tedrick@ernie.berkeley.edu (Tom Tedrick) (03/16/86)
Can someone explain Tarski's definition of truth to me? I never quite understood it. In the last graduate course in mathematical logic that I took, many years ago (which convinced me that mathematical logic was not the field for me), I remember Prof. Vaught walking up and down in front of the class, saying "The statement 'Grass is green' is true iff grass is green", and smiling a funny smile. Is it possible for someone to clue me in as to what is going on, or is this as difficult to explain as a koan ... It was especially disturbing to me because I had spent a month the previous summer convincing myself that I could prove that it was impossible to define truth ... -Tom tedrick@ernie.berkeley.edu
torek@umich.UUCP (Paul V. Torek ) (03/17/86)
That's about all there is to it, Tom -- Tarski's theory of truth isn't much of a "theory". `"Grass is green" is true iff grass is green' about sums it up. On Tarski's view, saying `"S" is true' is redundant; one can say the same thing by saying `S' (where S is a proposition). --Paul Torek torek@umich
weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (03/17/86)
In article <12411@ucbvax.BERKELEY.EDU> tedrick@ernie.berkeley.edu (Tom Tedrick) writes: >Can someone explain Tarski's definition of truth to me? I'll give it a shot. >In the last graduate course in mathematical logic that I >took, many years ago (which convinced me that mathematical >logic was not the field for me), I remember Prof. Vaught >walking up and down in front of the class, saying >"The statement 'Grass is green' is true iff grass is green", >and smiling a funny smile. Two points. First truth in logic refers to truth in some model. Not to any absolute truth. Second, any logical sentence can be broken down into well determined atomic formulas. Each of these atomic formulas has a definite truth value within the model. (That's what makes them atomic.) Any logical sentence can be broken down into it's atomic parts, and they are then checked in the model in question, and put back together by following the meaning of the logical symbols joining the atomic statements. The statement "'Grass is green' is true iff grass is green" is a standard way of illustrating the process I just outlined where the model is by default the real world. Prof. Vaught smiled because it is humorous. >It was especially disturbing to me because I had spent a month >the previous summer convincing myself that I could prove that >it was impossible to define truth ... That refers to Tarski's famous theorem about truth. This refers specificly to number theory, and says no formula in the language of Peano Arithmetic can capture the essence of truth for the standard model of Peano Arithmetic. There are, in contrast, non-standard models of Peano Arithmetic which have internal truth definitions. Too bad Prof. Vaught didn't cover this theorem. (Usually done right after you finish Godel's first incompleness theorem.) Because he would have then smiled and told you that Tarski's other great accomplishment, in addition to defining truth, was his proving the undefinability of truth. ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720
ladkin@kestrel.ARPA (Peter Ladkin) (03/18/86)
In article <12411@ucbvax.BERKELEY.EDU>, tedrick@ernie.berkeley.edu (Tom Tedrick) writes: > Can someone explain Tarski's definition of truth to me? > I never quite understood it. > The statement 'Grass is green' is true iff grass is green There's much more to Tarski's theory than this, but to start here: The left hand side ascribes truth to a certain sentence, and the right hand side describes a fact about the world. The statement asserts that the sentence is true iff the fact holds. This is not a *theory* of truth, but a condition of adequacy that any theory of truth has to fulfil. Tarski wanted to ascribe truth to sentences, via the notion of satisfaction of formulae (sentences with object-place-holders). He was able to explain how a sentence with multiple quantifiers can inherit its truth value from its parts, something which had puzzled logicians from Aristotle onwards. The best way to start is to read the original article, "The Semantic Conception of Truth" in his collection "Logic, Semantics, Metamathematics", easily obtainable from the UCB Philosophy Library. Peter Ladkin
ladkin@kestrel.ARPA (Peter Ladkin) (03/19/86)
In article <12454@ucbvax.BERKELEY.EDU>, weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) writes [regarding Tarski's truth defn]: > Two points. First truth in logic refers to truth in some model. > Not to any absolute truth. Second, any logical sentence can be > broken down into well determined atomic formulas. Each of these > atomic formulas has a definite truth value within the model. I'm not sure I agree with your first point. Logical validity is defined by quantifying over all models, by Tarski. Validity is not *absolute*? Secondly, atomic formulas do not have a definite truth value within the model. They have truth values *under an assignment*. A formula can only have a definite truth value in your sense if its universal closure is true, or if the universal closure of its negation is false. Peter Ladkin
weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) (03/20/86)
In article <5961@kestrel.ARPA> ladkin@kestrel.ARPA (Peter Ladkin) writes: >In article <12454@ucbvax.BERKELEY.EDU>, weemba@brahms.BERKELEY.EDU >(Matthew P. Wiener) writes [regarding Tarski's truth defn]: >> Two points. First truth in logic refers to truth in some model. >> Not to any absolute truth. Second, any logical sentence can be >> broken down into well determined atomic formulas. Each of these >> atomic formulas has a definite truth value within the model. > >I'm not sure I agree with your first point.... >Secondly, atomic formulas do not have a definite truth value within.... Yes, Peter, your points are correct. I was deliberately simplifying the explanation, since otherwise the point I was trying to get through to Tom would have been lost. Based on his later thanks, I think I did what he and I wanted. You are welcome to post the full definition--I did not think those who know better could have been confused. I've written a few long and semi-detailed mathematical articles, and they sure take up time. But thanks for the clarification anyway, since I might have proven to some people elsewhere that I don't know any logic. :-) As it is, I'll try to post accuracy disclaimers when necessary. ucbvax!brahms!weemba Matthew P Wiener/UCB Math Dept/Berkeley CA 94720