Convention-T and the T-Schema
In his paper "Naturalism, Realism, and Normativity" published recently in the Journal of the American Philosophical Association, the late Hilary Putnam does an admirable job of disentangling Tarski's Convention-T from Tarski's T-Schema. For too long, orthodox interpretations of Tarski's theory of truth have accepted that Convention-T and the material adequacy condition are the same, indistinguishable from one another. Putnam seems to suggest an alternative to the orthodox interpretation that captures the distinction between the formal semantic theory of truth that Tarski went to great lengths to uncover in his work and the ordinary non-technical notion of truth he left aside because it was too ambiguous for us to make any headway (cf. §1 of Tarski 1936/1983b and Pt. II of Tarski 1944). Putnam correctly points out that Convention-T is not a 'convention' but a 'criterion' of a formal language where its ideal form removes any mention of truth in order to avoid the vagaries of ordinary or natural language. Under this interpretation, material adequacy, though distantly related with Convention-T, turns out to be associated with the non-technical folk theoretic notion or conception of truth.
Tarski accomplishes regimentation of the truth-concept by specifying the structure of the languages for which truth is defined and by specifying a formal criterion of material adequacy (i.e., of whether the formally defined concept tracks the ordinary use of truth) in terms of whether all instances of the equivalence
[T] X is true-in-L if, and only if p
follow from the proposed definition. Tarski’s formal criterion of material adequacy permits the analysis of sentences into subsentential constituents, and it dissolves correspondence relations into appropriate semantic components: names refer to or denote objects and predicates apply to or are satisfied by objects. ‘X’ is the name or description of a sentence in L, an object language, and p is a translation of the object language sentence into meta-language. Call the equivalence, i.e., [T], ‘Convention-T.’ Convention-T imposes a condition on defined truth predicates that Tarski often talks about grasping the intuition behind the “classical,” “old notion,” or “Aristotelian conception” of truth (cf. Tarski 1936/1983b, 155; 1944, 342f.). Any theory of truth, according to Tarski, must entail, for any sentence X in a given language, a sentence of the form [T].
The orthodox interpretation of Tarski’s semantic conception of truth has it that Convention-T is the material adequacy condition. Convention-T only applies to formal languages because there is no systematic way of deciding whether a given sentence of natural language is well-formed and natural languages are capable of describing semantic characteristics of their own elements. Since, for natural language, the latter permits semantic paradoxes, like the liar, to arise and the former allows ambiguity, formality remains a part of material adequacy.
While Convention-T applies to formal languages, there has to be a means of extending it to natural language. Tarski recognises the difficulties faced by attempts to clarify concepts using only the resources of ordinary language because “like other words from our everyday language, [true] is certainly not unambiguous” (Tarski 1944, 342). The urge toward formalisation is the urge toward imposing discipline upon our concepts. The extension of Tarski's concept of truth is given by the axioms derived from the T-schema: "snow is white" is true, if and only if snow is white. The T-schema is used to give an inductive definition of truth which lies at the heart of any realisation of Tarski’s semantic conception. For any instantiation of the T-schema, Tarski must have realised that it would not suffice for a general definition of truth because the instance could only cover whatever the schema’s content is. For example, we could imagine an innumerable number of T-schema sentences:
“Grass is green” is true if and only if grass is green.
“Ewa Kopacz is prime minister of Poland” is true if and only if Ewa Kopacz is prime minister of Poland.
“Golf was created by Scots” is true if and only if golf was created by Scots.
No matter how many sentences we imagine using the T-schema it seems insufficient to generalize from these examples to a formal rendering of a theory of truth. So, Tarski must have devised the semantic conception of truth, using the two formal conditions: formal correctness and material adequacy, in order to generalize over all instances of the T-schema.
The T-schema does represent a partial definition of truth, and it is something that should be accessible in natural language since the T-schema is expressed in natural language. The T-schema, then, should be something accessible to ordinary persons since they are at least minimally competent in the use of natural language. Its open accessibility seemingly requires us to consider whether ordinary persons believe the schema to be intuitively accessible.
Neither the nature of truth nor the nature of the truth concept is settled by the formally correct extensional account. Despite the fact that Tarski sometimes hints at the compatibility of the extensional definition with a correspondence story about the nature of truth, there is sufficient extant critical literature to withhold final judgment on whether Tarski's formalized semantic theory of truth requires a correspondence theory, or vice versa (cf. Field 1972; 1986; Kirkham 1992; Künne 2003; Popper 1972). Nevertheless, for those who return periodically to Tarski’s early paper (1936/1983b) coming now with an ear toward recent work in experimental philosophy, there is a further vein of thought to be mined: an attempt to describe the ordinary folk notion of truth and thereby make clear how to satisfy the material adequacy condition using methods adapted from the empirical sciences.
In his early work, Tarski writes:
Every reader possesses in greater or less degree an intuitive knowledge of the concept of truth and he can find detailed discussions on it in works on the theory of knowledge. (Tarski 1936/1983b, 153)
Who are the readers that Tarski has in mind? Are they philosophers? Or, are they philosophers and non-philosophers? What does it mean for any one of them to possess a greater or lesser degree of an intuitive knowledge of truth? Clearly, we cannot stake any claim about Tarski’s taking seriously the ordinary person’s conception of truth because he might be referring to every reader of his texts. Moreover, we cannot assert that Tarski’s limiting his analysis to the conception of truth upheld by professional philosophers since all of his readers would not be limited just to educated philosophers. His view is not that everyone, even non-experts, is capable of having an “intuitive” grasp of the formal concept of truth, but that they have partial grasp of truth. His focus is how “every reader” has the capability of understanding the intuitive sense of truth. This interpretation of what Tarski claims at the beginning of (1936/1983b) leaves open the possibility that some readers, perhaps less educated ones or perhaps more educated ones, do not share in the intuitive sense of truth compatible with the ‘old notion’ or the ‘classical conception’ inherited from Aristotle. This is not to say that the ordinary notion is representative of the formal analysis. Tarski conceived of the ordinary notion in a certain manner which is independent of Convention-T.
First, it is notable that (1936/1983b) §1 is entitled: “The Concept of True Sentence in Everyday or Colloquial Language.” The title of this section seems to signal in Tarski at least an awareness of and possibly respect for the average person’s understanding of truth if we assume that the ordinary person’s conception is an accurate reflection of the ‘old notion’ as understood especially by its Aristotelian incarnation. Toward the end of the section Tarski reveals that a formally correct definition, one based upon the laws of logic, cannot be equivalent to the ordinary person’s conception of truth. Tarski says:
If these observations are correct, then the very possibility of a consistent use of the expression ‘true sentence’ which is in harmony with the laws of logic and the spirit of everyday language seems to be very questionable, and consequently the same doubt attaches to the possibility of constructing a correct definition of this expression. (Tarski 1936/1983b, 165)
Tarski seems to have used this section of the text to argue that the formal correctness condition of truth is distinct from the “spirit” of how non-philosopher's use the term ‘true’, but nowhere does Tarski summarily dismiss the everyday or common-sense usage of the term.
One might object that a theory of truth is one not based upon the folk view of truth because non-philosophers tend to be unable to understand the concept of truth in a competent way that is prevalent among professional philosophers. Any language user is minimally competent in the employment of language, even someone who might be suffering from some kind of cognitive impairment that prevents them from developing sophisticated use of language. So, it seems unusual to distinguish between an ordinary language user and an extraordinary language user. The purpose here is not to suggest that there is a strong distinction between the two but that one might support such a distinction because philosophers are expert users of language, i.e., “extraordinary users of language”, and ordinary people are not. One might respond to this claim, as one reviewer of this paper has deftly pointed out, by recommending that we view the “users” of language as one continuous or seamless web.
Of course, this does not mean that Tarski accepted the non-philosopher’s conception of truth as the foundation for any view on the nature of truth. Tarski was not an “ordinary language philosopher!” (Even ordinary language philosophers might have had difficulty with the non-philosopher's notion of truth. There's nothing that prevents the ordinary language philosopher from abandoning certain views of truth. More on this in a later blog post.) If Tarski took seriously the everyday use of the term ‘true’, then it was a matter of providing him with some insight of how the term is employed in natural language by linguists, philosophers, and non-philosophers, and coming to better understand the deficiencies in it – i.e., that it leads to paradox or prone to mistaken use on some occasions.
Contrast his discussion in the early work with his more popular (1944) article where there are two distinct discussions of the concept. The first, and oft quoted, states:
We should like our definition to do justice to the intuitions which adhere to the classical Aristotelian conception of truth-intuitions which find their expression in the well-known words of Aristotle's Metaphysics:
To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true. (Tarski 1944, 342)
While the “old notion” to which Tarski refers here is consistent with the Aristotelian conception mentioned above, he believes that “our definition” should “do justice to the intuitions which adhere” to it. The intuitions presumably any person should have about truth should be consistent with this Aristotelian conception. But Tarski dutifully reminds us that,
The word “true,” like other words from our everyday language, is certainly not unambiguous. And it does not seem to me that the philosophers who have discussed this concept have helped to diminish its ambiguity. In works and discussions of philosophers we meet many different conceptions of truth and falsity, and we must indicate which conception will be the basis of our discussion. (Tarski 1944, 342)
The everyday use of the term “true” is not unambiguous and precise, according to Tarski. Philosophers have not “diminished” the ambiguity associated with the everyday use of truth in natural language. So, Tarski settles on the classical conception or “old notion” of truth.
Tarski indicates that he does not want to become embroiled in the debate over which ordinary conception of truth is correct (Tarski 1944, 355), but, in §17 Tarski returns to the question of how we ought to think about the matter. Tarski is aware that some do not share his view that there is some compatibility between the ordinary notion of truth and elements of Tarski’s semantic conception.
[S]ome doubts have been expressed whether the semantic conception does reﬂect the notion of truth in its commonsense and everyday usage. . . . I happen to believe that the semantic conception does conform to a very considerable extent with the common-sense usage although I readily admit I may be mistaken. (Tarski 1944, 360)
Tarski expresses his view that the semantic conception “conforms to a very considerable extent” with the ordinary person’s notion of truth, but he also suggests that if some maintain that the ordinary concept of truth is different from it, then the issue “can be settled scientifically . . . with the help of the statistical questionnaire method” (Tarski 1944, 360). Here, Tarski seems to question whether that disorderly nature of anecdotal evidence is sufficient for coming to terms with the colloquial view of truth. Only Arne Næss, a Norwegian philosopher, had considered incorporating a systematic and well-organised empirical study of the non-philosopher’s view of truth.
Interestingly, and perhaps not surprisingly, Tarski cited the work of Næss (1938a) as an example of empirical work relevant to settling the issue. Næss published two works in 1938 on the non-philosophers or common-sense conception of truth (1938a; 1938b). Tarski (1944) cited the monograph (1938a), but omitted a citation of the article published in Theoria (1938b). One might speculate that Tarski was reluctant to include a citation to the article in Theoria because it lacked the statistical and scientific rigour of the monograph.
Tarski does not give up on his intuitions about the ordinary notion of truth, but he seemingly abdicates ultimate authority over the question of whether the assumption of the classical conception of truth is actually capturing the content of the ordinary notion of truth. So the issue becomes an empirical one: is the ordinary notion of truth the classical conception? To get a better understanding of this Tarski believes we must turn to the empirical research of Næss. This would suggest, then, that the T-Schema is something empirically discoverable, which is distinct from the formally correct understanding of Convention-T.