I am indebted here to conversations with N.J.J. Smith, and some correspondence years ago with Susan Haack. My main contention here should not be attributed to either of them though - at least, not on the basis of this post.
I believe that the Tarskian device of using sequences of numbers in the definition of satisfaction in first order logic, and in turn of truth, is stupid. (For information about this device, see here or just Google it.) Either it is the product of philosophical confusion, or it panders to philosophical confusion.
Some believe otherwise. For instance, Peter Milne in his 'Tarski, Truth and Model Theory' calls it a 'stroke of genius':
This co-ordination of variable [sic] with objects through their indices allows the work that would otherwise be done by explicitly semantic assignments of values to variables to proceed without appeal to semantic notions.
There is, I think, a very strong inclination to feel that some sleight of hand is being effected here, that in some respect the wool is being pulled over one's eyes. This inclination is to be resisted. What Tarski does here is perfectly abobe board; not only that, it is, I believe, a stroke of genius.
Curious. (Thanks to N.J.J. Smith for the reference.)
Also, a former teacher of mine (whom I won't name) called the idea of using sequences of numbers 'a great insight'.
This bothered me for years, and I was hesitant in forming my current view of the matter, thinking that perhaps if I only understood better I would think otherwise.
N.J.J. Smith in a talk at the recent 2015 AAL Logic Conference made an interesting and amusing analogy concerning the device, and used the word 'coy' in relation to it; it is as if, at a 1950's dance, instead of pairing boys with girls (which would be vulgar), boys and girls alike are paired with numbers, and then the numbers paired. (This got me thinking about the matter again after putting it aside years ago, and led to this post.)
Smith's talk was not, in the main, about criticizing satisfaction by sequences, and this was (I think) more or less an aside. (His purpose was rather to argue that satisfaction-based definitions, whether by sequences or variable assignments, are inferior to another method when it comes to capturing the content of 'true'.)
Smith's remark here is on a similar track to my criticism, but is different. What Smith said may make it look like the conclusion to draw is that Tarskian satisfaction by sequences is just as semantic as variable assignment. Well, that's true, but I think the real insight comes from looking at it the other way: a variable assignment is no more semantic (in any sense which may create difficulties and philosophical problems pertaining to meaning) than the business with numbers.
Whenever you have two objects, there will be, out there in function (or set) space, all the possible mappings, and the notion of such a mapping is not semantic in any strong sense. (If you define the semantic as any kind of relation between symbols and other things, then it will be, but this just goes to show that that's not a good definition. To be semantic in the sense of having to do with meaning, the relation has to be of a special kind, with all sorts of surroundings.)
Thus, the Tarskian conjuring trick is philosophically ill-motivated. Any relevant misgivings about function- or set-theoretic assignments - which Tarski does indeed succeed in avoiding - are based on confusing these with real semantic relations. The avoidance has no real value at all.
Now, am I saying Tarski is guilty of stupidity? No - I am just saying that the device itself is stupid. Tarski's adoption of it may indeed be quite shrewd, given his audience. In that case, it may be argued to be intellectually irresponsible - by pandering to confusion instead of combatting it. (I take the last two verbs in this formulation, which I think very apt, from an email from Smith wherein he comes to understand my position.)