² The second article in the series, part II of the present paper, is forthcoming in this Journal.

³ See XVI 210 and XVII 281.

⁴ See Tarski's 28516 and Carnap's XIV 237.

⁵ See XIII 154(3) and the discussion of the relation of models to state-descriptions in XVII 214(3) [cf. footnote 26].

⁶ These are, presumably, the non-standard models discussed in XVI 145. But, as Skolem pointed out in the review, Rosser and Wang give no general definition of standardness. They have only some sufficient conditions for non-standardness. The distinction is made precise for a type theory of order to by Henkin in XVI 72(2).

⁷ Sections 6–8 form part II of this paper, cf. footnote 2.

⁸ The procedure of XIII 154(3) is here simplified by allowing only a single individual domain. The generalization is quite easy, as can be seen in the previous paper, but the notation becomes very cumbersome.

⁹ This will be discussed in some detail in section 3.

¹⁰ In XIII 154(3) I defined values only for wffs, but here the definition is given for all special formulas. There is no difficulty in this extension, and it is part of a scheme of unification to be developed in section 3.

¹¹ Only those syntactic constructions were discussed in this paper that are not completely standard. Thus, e.g., it was not pointed out that axioms are wffs, and that rules allow us to infer wffs from wffs (see XIII 154(3)). But these facts are assumed in this definition, and elsewhere in the paper.

¹² Three of the conditions on models in XIII 154(3) were here omitted: The last two conditions of definition 2 and the last condition of definition 6. These were conditions of convenience, intended to eliminate some non-intended models. There is no longer any need for such conventions, because the author can always refuse to recognize the non-intended models as interpretations.

¹³ The reasons for this definition are given in XIII 154(3).

¹⁴ The proof in XIII 154(3) made essential use of the last condition in definition 6, which was here omitted. Hence a new proof is here given, not using the condition.

¹⁵ The proof can be carried out using so few theorems because many-valued systems were excluded. If the author desires to allow a system with many truth-values, he must modify the definition of R_o and make some other corresponding changes. It is also possible to get systems very similar to many-valued systems by keeping the present procedure but making the many truth-values a new individual domain.

¹⁶ Naturally, anyone working in this field is deeply indebted to the pioneer work of Tarski and Carnap. The degree of indebtedness can only be measured by seeing how far I have followed their respective approaches. Since this in itself would take a great deal of space to discuss, I will not try to undertake it in the present paper.

¹⁷ It is understood that linguistic information about is not to be classed as factual.

¹⁸ Although the present usage of ‘analytic’ and ‘synthetic’ is due to Kant, the distinction is entirely clear (using ‘necessary’ and ‘contingent’) in Leibnitz's writings.

¹⁹ At least this appears to be the distinction he makes in XIX 136, p. 130.

²⁰ Although the complete definitions would include references to , as in definition 1, these references will be omitted where there is no danger of misunderstanding.

²¹ The term ‘analytically equivalent’ could be substituted for ‘equivalent,’ but I think that ordinary usage justifies the shorter term. When we assert that A is equivalent to B, we normally mean that either is a logical consequence of the other, which is an equivalence in my sense of the word. ‘Material equivalence’ is a more recently coined word, describing a closely allied relation: When we assert the material equivalence of A and B, we are asserting that A ≡ B is true; when we assert that they are equivalent, we assert that A ≡ B is A true. Similar remarks apply to the term ‘implies.’

²² A formal example will be given in part II.

²³ These are real translations, not quasi-translations. The difference between the two types of concepts lies in the individuals that hold the relation to the linguistic entities, but these have “dropped out” in the definition of translation. The same will hold true of the definition of truth to be given below. The question of translations will be discussed at length in part II.

²⁴ Cf. IX 68, p. 344.

²⁵ Cf. section 2. Since all the interpretations have the same R_i, the assignment to universal quantification is also uniquely determined.

²⁶ The role such axioms should play was discussed in my XVII 214(3). The term ‘meaning postulate’ was not used there; but I adopted this term later, which is due to Carnap. The present treatment of meaning postulates is an extension and to a certain degree a correction of the earlier treatment. — It is important to note that it is only necessary to have the meaning postulates among the axioms, it is not necessary to distinguish them from other axioms.

²⁷ In particular, both “logical” and “extra-logical” assignments may be made to the extra-logical constants of . [This is fortunate since the terms in quotes (referring to ) can only be defined in .] Hence I can postpone the discussion of the types of assignment till part II, where appropriate machinery for is introduced. I have already noted a similar point for translations.

²⁸ See especially definition 10.

²⁹ The usage of ‘phrase’ is questionable. But if we think of a constant as a word in , and a closed wff as a sentence (in the ordinary sense), then the word I am seeking must describe a basic grammatical category of which a single word and an entire sentence are two extreme cases. It seems to me that ‘phrase’ comes nearest to fulfilling this role.

³⁰ Since a phrase is a closed special formula, it has a definite value in every model.

³¹ The value of a constant is the element assigned to it by the model.

³² The translation of a constant may be a phrase that is a combination of many constants. After all, what is a primitive constant of a language is highly arbitrary. It is easy to find examples of single words in one natural language which can only be translated as a long phrase in another. The translation of a sentence must be a phrase of type o, but it is an arbitrary matter whether this is recognised as a wff. (I cannot think of an analogue for this in natural languages.) But it is clear from the schema of translation given above that the translation of a phrase is always a phrase. I find this to be a conclusive reason for thinking of phrases as the fundamental units, not constants and sentences.

³³ This is the weakest acceptable criterion of synonymy. It corresponds to Carnap's L-synonymy. It will be extended in part II to phrases taken from two different languages — this will be my definition of free translation. Many authors use a stronger criterion for synonymy, corresponding to literal translation.

³⁴ Cf. XIX 134(2), pp. 22–23.

³⁵ Always putting the same constant for all occurrences of a variable.

³⁶ Cf. XV 68 and XVI 72. The proof is too complex to be repeated here.

³⁷ For the reasons for this choice see XIII 154(3), section 7.

³⁸ Although this theorem has the same restriction as theorem 27, it will help to clarify our intuition.

³⁹ This theorem is restricted, because the concept of decidability presupposes that contains a negation.

⁴⁰ It is understood that the logical constants of have the same translation whether they are in , or in , and that they are logical constants of ; hence the only interpretation of is that determined by M* of . Thus a wff without extra-logical constants is analytically true in if and only if it is analytically true in .

⁴¹ Since C_o has no extra-logical constants, it has the same value in all interpretations for a given assignment.