Alfred Tarski’s great technical innovation in his Wahrheitsbegriff (1933/36) was the introduction of the auxiliary notion of satisfaction of a first-order formula by a function assigning objects from some model ‘s domain to the variables of the language. Once the notion of satisfaction is in place, one can then define truth of a sentence in a model , by saying that is true just in case it is satisfied by some (equivalently: every) assignment. It is this latter notion of truth that is of real mathematical and philosophical interest, of course, but it cannot be defined directly by recursion because in a first-order language (contrary to the propositional case), sentences are not built up from other sentences by means of constructors such as connectives. Rather, it is formulas in general that are so built, with sentences providing a special case. The characteristic clause of satisfaction is the one for quantified formulas:
- A formula is satisfied by an assignment in if and only if the formula is satisfied, in , by every assignment that that differs from at most in the value .
The detour through satisfaction, while technically interesting, it is just that — a detour. It can also sometimes get pedagogically problematic, as students usually grasp the importance of truth while not always appreciating the necessity of satisfaction.
There is an alternative approach, which does away with the notion of satisfaction, but uses a variant notion of a model. According to this approach, we only allow models that are “rich” in that every object is the denotation of a constant in the language. The truth clause for the universal quantifier is then as follows:
- A sentence is true in if and only if is true in for each constant .
(I believe this definition is fairly standard in some textbooks, but I can’t track down a definition now, any references appreciated). While this definition sidesteps the notion of satisfaction, there are some drawbacks:
- The definition only applies to “rich” models, while models in which some elements are not denoted by constants are also very natural. Moreover, we are forced to consider truth of sentence in a model having a different signature than itself.
- The language changes with the model: for each model , the definition applies only to the expanded language introducing new constants for each member of .
- The definition forces us to consider uncountable languages, in order to assess truth in uncountable models.
- The expanded signature affects the available automorphism of a model. For example, the model has infinitely many automorphisms in the signature with the relation only, but only one if we add the constant .
So here is a way of combining these two approaches in such a way that:
- There is no detour through satisfaction.
- The language is countable and independent of the model in which a sentence is evaluated for truth (although the language will be expanded, just not in a way determined by the model).
Fix a basic language , and let be a countable set of new individual constants. For each let (so that ). We define the notion of truth by induction on -sentences, simultaneously for every . The quantifier case:
- An -sentence is true in the -structure if and only if the -sentence is true in every -expansion of (i.e., if is true in every -structure that differs from only in that it assigns a denotation to ).
So, when is an -sentence, the definition gives us a direct account of the truth or falsity of is a model having the same signature.