The Construction of The Theory of PM
As noted in the criticism of the theory by Kurt Gödel (below), unlike a Formalist theory, the "logicistic" theory of PM has no "precise statement of the syntax of the formalism". Another observation is that almost immediately in the theory, interpretations (in the sense of model theory) are presented in terms of truth-values for the behavior of the symbols "⊢" (assertion of truth), "~" (logical not), and "V" (logical inclusive OR).
Truth-values: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw (pure) Formalist theory would not provide the meaning of the symbols that form a "primitive proposition"—the symbols themselves could be absolutely arbitrary and unfamiliar. The theory would specify only how the symbols behave based on the grammar of the theory. Then later, by assignment of "values", a model would specify an interpretation of what the formulas are saying. Thus in the formal Kleene symbol set below, the "interpretation" of what the symbols commonly mean, and by implication how they end up being used, is given in parentheses, e.g., "¬ (not)". But this is not a pure Formalist theory.
Read more about this topic: Principia Mathematica
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