By Haskell Brooks Curry, Robert Feys, William Craig
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Extra resources for Combinatory Logic, Volume I
Now, where does semantics begin ? I t must be admitted that there is a major break between aletheutics and onomatics; for if a metatheory of the latter sort is formalized, there will be obs to be interpreted as expressions (or ways of constructing them), as well as obs whose interpretations are the designata of these expressions, which would not be the case for an aletheutical theory. On the other hand a large part of Parts B and C of Carnap’s book is concerned with aletheutics; in particular $5 9, 14, 18, and several other sections deal with theorems about an arbitrary aletheutical system.
2. Metasystems The notion of a syntactical system was dismissed with a few remarks in 5 L)S because combinatory logic, which is our principal business, is presented as a n abstract system. But, since by far the greater part of the current literature on formal methodology is written from a semiotical point of view, it will be appropriate to add here a few supplementary remarks. We refer to the literature for the details: but we shall discuss here certain questions in which it seems to us that the literature needs amplification, if not correction.
Inmodern logic this is done almost universally. In that case it is natural to have a different conception of some notions connected with formal systems. This differencecan be illustrated in regard to the term ‘axiom’. In 5 B an axiom is an elementary statement; it is expressed by a sentence in the U-language. In a logistic system thisstatement asserts a certain ob (formula). It is quite common to apply the term ‘axiom’ to the ob (formula) rather than to the statement which asserts it. Analogous differences pertain to some other notions.