So today in my philosophy seminar we talked about Tarski’s ‘definition of truth.’ This topic never fails to get my blood boiling because philosophers just can’t seem to avoid ascribing great philosophical importance to what is really a totally trivial (philosophically) statement. I don’t know if it is reflexive respect for hard subjects like mathematics, simple confusion caused by the subtle distinctions involved, or just the misleading terminology but, like some kind of philosophical crack, no matter how many times philosophical uses of Tarski’s ‘definition of truth’ are slapped down someone always comes back for another hit. So in case you don’t already understand why Tarski’s definition of truth can do no serious philosophical work I plan to rant about it here, but not in this post. Since many philosophers seem to get blinded by the math in this post I will try and explain the mathematical background. Hopefully this will be generally understandable but if you haven’t seen the T-schema before I suggest you check out the wikipedia article on the semantic theory of truth and the Stanford Encyclopedia of Philosophies entry on Tarski’s Truth Definitions. The more intrepid reader may also want to check out a brief outline of Tarski’s paper or for those of you with JSTOR access Tarski’s 1944 paper.

In the following paragraphs I will be employing the notions of a model. Unfortunately this is a necessary technical notion but a definition can be found over on wikipedia. In case you don’t want to parse the formal definition the basic idea is that a model is a mathematical world or state of affairs. One way to think of it is something like an ersatz possible world and just like a possible world things are true or false in a model (don’t take the analogy too far, mathematical truth is necessary after all). A model is a possible way certain relations, functions and constants could work together and statements are to be true in a model just if they describe the ways things really are in that model. Terminologically saying M models a sentence s just means s is true in M, also written M |= s (M models s).

For instance we might have a model M in a language with one unary relation symbol R and five constant symbols a,b,c,d,e. The model in this case just tells us which constants the relation R holds of. So in this simple case we might specify M by saying R_M holds of a,c,d and doesn’t hold of b,e. Then for this example of M we would have M |= “R(a) & R(d)” but M wouldn’t model “R(b)”. Also, if we pretend that A is really upside down (forall symbol), v denotes or and ~ not then M |= “Ax(R(x) v ~R(x))” (trivial tautology) but M doesn’t model “Ax(R(x))”. Note that a model (unlike just a list of sentences) has to give a consistent account of the way things work. One cannot both have M |= “R(a)” and M |= “~ R(a)” (unless we start talking about para-consistent logics or other silly things).

So the basic mathematical problem which Tarski’s definition of truth solved is something like the following (rephrased in terms of set theory instead of the more abstruse higher order logics Tarski initially had in mind). We are working inside set theory (the meta-language) and wish to define a formal language (the object language) and truth for that formal language. In particular say we want to talk about sentences in the language of number theory (L(N) ) and models (or structures if you prefer) in that language. Now defining truth for sentences in the language of number theory means giving a definition for the relation M |=_L(N) s (M models s) where M is a model in the language of number theory and s is a sentence in that language. Such a definition is necessary because formally a sentence in the language of number theory is just a certain sequence of symbols satisfying syntactic constraints and only by defining a notion of satisfaction can we give it semantic meaning. Of course such a definition should reflect our intuitive idea of what it means to be true, i.e., if 2_M +_M 2_M = 4_M then we should have M |=_L(N) “2+2=4″.

Now if all we want to do is define the set of sentences which is true in some particular model M there is no problem. M is a set and M gives us the atomic sentences which are true in M so by normal set theoretic methods we can get a set T(M) (the set of true sentences in M). Where things get complicated is if we want to define a relation |=_L(N) which holds for any model M and sentence s in L(N). Since the collection of models in L(N) is a proper class there is no set giving us the relation |=_L(N). We are stuck with the fact that |=_L(N) is a proper class and if we want to work conveniently with that proper class we would like to find a formula |=_L(N)(M, s) (slight abuse of notation using the symbol to denote both the formal definition and the relation) in the language of set theory (meta-language) which holds iff M |=_L(N)s. Tarski’s definition of truth lets us build such a formula, or in other world it lets us define truth (at least for first order languages) in a formal language by a finite list of conditions (we can also get rid of the explicit relativization to a particular language and make it a three place relation between a model, sentence and language).

Tarski’s key insight is that we can use a recursive definition to define truth. In particular assuming we have chosen to let the symbol & in our object language (L(N) in this example) play the same semantic role as ‘and’ in our meta-language (language of set theory) then for any well formed formulas x and y in the object language we can define M |=_L(N) “s & t” as follows:

M |=<sub>L(N)</sub> "s & t" iff M |=<sub>L(N)</sub> "s" and M |=<sub>L(N)</sub> "t"

The same trick can be played with all our logical connectives and functional symbols giving us a definition for truth in M. Of course such a definition only works if we somehow know for what atomic (can’t be broken up into smaller components) sentences x, M |=_L(N) “x”. However, in the case of truth in a model these facts can be read directly off of the model. Thus we end up with a finite definition of the relation |=_L(N) which looks something like this:

M |=<sub>L(N)</sub> r iff

    1) r = "s & t" and M |=<sub>L(N)</sub> "s" and M |=<sub>L(N)</sub> "t"
    OR
    2) r = "s v t" and M |=<sub>L(N)</sub> "s" or M |=<sub>L(N)</sub> "t"
    OR
    3) r = "(Ax) s" and ....
    .
    .
    .
    OR
    LAST) r is atomic and M says r is true 
    (for atomic sentences this can be read right off the model) 

Importantly nothing in Tarski’s solution in this case presupposes that M is a set, we could just as well substitute in the defining formula of a proper class.

The philosophical usage of Tarski’s definition of truth is really is sorta special case of the definition of truth in a model I gave above. The special case is where the “model” in question is actual truth, i.e., the same universe we are evaluating our meta-language in. Technically we are no longer really evaluating truth in a model but the basic idea stays the same. We can still define the semantics of our object language via a finite recursive definition in our meta-language except now the mentions of our model are just replaced with actual truth. So for instance the example given above becomes (where “s & t” is a statement in our object language”.

"s & t" is true iff "s" is true and "t" is true

The rest of the definition can be filled out similarly. Though of course this only defines truth if we presuppose we know when atomic sentences in our meta-language are true. Now if we suppose that the object language is actually a fragment of our meta-language and the symbols are to have the same meaning in both, so ‘and’ in the object language is meant to have the same semantic meaning as ‘and’ does in our meta-language, we can follow through the definitions and get Tarski’s famous T-schema.

T: "P" is true iff P

Where “P” is a statement in the object language and P is a statement in the meta-language. For example one instance of this schema is the following.

"Snow is White" iff Snow is White.

However, unlike in the formal case in natural language we no longer have a restricted finite list of logical symbols (symbols like & which are interpreted as part of the logic). Thus the definition of truth is no longer even a finite definition in terms of the atomic truths. Additionally it is unclear what use such a schema could have in natural language. Unlike the formal case it isn’t clear that there is a special role to play for finite syntactical definitions. Most importantly though the utility of Tarski’s definition of truth is that it lets us finitely define truth in an uninterpreted object language given an already understood meta-language. Hopefully you already have suspicions about the philosophical utility of such a ‘definition’ and in my next post I will give a full fledged argument as to why it doesn’t tell us anything we don’t already know.