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Tarski Statements as Rationalist Exercise

11 Post author: Vladimir_Nesov 17 March 2009 07:47PM

Related to: Dissolving the Question, The Second Law of Thermodynamics, and Engines of Cognition, The Meditation on Curiosity.

The sentence "snow is white" is true if, and only if, snow is white.

-- A. Tarski

Several days ago I've spent a couple of hours trying to teach my 15 year old brother how to properly construct Tarski statements. It's quite nontrivial to get right. Learning to place facts and representations in the separate mental buckets is one of the fundamental tools for a rationalist. In our model of the world, information propagates from object to object, from mind to mind. To ascertain the validity of your belief, you need to research the whole network of factors that led you to attain the belief. The simplest relation is between a fact and its representation, idealized to represent correctness or incorrectness only, without yet worrying about probabilities. The same object or the same property can be interpreted to mean different things in different relations and contexts, indicating the truth of one statement or another, and it's important not to conflate those.

Let's say you are watching news on TV and the next item is an interview with a sasquatch. The sasquatch answers the questions about his family in decent English, with a slight British accent.

What do you actually observe, how should you interpret the data? Did you "see a sasquatch"? Did you learn the facts about sasquatch's family? Is there a fact of the matter, as to whether the sasquatch's daughter is 5 years old, as opposed to 4 or 6?

Meaningfulness of these questions is conditional on their context, like in the notorious "when did you stop beating your wife?". These examples seem unnaturally convoluted, but in fact every statement suffers from the same problem, you must cross the levels of indirection and not lose track of the question in order to go from a statement of fact, from a belief in your mind, to the fact that belief is about. First, you must relate "sasquatch" on the TV screen to an actual sasquatch, which fails if there isn't one, and then you need to relate the sasquatch to his daughter, which again fails if he lies and doesn't have one.

By contemplating plausibility of the surface interpretation of a statement that doesn't yield to that interpretation, you fail before you even start. If the surface interpretation of the data doesn't apply, the data doesn't say anything about your interpretation. To form accurate beliefs about something, you really do have to observe it. When you misinterpret the data, the misinterpretation comes from you, not from the data, and so the data doesn't say anything about the thing you misinterpreted it to mean.

Every piece of evidence tells something about the world, every lie speaks a hidden truth. By correctly interpreting the data, you may learn about the process that generated it, and further your expertise in correctly interpreting similar data. If your mind lies to you, it is an opportunity to learn the algorithms that constructed the lie, to right a wrong question and come out stronger from the experience.

Consider a setting with a note lying near an apple stating "the apple is poisonous". There are two objects in this scene, the apple and the note. We associate an interpretation with the note, a statement of fact "the apple is poisonous". This statements corresponds to the note pretty much unambiguously. The statement implies that we should associate a non-obvious interpretation with the apple, "poisonous thing". The object (apple) and a newly introduced interpretation ("poisonous thing") are related to each other by means of our interpretation of the note. The interpretation of the apple as being poisonous is valid if and only if the statement of fact we read in the note is true. And this is what's stated by the Tarksi statement for this situation. The Tarski statement relates two things: the truth of interpretation of data, and the interpretation of the fact this data is about. It commutes two pathways by which we arrive at the fact stated by the data. First, the fact is an interpretation of the state of the world. We start from the apple, and go to the associated "poisonous thing" interpretation. And second, the fact is referred to by the interpretation of the data. You start from a note, proceed to interpreting it to say "the apple is poisonous", and interpret the statement a second time to extract from it the relation between the apple and the statement "poisonous thing". And so you state: the sentence "the apple is poisonous" is true if and only if the apple is poisonous. Or, in other words: the sentence "the apple is poisonous" is true if and only if the interpretation "poisonous thing" applies to the apple.

Each of the transitions between the levels of indirection may turn out to be wrong. Of reality, we only see the note and the apple, in this model we are pretty sure they are present. All the rest are the constructions we associated with them, following interpretation of the evidence. First, we construct interpretation of the scribbles on the note. To do that, we must understand that the note is intended to read literally, in the language in which it's written, not in some code, so you won't see "Tom is a spy" when you read the note. Second, you interpret the statement that you read from the note to refer to this particular apple, and to a particular property, "poisonous thing". The note could be left on the scene by mistake, referring to a different apple. There is no poison in the scene as you see it, the poison is only in the form of your interpretation of the statement read from the note. And then you decide whether the statement is true, and if you decide that it is, you make a new connection between the property "poisonous thing" and the apple, you start interpreting the apple itself as "poisonous thing".

We build our knowledge about the world step by step, and every step hides a potential error. Consciously inspecting the suspect steps requires constructing a model of these steps, with all the necessary parts. Rational analysis of beliefs and decisions in the real world requires not just understanding of mathematics of probability theory and decision theory, but also the skill of correctly applying those theories to the real-life situations. Learning to explore the relations between the truth of statements of fact and perception of the facts referred to by these statements may be a valuable exercise. Constructing Tarski statements requires mathematical thinking, keeping in mind many objects and relations between them, interpreting the objects according to the intention of current inquiry, constructing new relations based on those already present, and integrating them into the understanding of the problem. At the same time, the domain of Tarski statements is the reality itself, "obvious" common-sense knowledge.

Comments (8)

Comment author: Cyan 17 March 2009 08:14:37PM 4 points [-]

"how often do you beat your wife?"

I never beat my wife. The notorious question you're looking for is, "when did you stop beating your wife?"

Comment author: hhadzimu 17 March 2009 08:21:32PM 5 points [-]

I actually think it's the marginally different "Have you stopped beating your wife?" which allows for yes/no answers only, except that neither will help you.

Comment author: Vladimir_Nesov 17 March 2009 08:27:34PM *  0 points [-]

The notorious question you're looking for is, "when did you stop beating your wife?"

Right; fixed.

Comment author: PhilGoetz 18 March 2009 11:57:04PM 2 points [-]

Perhaps I would understand what you're getting at if you explained what a Tarski statement is. Google doesn't know.

Comment author: Vladimir_Nesov 19 March 2009 01:17:20AM *  1 point [-]

"Tarski statement" is a naturally made-up expression, just like "Litany of Tarski", that describes the statements of the form given in the citation in the epigraph and later in the article. For example:

[T]he sentence "the apple is poisonous" is true if and only if the apple is poisonous.

It's even used in a few papers as the Google test shows.

Comment author: Johnicholas 18 March 2009 11:46:27AM 1 point [-]

You mentioned something commuting with something else. Can you draw a diagram or write an equation of this? It doesn't have to be complicated.

Comment author: Vladimir_Nesov 18 March 2009 01:51:34PM 1 point [-]

By that I only meant what I described in the same paragraph: that the relation between the apple and its property is the same one in two contexts: when you start from considering the apple itself, and when you start from the note. The same fact, integrating the evidence from two sources. It's an loose analogy with commuting in category theory, you arrive at the same result no matter which path you followed.

Comment author: Johnicholas 18 March 2009 02:40:03PM 3 points [-]

Yes, I'm familiar with category theory commutative diagrams. Still, I think it would be beneficial to draw the diagram (or write the equation - I can draw the diagram from the equation), because it requires you to name the elements, the dots and the arrows.