Similar information theory books

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Additional info for Combinatory logic. Vol. 1.

Example text

I t may even happen this second formal system is identical with the first. Thus, referring to the system No,we can define a k b as meaning that a and b are the same ob; it then turns out that the epistatement a f b is equivalent to the elementary statement a = b. 4 If the epistatement in such a case is precise and definite, then it is a matter of choice whether we bother with the elementary statement at all. I n particular, for a decidable formal system the theory proper can be abolished, and the system considered as a pure morphology.

See footnote 20. 1Dl LINGUISTIC ASPECTS OF A FORMAL SYSTEM 23 and the subject matter to which it is applied, the former takes into account our purposes in studying that subject matter. When a formal system is considered in connection with an application there are two sorts of truth concept to be distinguished. The truth of an elementary theorem of the formal system is determined by the abstract nature of the theory itself. Validity and acceptability are properties of the system as a whole in relation to a subject matter; if the subject matter is empirical, they are empirical also.

Then if 6 is the system No,the primitive frame for (5‘ will be as follows: 0 is a basic ob. If u is a basic ob, then a’ is a basic ob. If u and b are basic obs, a 0b is a formula. If A is a formula, I- A is an elementary statement. 1- o n o . If a and b are basic obs and 1- a c]b, then 1- a’ 06’. g. (0 0’)’,but no change in the elementary theorems. This cannot be affirmed in general. This process applies also to systems allowing connectives, which were excluded at the end of 3 B1. In fact, we can let a connective in 6correspond to an operator in 6‘ which combines formulas to give other formulas.