On Sep 26, 2008, at 7:05 PM, John F. Sowa wrote: Pat, That is true: PH> Because a language is itself formalized, that does not require or even imply that it can only speak of mathematical entities.
Tarski's main example was the assertion that snow is white, and
he was talking about snow, not a mathematical simalcrum of snow.
First, Tarski's example "Snow is white" was an unfortunate choice
Indeed it was, for all the reasons you note. Im sure if he had the opportunity to go back a rewrite history, he would now choose a different example. Still, I am pretty confident that he did intend it to refer to actual snow, the cold white stuff, and not something Platonic and "mathematical". Which was my only reason for mentioning it. because it introduced several complex issues he did not explain in
that paper: (a) continuous substances such as snow, (b) a generic
statement without the use of quantifiers, (c) the question of how
FOL could be used to talk about continuous stuff, and (d) the
question of how a statement in ordinary language could be related
to the formal notation for which he defined his semantics.
Second, even in that paper, Tarski discussed the problems about
natural languages and explicitly admitted that a great deal more
work would be needed to apply his approach to those languages
and that he wasn't sure whether such an application was possible.
In that paper and in his later work (1944), he did not get into
details about many issues that are critical for relating any
symbols (formal or informal) to things and events in the world.
He certainly did not discuss *symbol grounding* which was the
subject of much debate a few years ago. Nor did he claim that
by using set notation the symbol grounding problem was avoided
or magically solved.
And he did not do this for a very good reason. He was talking about semantics, and these topics, fascinating and important as they are, are not about semantics. They are a DIFFERENT TOPIC. That is the point I have been trying to get you to understand. Tarski understood it.
In my earlier note, I mentioned Carnap, Popper, and others who
recognized that some such methodology is a necessary component
of a theory of meaning. It can be a supplement to Tarski's
models, but it is a necessary supplement.
Necessary for what? For ontology engineering? If so, why? PH> Maybe this misapprehension is the source of your blind spot with regard to semantics.
I respect your technical knowledge, but I believe that you're the one with the blind spot about the need for some kind of mapping to ground the symbols of a formalism.
I'm all for such a mapping. It is exactly the Tarskian interpretation mapping. The grounding problem is concerned with how to restrict the (large) set of possible Tarskian interpretations to a subset on which certain terms only denote certain kinds of thing. This is a deep and complex issue, but it not best approached by trying to replace or supplement Tarskian interpretations by some *other* kind of semantic mapping. The semantics we have is just the semantics we need. Its not broken, so stop trying to fix it. JFS>> Without [a methodology for relating symbols to reality], a Tarski-style model has no relationship whatever to anything
that exists in the world.
PH> WRONG. The fact that Tarski's semantic metalanguage uses set theory, does not entail that it can only speak of mathematical
abstractions... Set theory is not restricted to speaking only
of sets of abstractions.
I agree that the phrase "a set of apples" is as legitimate in English as the phrase "a bushel of apples".
That's not what I said. But there's a big
difference between talking about integers and apples
I didn't say anything that was in opposition to this, either. :
1. Peano's axioms for arithmetic determine criteria for finding
an integer n that has a property P(n) and for determining
whether two integers n and m are the same, n=m.
2. But I can't accept a proposed model with a physical entity
x and no methodology of any kind for determining whether a
property P(x) is true in the actual world or whether two
variables x and y represent the same entity in the world.
No "methodology" is required, beyond simply stipulation. You seem to be asking for too much, and blaming Tarski for not getting it. In other emails in this thread I've given several examples of Tarskian models made up of pieces of the real world, but lets take apples. Suppose I have a bowl with some fruit in it. The apples in this bowl will be my universe. Looking at the bowl, you hold up a rather soft and discolored object and challenge me: is this an apple? And I may even be in some doubt. But I will simply make a decision: um... Yes, it is an apple. That gives me one Tarskian interpretation. Or, I might have decided the other way: um... No, thats not an apple. That also gives me a Tarskian interpretation: different from the first, of course, in its extension of the predicate Apple. Which is of these is really correct? I don't know, and I don't NEED to know, in order to do semantics. All I need, is to be able to describe how truth in my formal language (which talks, in this case, of Apple-hood) is related to how that language is related to circumstances on the actual world being described. It may well under-describe it, in the sense of allowing many alternative ways of being true with respect to the world. In fact, this is the normal case. And this may well arise from there being some doubt about what exactly are the criteria for being a Apple. None of this is grounds for rejecting the Tarskian semantical machinery.
So, one might respond, what of real truth? If a banana or a dust bunny can be counted as an Apple for purposes of determining truth, then surely the semantics isn't doing its job. And this would be a good objection, IF the formal representation al system claimed to represent the notion of "apple". If, that is, Apple were a 'logical symbol' (like AND or FORALL). But it isn't. In the formal logic whose semantics is being described, Apple is just a predicate: true of some things, false of others. Any extra 'meaning' it has must come from the axioms that are asserted which use that predicate, from an Apple ontology. And of course the semantics of the underlying logic does not specify the content of any axioms written in it (if it did, there would be no need to write them in the first place.) However, this is not an argument for abandoning this semantics and replacing it with another, more Apple-aware, semantic theory; nor for looking for some supplementary semantics which will nail down the intended meaning of 'Apple' more adequately. It is an argument for, guess what, doing ontology engineering. And such engineering relies on the semantic theory of its underlying formalism, but does not set out to replace it.
Without such "grounding", it is absurd to claim that any
claims of truth or falsity about the world are credible.
Nonsense. I don't need to have solved 'grounding' in order to describe an interpretation. I realize that point #1, which can be solved by an explicit
enumeration with a countable universe, becomes more difficult
to formulate with an uncountable universe. But for physical
domains, I'd like to see a methodology for #2 that acknowledges
the symbol grounding issues
As understood in the cognitive science literature, the 'symbol grounding problem' is how to attach referents to proper names - why/how does 'Cymri' mean Wales, the country? - and to representational names which identify objects of perception, broadly conceived - by 'my apple' I mean that (while pointing to something we all can see). These are interesting and hard questions, but they are not part of a semantic theory. Approaches to them will involve using a semantic theory, but it is just bad science to insist that hard issues be completely solved before proceeding with solutions to relatively easy problems.
and presents at least a partial
answer to the two questions about the world: P(x)? and x=y?
I agree we have to have the 'x=y' question answered before we can even begin to speak about anything. However, I don't see why you feel this is basically harder for the real world than for anything else. After all, we do individuate the real world all the time. I don't need a methodology for doing this 'scientifically' or 'correctly', only a fundamental contract that it will get done somehow. Different ways of carving the world at its joints will give different universes of discourse, of course, and hence different interpretations, and presumably these differences will be revealed by writing ontologies: some ways of carving will make some ontologies true but not others. We will not converge upon a single agreed 'truth' by agreeing to talk about the same reality. But we will be disagreeing, if we disagree, about that one reality. PH> I claim, and I believe it is impossible to refute this claim, that a Tarski-style model can BE an actual part of the real world.
All I'm asking for are some quotations that acknowledge the issues in point #2 above and suggest some methodology.
You won't get them, because these aren't semantic questions. You are asking for too much. If that means you can't allow yourself to do semantics as it is understood by everyone else, then that is your problem.
You can start with Tarski's 1944 paper, which I posted on my web site several years ago: http://www.jfsowa.com/logic/tarski.htmIf you can't find clear evidence there, then please send some clear quotations from some other source that explain the matter. PH> Do you think it is incoherent to speak of, say, sets of people or sets of rivets or sets of galaxies?
It's not incoherent. But when people use such language, they can usually state, if asked, how an entity x with a property P(x) could be found and how to answer a question of the form x=y?
I can do that for rivets and galaxies (and apples) , or I know how to find people who can. Butif I could not, then I would just say, toss a coin. You will then get a lot more interpretations, of course, by tossing the coin different ways. Perhaps some of them will be ruled out by your axioms, in which case you are well on the way to having a useful ontology. Airline personnel, for example, talk about reservations, flights,
and passengers in a database, even though the database actually
contains unique identifiers for the entities, not the entities
themselves. But they maintain a clear methodology for relating
the symbols in the DB to the physical objects and events (e.g.,
a phone number to call a passenger in case of a cancellation).
True, but so what? What if they did not? That would not detach their representations from the real world; it would, rather, allow more ways to 'attach' it. It would allow more Tarskian interpretations to satisfy their representational ontology. The Tarskian semantics make this notion of having a tighter or looser grip on reality quite precise, which is exactly what one would expect a semantic theory to do in such a case.
BUt let me turn this example around and ask you a question. On your account of how Tarskian semantics should be understood, a Tarskian model cannot possibly comprise a part of the real world. It is always a mathematical abstraction, which needs to be supplemented by some other mapping from its abstract entities to the real things. So in this case of the airline, what is this second mapping, and how does it affect the truth of the airline's database of reservations? PH> I have found quotes from Russell, Quine, Church, Tarski, Carnap and others all making it vividly obvious that they had what one
might call the 'realist' view of sets.
Yes, Russell was notorious on that point. He claimed that a "proposition" about Mont Blanc actually contained the physical mountain with all its rocks and snow. The fact that Frege was incredulous about Russell's claim indicates that Russell was mixing physical and abstract entities in an unusual way. PH> You are simply stuck with a limited and mistaken grasp of the scope of set theory... John, you simply DO NOT UNDERSTAND
semantics. I give up on you. Remain mired in your ignorance.
No. I believe that you have taken a system that Tarski developed for "formalized languages" and applied it to the real world
We are TALKING about formalized languages. And (as I already said, and you apparently agreed, see the first line of your reply) the fact that the language is formalized does not mean that what it describes has to be. Formal languages do apply to the real world. They always have done. without including a methodology for answering the kinds of
questions that are routinely answered for mathematical systems,
such as P(x)? and x=y?
They are routinely answered for all kinds of things, John. Human communication would be impossible if they were not. The answers do not have to be correct, you see, only to be there. Where is the exact edge of Mount Everest? I don't know, but this lack of knowledge is not an obstacle to giving a Tarskian semantics for "MountEverest". It simply acknowledges that there are many ways of carving up planet earth to yield a Tarskian interpretation, and they are all equally 'right'. Each of them is well-defined, if you like to arbitrary, even impossible, degrees of accuracy: in each one of these interpretations, Mount Everest contains an exact number of protons. It doesn't matter that I can't determine which one of these is 'right', or that I lack a methodology for determining that meaningless question. They are all possible interpretations: that is enough.
Pat
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