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In "The Semantic Conception of Truth and the Foundations of Semantics" (1944) Alfred Tarski asserts that a satisfactory definition of truth must be both formally correct and materially adequate. A theory of truth is formally correct iff it does not contradict the rules of the language in which it is given (the metalanguage describing the object language). Furthermore, a theory of truth is materially adequate if it entails all the equivalences (T) in the object language of the form "X is true if, and only if, p," where 'p' is an arbitrary sentence in the object language and 'X' is the name of that sentence in the metalanguage.

The definition of truth Tarski finally decides to put forward is "a sentence is true if it is satisfied by all objects, and false otherwise." For example, if snow satisfies the sentence "Snow is white," then the sentence is true. My question is, how can it be shown that this definition of truth is formally correct and that it implies all equivalences (T) in the object language? It may be that the treatment of the theory I am reading is just an outline, or that I am misunderstanding something, but either way it would be of great help if someone could point me in the right direction.

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  • The "exemplification" of Tarski's theory as : "a sentence is true if it is satisfied by all objects, and false otherwise" is misleading. Commented Jul 15, 2014 at 6:25
  • From Tarski's Truth Definitions : "The definition of True should be ‘formally correct’. This means that it should be a sentence of the form : For all x, True(x) if and only if φ(x), where True never occurs in φ. [...]. The definition should be ‘materially adequate’ (i.e. ‘accurate’). This means that the objects satisfying φ should be exactly the objects that we would intuitively count as being true sentences. Commented Jul 15, 2014 at 6:41
  • You can see also Axiomatic Theories of Truth Commented Jul 15, 2014 at 7:17
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    @MauroALLEGRANZA Everything in the question in quotes is taken directly from the essay by Tarski. As such I am confused as to how it is misleading
    – Exit path
    Commented Jul 17, 2014 at 20:47
  • The way I understood the requirement for formal correctness and material adequacy was not as a definition but as a condition that must be met by any definition of truth for it to be satisfactory.
    – Exit path
    Commented Jul 17, 2014 at 20:50

1 Answer 1

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Alfred Tarski asserts that a satisfactory definition of truth must be both formally correct and materially adequate.

I can't speak for philosophical logic, but for mathematical logic, Tarskis assertion became a definition; in the sense that these are good requirements to have for a mathematically tractable definition of truth; that this is a good definition (mathematically) is shown by the consequences of his definitions and theory; and this entailed the development of Model Theory.

Here, Godel showed in his completeness theorem that:

is a fundamental theorem in mathematical logic...[it] establishes a correspondence between semantic truth and syntactic provability in first-order logic.

That is a proposition p is syntactically true, that is derivable by the laws of inference if and only if p is also semantically true, that is satisfied by its model(s).

The forward direction is called completeness & the backwards direction soundness - in mathematical logic at least.

To give a simple example of this, take the theory of (elementary) Euclidean Geometry (EG), and first-order logic, and modus tollens as the law of inference. Then every arithmetic theorem that one can prove formally by the rules of logic, inference and the axioms of PA, can be shown to be satisfied by the actual plane; and the converse also holds.

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  • What are you meaning with : "Then every arithmetic theorem that one can prove formally by the rules of logic, inference and the axioms of PA, can be shown to be satisfied by the standard model - the actual integers; and the converse also holds" ? The "converse" must sound like : "if a statement is satisfied in the standard model, then it is formally provable by rules of logic from f-o Peano's axioms" ... If so, it is not correct : G's Incompleteness Th proves precisely that there are "intuitively" true arithmetical statements (i.e. true in the intended model) which are not provable. 1/2 Commented Jul 15, 2014 at 6:22
  • Thus, by G's Completeness Th, they are false in some non-standard model. 2/2 Commented Jul 15, 2014 at 6:23
  • @Allegranza: Ok, I slipped up. Do you know of a simple and natural example that satisfies the conditions for G's completeness theorem? I reached for PA forgetting his limitative result. Commented Jul 15, 2014 at 6:40
  • Completeness Th, says that all logical consequences of axioms are provable, and vice versa. To be log cons of axioms means : true in all models (and not only in the intended one). Thus, all "usual" theorems provable from f-o Peano's axioms, like : for all x, exists y ( x < y) is clearly true in all models (non-standard included). The "strange" formula of G's Incompleteness Th, being non-provable, must be not logical cons of the axioms, i.e. true in the standard one (we "see" this through the proof of G's Th) but false in some non-std one... Commented Jul 15, 2014 at 6:45
  • @Allegranza: sure; but shouldn't one qualify all, sometimes the models are kripke, topological, categorical etc; for the purposes of my answer, a simple concrete example is what I was looking for; I vaguely recall coming across some - does Tarskis euclidean plane geometry fit the bill? Commented Jul 15, 2014 at 6:57

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