kind of structural isomorphism, which I think is what early Wittgenstein was thinking about the way in which propositions are structured, matching the structure of the world.

And you have to imagine here sort of a network of the most primitive propositions, and then they're all linked together into more complex propositions and they match the texture of the world, where the facts, atomic facts, are built up into more complex facts in exactly the same way. And you can sort of lay that one fabric over the world, let's say, and have them match up.

6:03 Mark: So the fact would be the metaphysical object and then it would be the sentence or proposition, the thing that is fundamentally in your mind or that people talk about, people express. Let's say that is the thing that has to correspond. You can't say the fact is true, that's redundant.

6:19 Seth: The fact is not a metaphysical object, mark it. The fact is something having to do with the world. The whole idea of correspondence is that there's a relationship between whatever we take to be the content of our consciousness, whether we call it propositions or beliefs

6:34 Wes: or propositions aren't contents of your consciousness. But we should get to that.

6:39 Mark: Every single thing one of us says is problematic. Go ahead.

6:42 Seth: That's right.

6:44 Dylan: I think every single thing you guys have said, there's been somebody who said it, at least according to the Stanford articles.

6:51 Seth: Yes, exactly. So maybe let's abstract a little bit from it and say, look, there's correspondence theories of truth, there's coherence theories of truth, there are different flavors of attempts to define what truth is that are coming out of long tail philosophical tradition. And also simultaneously, what's happening, turn of the century, let's call it. And so Tarski, even this is problematic, right? Is that Tarski thinks he's cutting through the swath and giving like a very clear, non metaphysical, non correspondence, non coherence theory of truth, which is why he calls it the semantic theory of truth. But at the same time he's a mathematician, and so he's not really that wound up about the metaphysical or philosophical consequences of what he's doing either. He's thinking more in terms of like set theory and other sorts of things, problems that he's trying to solve.

7:49 Wes: I just want to say, as the part of this preparatory background we're giving, since we're talking about truth, and we're going to be talking about truth as a property. We want to say what it's a property of. And over the history of analytic philosophy or philosophy in general, it's been treated as a property of different things. So, yes, it may be a property of a belief, of my particular belief about the world. But when philosophers get to the point where they talk about propositions, we're in the realm of sort of quasi Platonic entities. This is why Davidson says we.

We're not really comfortable with these things, propositions, these are the entities that. Right. Mark and I can have the same belief, which is to say that we can have two different belief tokens that are the same in the sense we believe the same proposition. But the proposition would be one thing, not linked to. It's not psychologistic. In other words, it's not the particular thing in my mind or Mark's mind in Tarski he's going to talk about. Truth is a property of sentences, which on the face of it sounds like what it's not, which is like, well, sentences as in the markings on the paper or something like that, where the property of the sentence I'm speaking now is that it has the letter O in it.

In fact, that's not what we mean when we talk about sentences in this context. We're talking about what we call interpreted sentences. So we are talking about meanings. So a property of the sentence that I'm speaking now is it employs the concept sentence, let's say. I don't know if that's exactly accurate, but you get the idea. I just think before we go into this and start talking about sentences, which we will. We want to give that context because it sounds weird.

9:35 Mark: I can't resist throwing in here if folks want to listen to. Seth and I did a close reading quite a while ago that's on the website on Heidegger's essay on the essence of truth. And I bring this up only because we just finished the whole St. John's, our last episode was about to really understand a concept, you have to go through its history. And Heidegger is one of the guys who did this exact thing in this way. And he actually, you know, Wes was just saying the conflict between some of the guys we're reading, it does refer to propositions or does it refer to sentences?

It certainly doesn't refer to things like that. Doesn't make any sense to say it refers to things. But that's actually what Heidegger's. To really understand the question of truth. The essence of truth. You have to look at it, the history of the word, and it was actually used first, he says, to refer to the genuineness of things. So, like, is it real gold, is it true gold, or is it fake gold? And like, if that's the key to understand anyway, these guys all would roundly reject that as. No, we're talking about semantics now. We're talking about the meanings of words.

10:36 Dylan: Basically, you're saying that all these guys reject that claim about what truth is. You're not saying anything about the historical argument, is that right?

10:47 Mark: I'm not sure what to make of the historical argument. It's maybe something we can reflect on as we go forward here. Does it help to think about that we only use that term about propositions and senses now? Because that somehow evolved from using it to talk about whether gold is real gold or fake gold. I don't find it illuminating, and I don't think Tarski would have either.

11:08 Dylan: Okay, you're making a crack at our previous podcast. That's what you were doing.

11:12 Mark: I understand. Okay, so historically, this is a 1944 article by Tarski.

11:19 Dylan: We'll put aside the fact that Tarski starts with Aristotle. Just saying.

11:22 Mark: Okay, there's a lot of historical things one could say, but as far as Tarski is concerned, yes, this article is from 1944, but this is actually kind of a later reflection and it's more philosophical. Then. He originally came up with this definition in a 1931 article, or 31 or 33. He wrote a few things at that time, and it was very much straight up just referring to artificial logical languages. In other words, it's an extension. If you want to go listen to our Godel episode or the one by Russell on math, you know, those guys were concerned. In fact, Tarski explicitly refers to Godel's incompleteness theorem, that in a consistent formal language, it's not possible that all the true sentences should be provable.

He's actually going to use that insight to come up with a parallel one about in his own theory. And he is concerned with some of the same things that Godel and Russell were concerned with, which is paradoxes. It's a problem if you have things like the liar's paradox, where you say this sentence is false. Any semantic theory of truth has to be able to make sense of why that is. That sentence, the nonsense. How could it exist?

12:29 Wes: Well, you can fully formalize the language, I think is the problem, if you allow for that paradox. Right.

12:37 Dylan: And that's one of the keys for Tarski Right. Is restricting a language so that its structure is exactly specified. That's the way that you manage to get a way of dealing with truth that works is you have to have it within a language whose structure has been exactly specified. And when I read that, and then I read other comments, that Tarski ends up being very important for people who are thinking about evaluating models and mathematical models and doing modeling in general. That makes complete sense to me that this way of thinking about the problem would work perfectly well for a constructed world like a mathematical model or a computer model, because you would be able to have a closed system in which all the rules were exactly specified and all the entities and had exact specifications. And therefore you can have a truth definition that works completely robustly within that universe.

13:34 Wes: So to be exactly specified is to, quote, characterize unambiguously the class of those words and expressions which are to be considered meaningful. To say which terms are primitive or undefined, and then say how other words are defined in terms of those primitives, and then, say, give criteria for what sentences are and whether they can be asserted. Primitive sentences which are axioms and then the rules of inference. That's all what we are doing when we give a specified structure to a language.

14:06 Dylan: And importantly, that specification, as you just said, Wes, can involve axioms and things that aren't defined, so to speak, but are just features of the language. And you take them that way and then you work from there.

14:21 Wes: Yeah, you essentially refer only to form. You're formalizing the language and in your. What will turn out to be a metalanguage. Yeah, you're strictly talking about the forms of the object language. So, for instance, sentence snow is white. The form of that sentence can be given a characterization in the metalanguage. So there exists an X such that X is W. For instance.

14:44 Seth: Can I go to the text? Because I feel like you guys have jumped off of a high springboard into the deep end. And I want to. I want to wade in a little bit, please. So this is from the Tarski paper, page 6, section 5. Truth as a semantic concept. I should like to propose the name the semantic conception of truth for the conception of truth, which has just been discussed. Semantics is a discipline which, loosely speaking, deals with certain relations between expressions of a language and the objects or states of affairs referred to by those expressions.

As a typical example of semantic concepts, we may mention the concepts of designation, satisfaction and definition, as these occur in the following examples. The expression quote the father of his country unquote designates or denotes George Washington Snow. Satisfies the sentential function or the condition X is white. The equation two times x equals one defines or uniquely determines the number one half, while the words designates, satisfies, and defines express relations between certain expressions and the objects referred to by these expressions.

The word true is of a different logical nature. It expresses a property or denotes a class of certain expressions, these sentences. So this is his major insight. This is the thing that gets everybody in a tizzy for the 20th century, that there are, again, different kinds, semantic relationships which define a connection between an expression and an object to which the expression refers or designates or defines or what have you. But truth, for Tarski, is that it's a property and not a designation or a referral or what have you. That is the nut of the Tarskian concept of truth.

16:54 Wes: Well, not a relation, I guess, would

16:56 Dylan: be the truth is not a relation.

16:58 Wes: This idea of these more typical semantic expressions like refers or denotes, for instance. Right, we see that. I mean, he says explicitly it's a relation between language in the world, basically between expressions and objects. I think precisely he wants to avoid true being a word of that, even though it's semantic.

17:21 Mark: Why is. Looking at the second one, though there. Snow satisfies a sentential function. The condition X is white. So it seems like you could say there's the set of all things that are white, and snow happens to be one of those things. That's not a relationship between snow and whiteness. He's saying true is a set, a property, denotes a class of certain expressions. So maybe snow is not an expression, but let's say snow is a verb just to make an expression. Snow in quotes is a verb. So can't we say that's not a relationship between snow and being a verb. It is the class, the class of all verbs. So the true is the class of all true sentences, and verb is the class of all, well, verbs.

18:07 Dylan: Well, I don't know about that. I think you would have to understand in this case that X is a noun. Right, because verbs aren't white.

18:16 Seth: Right?

18:17 Dylan: Yes, I want to get to that one. Can we do the third one first? Because that one, to me, is the one that makes the most sense.

18:27 Wes: Can I just say one thing before we do that?

18:29 Dylan: Yeah.

18:30 Wes: So he's saying these basic semantic notions. Although, Mark, you're right, satisfies is a special one, because he's going to define truth later in terms of satisfaction and not the others. So that's very important.

18:44 Dylan: What are the other ones?

18:45 Wes: Denotes, for instance, he does not want to Define truth in terms of denoting.

18:50 Dylan: Oh, he doesn't include designation, satisfaction, designation and definition don't all work.

18:56 Wes: Well, he doesn't want to define truth in terms of denoting. This is really what Field is onto with that paper. He wants to get down to a definition purely about satisfaction. And we can talk about why that is later. But the main idea here at this point, he's saying is that when I define truth, I'm going to be defining it in terms of these other semantic notions. That's what makes it the semantic conception of truth. Okay, so before we kind of glossed over the material criterion, the famous convention T. Right. Which is.

19:32 Mark: I don't think we glossed it. I thought we just hadn't gotten to it, but go ahead.

19:36 Wes: Well, is it okay to back up to that, or do we want to just finish this, what we were doing here before?

19:41 Mark: I think it might be easier to understand the math, the condition T if we get at what these basic things are.

19:48 Wes: Okay. What he calls the material criterion is just means we want to capture the way this word is already used when we give our definition. And that's why he brings up Aristotle. But then he wants to give a more formal sort of not a definition, because this convention, we haven't reached the definition yet, but a. What they call a schema, which will serve us well when we get to the point where we have the definition. And that, as we said, that schema is this idea. Snow is white, for example, is true if and only if snow is white. And this idea that it's in quotes, right, is important because we're essentially referring to the sentence.

We're not asserting it, but we're referring to it in some way, or we're naming it. So when we point to the sentence itself and then ascribe to it the property true, and then if and only if. And then we assert the sentence without the quotations. So the definition of truth ultimately is going to be material adequate. In other words, it's going to capture our common intuitions about truth if all equivalences of the form T can be asserted to follow from it. So, for instance, if our definition of truth works, we're never going to get a case where we say, quote, unquote, snow is white, if and only if snow is not white or something like that.

We're always going to satisfy this intuition that we have between the ascribing a property true to a sentence and then being able to assert the sentence.

21:17 Mark: Does anyone know the background of that term materially adequate there? Because I mean, the way that you started to introduce Wes, it made it sound like, oh, it has to go with our intuition. That's what makes it material adequate.

21:27 Wes: Well, he says that he defines material adequacy himself. That's the only reason I know what it means.

21:32 Mark: The way that I had read it is that it's materially adequate because of the last thing you said, which is that it defines every single true sentence as true. In other words, if it only applied in some cases in the language, it would not be materially adequate even if it worked, you know, very well for a lot of cases. And it seemed like a good theory. It has to the sentence. We already said that this only works for languages in which every single element in it is unambiguous. In other words, it really is. Every single sentence that is, that we're going to admit as grammatically well formed.

22:06 Seth: They're all fully defined.

22:08 Mark: Yes, fully defined, I think. So it has to either be true or be false. And so for it to be materially adequate, then it has to for sure catch all of them. It has to make all of the true ones true and all the false ones false.

22:21 Wes: Can I just defend really quickly where I got this from? Just on page 341, we must first specify the conditions under which the definition of truth will be considered adequate from the material point of view. The desired definition does not aim to specify the meaning of a familiar word used to denote a novel notion. In other words, he's not just arbitrarily defining a new word to mean such and such. On the contrary, it aims to catch hold of the actual meaning of an old notion. So material adequacy just means. Here I'm not just arbitrarily defining something.

I am actually trying to get hold of something that we already, you know, a concept that we already use intuitively, or at least to satisfy that on some very basic level. I don't have to capture all of it, but I don't want to contradict that. I don't want to say something that is completely out of line, for instance, with what Aristotle said about truth.

23:12 Dylan: And in some ways he wants to preserve the piece of correspondence theory that we want to cling most to. Well, maybe that statement is too strongly.

23:22 Wes: Yeah, I know it's hard to say. It seems like it, but he might object to that. But it does seem like that there's

23:28 Dylan: a piece of correspondence theory that feels intuitive about it, that we want to say that something is true if it corresponds to reality. And that kind of intuitive Sense is what I think. That seems to me that he's trying to preserve, but notched down.

23:45 Wes: Right.

23:45 Dylan: He wants to remove the corresponding to reality part at all, make it metaphysically neutral.

23:51 Mark: Yeah. The passage you just read, Wes points out kind of what a frustrating writer he is. Let me just read that again. In order to avoid any ambiguity, this is like only, you know, this is on page one. This is like the second real paragraph of the whole thing. In order to avoid any ambiguity, we must first specify the conditions under which the definition of truth will be considered adequate from the material point of view. The desired definition does not aim to specify the meaning of a familiar word. Blah, blah, blah. Okay. So that in other words, you can see that the second sentence there as explaining what he means by a material point of view, or the way that I read it was that he was using material point of view as a technical term, assuming you already know what it meant, and then going on to make a separate point, though related, about how it's supposed to capture the familiar meaning of the word.

So I did not see him as defining material point of view. I just saw him as using it there. And I'm not. I don't know which of us is right, but I think he's a bad writer. The fact that we don't know that he could have put a colon there and then it would have been clear.

24:55 Wes: It seemed pretty clear to me. Because he says material adequate and formally correct. And then he gives a paragraph about material adequacy. And then he says, secondly, we must determine on what the formal correctness of a definition depends. Just the whole structure of that section is. Introductory paragraph, materially adequate, formally correct paragraph on what we mean by material adequacy. Paragraph and what we mean by formal correctness.

25:17 Mark: Yeah. See, if you like this definition of formal correctness, then secondly, we must determine on what the formal correctness of the definition depends. Thus, we must specify the words or concepts which we wish to use in defining the notion of truth, and we must also give the formal rules to which the definition should conform. Speaking more generally, we must describe the formal structure of the language in which the definition will be given. I guess that just sounds so weird because it's just not a way we're used to thinking.

25:45 Wes: He's just saying we have to put it in the formalized language that we have specified in. The way you specify language is right. So just for listeners, the sort of prototypical formalized language is logic, and logic is formally specified in the sense that the formal adequacy of the definition depends on us performing this first step where we formalize the language that we're using and say what the words mean, say what the axioms are, say what the most primitive terms are, blah, blah, blah. All that stuff we talked about earlier,

26:15 Mark: what we didn't do is motivate briefly, before we get into the details here, why anybody would care about a formal language unless they're just doing math. And this is a lot of, I think, what people reacting to Tarski are talking about. Tarski has some commentary on it later in the paper himself, but he's specifically saying, this concept of truth that I'm about to give, this definition of truth, it is language dependent, and it only applies to these kind of formal languages that I've laid out. Later he says, well, I think it should be able to extend to, say, specialized technical vocabulary in certain sciences.

26:49 Dylan: But natural languages might not work.

26:51 Mark: Yeah. And this is just part because I think back in our old episodes, like our first Carnap episode and Wittgenstein's Tractatus episode, we were talking about this desire that people had toward the beginning of the 20th century. You know, ordinary language is too messy. It gives rise to all these philosophical problems. What we need is clarity when we're actually trying to talk accurately about anything. Maybe we should not use ordinary language in its horrible histories that has stuff like that piece that I mentioned from Heidegger. In it, we need to use formal systems.

And then I guess the hope is we can kind of extend it out insofar as the ordinary language resembles the formal system, then we can say something about ordinary language. But the goal is never really to say something about ordinary language. It's to get at truth through a more exact language than ordinary language will allow us.

27:42 Wes: Just real quick, the reason why you don't use an ordinary language is because of ambiguity. They want to avoid ambiguity. So you want every term, every name to be unambiguous, to mean one thing and not the other. So a good example in English given in one of the other papers, John takes grass, for instance, can mean that he's taking grass from the lawn, or it can mean he's smoking marijuana. The whole idea of formalization is to avoid all that ambiguity. You simplify, and you're not dealing with any of that messy stuff that we have in natural language.

28:14 Dylan: And in fact, Tarski's strong claim is that truth is a characteristic only of such formal languages. If you don't have a formal language that satisfies the condition of having all

28:27 Wes: the elements uniquely specified well, you can't define truth.

28:32 Dylan: That was the point that truth isn't a property anymore of it because you can't define it.

28:36 Wes: Well, it can be a property, but you can't define it. The language has to be not semantically closed. Right. And it has to be formalized, otherwise you can't define it.

28:45 Dylan: I guess from Tarski's point of view, I don't know that there's a difference there, right, between it not being able to be defined and it not being a property.

28:53 Wes: Well, he's not saying we can't call sentences true in English. He's not saying, oh, we can't assert that English sentences are true because it's not a formalized non semantically closed language.

29:02 Dylan: This is the point that's going to. Maybe we should bracket off and talk about more later is exactly the way in which he means that you can still talk about truth and the limits on which you have in a natural language. Because if it's not a formal language, it's not clear to me what it means to then be talking about it in a natural language, given his definition, like it's not clear, it has to

29:25 Mark: be some sort of metaphorical extension.

29:28 Dylan: Yes, a metaphorical extension, Yeah. I mean by truth, this kind of

29:31 Mark: thing, the formal language is kind of like a simplified model. And insofar as natural language resembles the formal languages, then yes, we can talk about truth in natural language. But that's not his business to actually, you know, this is why the difference between early Wittgenstein and later Wittgenstein and J.L. austin and the folks we talked about. There was a industry wide shift from this formalization project to actually analyzing natural languages and try to make sense of the logic that is inherent in them rather than this normative logic of a formal language, that the relationship to natural language is by necessity unclear, because natural languages are unclear and that's the point of them.

30:12 Seth: This is a quote from Tarski, page 342, section 2. The predicate True is sometimes used to refer to psychological phenomena such as judgments or beliefs, sometimes to certain physical objects, namely linguistic expressions and specifically sentences, and sometimes to certain ideal entities called propositions by sentence. We understand here what is usually meant in grammar by declarative sentence as regards the term's proposition. Its meaning is notoriously a subject of lengthy disputation by various philosophers and logicians, and it seems never to have been made quite clear and unambiguous for several reasons.

It appears most convenient to apply the term true to sentences and we shall follow this course. Consequently, we must always relate the notion of truth, like that of a sentence, to a specific language. For it is obvious that this same expression, which is a true sentence in one language, can be false or meaningless in another.

31:10 Mark: Yeah, so it seems like he's given up right here on the beginning of the paper on a general theory of truth that would apply to all languages. Just because, ah, that term proposition that's supposed to be, you know, the thing that is expressed both by snow is white and Schnee is Weiss. That's too hard for me to understand. That notion proposition. Oh, people argue about that. Forget about that. We have to stay to something like a sentence that we can actually understand. So this is the logical positivist sort of attitude that is getting into his, you know what, he considers it to be a legitimate philosophical problem. Right at the beginning.

31:46 Dylan: Do we want to go back to those three examples of truth as a semantic concept? We read those three and I said that the third one, which involves a mathematical equation, is the one that seems to me most plain and also to me reveals a lot about the way he's thinking about this problem and just repeat it. The equation two times x equals one defines the number one half. And I think that you could rewrite that in the same way as the previous sentence. You could say the number 1/2 satisfies the equation 2 times X equals 1. So to me, what's interesting about that is it plainly associates it being true, the true statement with the relationship between that equation being satisfied by a particular value, as opposed to saying that the truth is somehow some kind of relation that's embodied in the equation.

32:43 Seth: Dylan, I understand what you're trying to get at there, but I think for Tarski that's not exactly going to work. Unfortunately, the triviality of the sentence snow is white unquote is true if and only if snow is white, which is essentially what this snow satisfies the sentential function quote X is white unquote, that relationship, this notion that snow is the X and so white can be predicated of snow is not the same thing as 2x equals 1 can be predicated of 1/2 that. That's why he's trying to make a distinction here, because it's unique in the latter case, no, because it's not the same sentential function definition. 2x equals 1 is not the same thing as predication that shows up in the second example. It's not the same.

33:32 Wes: 2x equals 1 is a sentential function.

33:35 Seth: It's not a predication.

33:36 Wes: That's Like a standard example of a sentential function. Just like X is white.

33:40 Seth: How?

33:41 Dylan: Because the equals is the is.

33:43 Wes: Pick up any textbook look up sentential function. They're going to use the math example before they use the X is white example.

33:50 Mark: And though it uniquely determines 1/2, in other words, 1/2 is the only answer you're going to get out of it, it is not the only one that establishes that definition.

33:59 Dylan: I can give you an infinite number of X's that will satisfy that equation. One half is one of them, just like snow is an X that is white.

34:06 Mark: So you're saying 1/2 and 2/4 and stuff like that?

34:10 Dylan: Yes, and many, many more complicated things. There's all kinds of things I could put in that X that will satisfy that equation.

34:18 Mark: Isn't the point here that this is what Wes was saying about why we're using a formal language is because, you know, we had in Frege the difference between sense and reference. We don't want to care about sense. We don't want to have problems. Like, is Superman the same as Clark Kent? Like, if Superman and Clark Kent refer to the same thing, they actually should be the same term.

34:37 Dylan: No, but all I'm doing is doing the same thing that's done in the previous line. Snow satisfies the sentential function that X is white. So, and I'm saying that 1/2 satisfies the sentential function 2 times x equals 1.

34:51 Seth: Oh, okay.

34:53 Dylan: And I could say 4, 8 satisfies a sentential function 2 x equals 1.

34:59 Wes: But by the way, even if there were only one object that satisfied the sentential function, like X is Wes or X is Cambridge, it's still a sentential function.

35:08 Dylan: Yes, that's a good point to make that it's not the multiplicity that makes it a sentential function, but it would

35:13 Mark: define it if there's only one. Unless we want to say X is Wes, you know, of course that's ambiguous. But if we want to say all the people named Wes, so in other words, X is Wes is like X is white, X is named Wes. Then clearly, even if there happened to be only one Wes in the world, if you then had a kid and named him Wes ii, then that shows that is Wes is not a definition. Whereas if you're taking Wes to define uniquely, that means anything else that falls under that would have to be you. I agree that a definition is a type of sentential function, but it's easy to say why. It's because everything that satisfies it has to be identical to everything Else that satisfies it.

35:49 Seth: Yes, A definition in this form is an assertion of identity. Because if you were to say, like X is white, snow is white, snow, which has a referring function, right, we predicate of it that it's white. But in order to get the same form X equals, if you think that equals is the same as is x is, then you'd have to essentially divide by two on both sides, right? And you would basically say one half satisfies the sentential function that X equals one half, unquote, and it becomes an identity relation, not a predication of a property.

36:23 Dylan: I don't think we want to go down this road of.

36:25 Seth: No, I really want to spend the next hour and a half talking about that.

36:29 Wes: These were just meant to be some innocent examples of semantic concepts.

36:33 Seth: No one is innocent.

36:34 Mark: It's good that we're dwelling on these, because these are other two field and I think Davidson too kind of dwell on what is the relationship between these things and true. These are in the section truth as a semantic concept, in the way that Seth actually just introduced it or reintroduced it I think, a little bit ago was that snow satisfies a sentential function. X is white. Well, that's to say that it is true that snow is white. But he specifically is trying to give these things without using the word true in them, even though you might look at it and say, look, these are supposed to express things that if you asserted them, you were taking them to be true.

Or the expression the father of his country designates George Washington. That is the same as saying it is true that the expression the father of his country designates George Washington or something like this. So you could accuse him of at least surreptitiously basing these on the notion of truth. You have to have the notion of truth before you have any of these. But the way he's presenting them, he's saying these are just examples of semantic concepts. True is also a semantic concept, but it is of a different logical nature. And as we said, he's actually going to base it on the satisfaction one so that he derives truth from satisfaction, not the other way around.

He doesn't derive denotation from truth or definition from truth or satisfaction from truth. Those things have to be all independently out there as sensible expressions, and then we can get truth.

37:58 Dylan: I agree with you, Mark, but that's why I like the math example in particular for understanding what he means by saying that truth is a characteristic of that relation. You're not saying that it's true that it's one half. You're saying that what I mean by something being true is that phrase we

38:17 Wes: should just be careful because we haven't gotten to the actual definition of truth yet. I just want to remind listeners of that because the actual definition of truth turns out to be even crazier than the equivalence T. Okay, it's mind blowing. It's less obvious what it is in Tarski's paper than when we get to the other discussions of it.

38:35 Dylan: Before we move to that, I want to insert some comic relief just because we have read some other quotes from him. So at the end of this section on truth as a semantic concept, he has these funny asides which I don't know if I've ever seen in papers before.

38:49 Wes: Grumpy Tarski.

38:51 Dylan: Yeah, so the last paragraph on page 345. It is perhaps worthwhile saying that semantics as it is conceived in this paper and in former papers of the author is a sober and modest discipline which has no pretensions of being a universal patent medicine for all the ills and diseases of mankind, whether imaginary or real. You will not find in semantics any remedy for decayed teeth or illusions of grandeur or class conflicts. Nor is semantics a device for establishing that everyone except the speaker and his friends is speaking nonsense.

39:23 Wes: Yeah, that's great.

39:24 Mark: I'm not a logical positivist. That's what he's saying there.

39:27 Wes: There's a lot of that stuff in this Grumpy Tarski going off on philosophy.

39:32 Dylan: Don't ask me to solve real problems.

39:36 Wes: So I think we can skip over. We talked about language with a specified structure. We can just gloss very briefly over the antinomy of the liar stuff. He wants to. In the same way Russell deploys a theory of types to avoid self reference paradoxes. He wants to talk about a metalanguage and an object language.

39:56 Mark: This sentence is false. The sentence I'm speaking now is false.

40:00 Seth: Yeah.

40:00 Wes: So that's what he calls a semantically closed language where you can sort of make those types of descriptions. And he's just going to arbitrarily say no, we're going to divide things up into a metalanguage where we can talk about an object language. The object language is not going to be able to do that. And so we're not going to get into those sorts of self referential paradoxes.

40:21 Dylan: I don't know that much about it, but apparently some people consider his preoccupation with the liar's paradox to not be warranted. That he's over concerned about it.

40:32 Wes: Yeah. We could probably skip ahead to section 11. With 11 we get the outline of the definition, and that starts on page 352.

40:41 Dylan: Incidentally, I love the characteristic of a paper like this where we have had a 11 pages before we get to the construction in outline of the definition of the thing that he's going to talk about.

40:54 Mark: Well, just remember that we had Russell in the Principia Mathematica that took hundreds of pages to establish that one plus one equals two.

41:03 Dylan: Fair enough.

41:05 Mark: Touche.

41:06 Wes: That's true.

41:06 Mark: I'm just saying this is the tradition that he's coming from.

41:08 Wes: It's not that 13 pages to get to the truth with a capital T.

41:12 Mark: I don't know how long the 1931 paper was.

41:16 Wes: So he basically just starts out by saying, I'm going to derive the definition of truth from that of satisfaction.

41:21 Dylan: Yep.

41:21 Mark: And he's going to do it in the meta language, not in the language itself. We can only do it about a language. In fact, it's maybe simpler to say we're doing it for German. We're talking in English. English is the metalanguage. And we're going to say Schnee is Weiss or something is our example, just so we don't confuse ourselves.

41:37 Wes: Yeah. So to talk about things in terms of satisfaction is to talk about things in terms of these sentential functions, like X is white. And then we can define sentential functions by way of what he calls recursion. We can define the procedure. We can define sentential function by a recursive procedure. Yeah, we basically show how the most simple sentential functions, F of X, for instance, can be used to construct more complicated functions by way of logical operators like or. And. And so I don't think we need to dwell on that. That's not that important either.

But then we get to define satisfaction where we use recursion. Again, we say which objects satisfy the simplest sentential functions, and then we give those logical operators again for the conditions of compounding the more basic ones into more complex ones. But to define satisfaction is basically just to say which objects satisfy sentential functions.

42:39 Dylan: So let's just give this example on page 353. We indicate which objects satisfy the simplest sentential functions, and then we state the conditions under which given objects satisfy a compound function, assuming that we know which objects satisfy the simpler functions from which the compound one has been constructed. So that's the recursiveness. Right. If you're going to talk about a compound function based upon simpler ones. So he gives the example. We may say that given numbers satisfy the logical disjunction X is greater than Y or X is equal to Y if they satisfy at least one of the functions X is greater than Y, quote unquote or X is equal to Y, quote unquote.

43:21 Wes: Yeah, at that point, we're talking about satisfaction in these general terms. We haven't yet said what satisfies a particular sentence like snow is white. We've talked about satisfying X is white in terms of what objects are, the extension of that function. But now we have to say, and this is the really interesting part, what it means to satisfy something which is already saturated, to use the logic, something that already seems to be satisfied in the sense that it already has an object in the variable. The variable has already been taken up by an object like snow is white. What does it mean to satisfy something where the variable already seems to have been satisfied?

44:02 Mark: Yeah, you might want to just say that that doesn't make any sense, because satisfaction only applies to functions. If it already says what the thing is, what the X is, then it's not a function anymore.

44:12 Wes: So let me just read his words once. The general definition. I'm glad you guys think this is as crazy as I do, but once the general definition of satisfaction is obtained, we notice that it applies automatically also to those special sentential functions which contain no free variables, that is, to sentences. Yeah, they're very special. They're very special functions which contain no variables. Here's sort of the meat of the whole thing. It turns out that for a sentence, only two cases are possible. A sentence is either satisfied by all objects or no objects.

Hence, we arrive at a definition of truth and falsehood simply by saying that a sentence is true if it is satisfied by all objects and false otherwise. In other words, the definition of truth. You get it in a particular language just by listing all the sentences of the language where there's satisfaction. It's not just that to say that snow is white is satisfied. I list all the instances of snow being white. I also point to the black cat and to the red car and basically everything in the universe. That's what this means. I give a list of every possible sentence and object, or really objects, and I show that every object satisfies snow as white, which in other words means that I don't come across an object which contradicts snow as white.

Did you guys get the same thing out of that?

45:39 Mark: A good analogy for me was with the empty set, where you say the empty set is a member of every set. So in the Same way this finished sentence, which has no variables in it, it satisfies every single function.

45:51 Wes: Or it's satisfied by all objects.

45:53 Mark: Yes, exactly. You would try to put X is gold into snow is white and hey, fits just fine. There's nothing in snow is white that contradicts X is gold. So it fits.

46:06 Wes: Well, if it's an object, it would be a particular piece of gold, I think. Or am I wrong about that? So if I had the sentential function X is white, I could go around looking for the objects that satisfy it and say, okay, here's A, here's B, here's C. But with the sentence snow is white, I list every single object in the universe. Like I said, the black cat, the red car, this, that anything satisfies it. If it's true, it's satisfied by every single object.

46:32 Mark: Just like the empty set.

46:34 Dylan: Does his footnote address this?

46:37 Mark: I didn't read any of the footnotes

46:38 Wes: in this paper, by the way. I had this image of Oprah Winfrey. You know how she does the. You get a car, you get a car, you get a car going around and saying, you satisfy snows white, you satisfy snow is white. How bad is that joke? Cut it out.

46:53 Mark: That's brilliant.

46:56 Wes: So is that insane?

46:58 Mark: As a definition, I mean. Yeah, it's true.

47:01 Wes: As a definition. Just create a list of everything. Because I don't want to refer to the word denotes or something. I'm just going to. Field is going to get at this, of course, with valence and all that stuff. But just create a big list, then call it a definition.

47:14 Dylan: Well, yeah, so he. In his footnote, I think he is referring to the difficulty that you're talking about, Wes. He says that there's a certain technical difficulty that arises. A sentential function may contain an arbitrary number of free variables, and the logical nature of the notion of satisfaction varies with this number. Thus the notion in question, when applied to functions with one variable, is a binary relation between these functions in single objects. When applied to functions of two variables, it becomes a ternary relation, and so on.

47:45 Wes: Well, he's just talking about relations like instead of something being white, someone being the father of someone. That's what it means to have two variables.

47:54 Dylan: Yeah, okay, that makes sense.

47:57 Mark: I understand Wes, though I agree that it's weird and counterintuitive and doesn't seem to be what we mean by definition to just list all the things. Like if I asked you what's the definition of sheep and you just named all the sheep.

48:09 Wes: No, not just named all the sheep. Named everything in the universe.

48:13 Mark: Well, but there's another thing. So the bottom of page 344, it should be emphasized that neither the expression T itself, which is not a sentence, but only a schema of a sentence, nor any particular instance of the form T, which is what we were just given, can be regarded as the definition of truth. We can only say that every equivalence of the form T obtained by replacing P by a particular sentence and X by the name of the sentence may be considered a partial definition of truth, which explains wherein the truth of this one individual sentence consists.

This general definition has to be, in a certain sense, a logical conjunction of all these partial definitions. So it is kind of like naming all the sheep, but in just naming one of the sheep, you're naming all the objects in the universe, and that kind of lets you name one of the sheep, and then you have to name the rest of the sheep, and then you at least have the definition of the sheep for this world with these sheep in it. That's just an analogy again, but that's how far from what we think a definition is.

49:08 Wes: The general definition has to be, in a sense, a logical conduction of all these partial definitions. So, yeah, that's just another way of saying we'd state every true sentence in the language. And that's the definition of satisfaction. No, that's the definition of truth. I mean, that's the kind of preliminary of the definition of truth. We have to recast that in terms of satisfaction. Yes, yes, right.

49:28 Mark: You've said that one sentence. Snow is white is true because of the listing of all these sentences. But just saying, I've now said what it is for this one sentence to be true doesn't actually define truth. You have to then say, oh, well, I've said what snow is white to be true. And I'm going to say what corn is yellow is true. And all the other sentences, you have to really list them all. And for each one of the sentences, as you said, you have to be complete, list all the other sentences.

49:54 Dylan: I think it's not quite as crazy as what you're saying, Wes. I agree with the way Mark is talking is that these objects, then it ends up being deeply recursive the way the previous example was. Right. He gave a rather simplistic example of a compound case with the logical disjunction, and then you would break it into the subparts.

50:16 Wes: Sorry, I think I've lost thread of what you guys are saying. I'm just still focused on this sentence. Hence we arrive at a definition of truth and falsehood simply by saying that a sentence is true if it's satisfied by all objects and false otherwise. What are you saying?

50:30 Dylan: It's not clear to me that all objects is not qualified in some way that we say all objects mean all objects in the universe.

50:38 Wes: I really do think it's all objects in the universe, especially based on the field thing where he's basically leveling this accusation. But say why you think it's limited when he says all objects? It's not.

50:49 Dylan: I guess part of it is that it just seems absurd otherwise. I mean, earlier we had something that amounts to a sentence is true if the condition in the quotation is the case and false otherwise. There's something about that that makes sense. It's understandable, at least as a way of talking about what you mean by true.

51:10 Wes: I think the way to get at this is to think about being satisfied as not being contradicted by. I think that's the sense in which all objects is important here and does refer to every object.

51:20 Dylan: That way of understanding all objects being everything in the universe. That does make sense, right?

51:25 Unknown speaker: Yeah.

51:25 Wes: It's just we take in a function which is supposed to have a variable, a free variable, and we think about satisfaction in terms of satisfying that free variable. Now we're thinking about satisfaction in this special way as satisfying a sentence in which the variable's been filled up. And then, of course, everything satisfies it in this trivial sense. Of course, if it's already had the variable filled, then, yeah, everything.

51:46 Mark: But it would seem weird if that's really trivial. We filled it up and it's true, so everything else satisfies it. But you don't think you have to then list all of them, all the things that satisfy it, in order to define what truth means. Like, that just seems intuitively to be very weird. Once you've established that the variable's been filled in, why would you have to pay attention to anything else? I think it's just because you don't want something else in the universe to contradict it. Right.

52:11 Wes: We're just saying what it means for it to be true.

52:13 Mark: So what it means to be true is that nothing else in the world contradicts it. Is that part of it?

52:17 Wes: Well, I don't know if you put it in that way. He's being very careful to say, well, what it means to be true here. He doesn't want to use these other semantic terms like denotes or to say, hey, there's a state of affairs out there to which it corresponds. That's why we get this very shady approach here, which is trying to avoid as much as possible any of that stuff. So, yeah, I think in the definition, I think if you were to say what it means for the sentence to be true is for the sentence to correspond to reality, the definition I might give.

But I can't do anything like that here. I can't give any general definitional form here because this general statement, a sentence is true if it's satisfied by all objects and false otherwise. That's the generalization. But normally I'm saying, what does it mean for some sentence s to be true for language? L. And if I'm defining that, I don't know, I don't know. I'm a little lost now, but maybe I was influenced by Field and also this earlier thing that you read about the logical conjunction. So this is Back to page 344. The general definition has to be, in a certain sense, a logical conjunction of all these partial definitions, where all the partial definitions are X is true if and only if P.

Or in other words, a partial definition of truth is snow is white, is true if and only if snow is white. That's what we call a partial definition of truth. And then to get the general definition, we create a logical conjunction of all the partial definitions, and the partial definitions are all the true sentences of the language. Okay, so that's his little preliminary gambit there. And of course, yes, he's not going to leave it at that. He's going to translate that into satisfaction, where we say the definition of truth has to involve the satisfaction by all objects.

Anyway, I don't know.