By F. H. George (Auth.)
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Extra info for Precision, Language and Logic
Similarly a rule of inference is independent if when it is omitted from the system, some of the theorems of the full system are no longer theorems of the remaining axiomatic system. We could also say more tersely that for an axiom, x say, to be independent is to say that M x is false" is consistent with the rest of the system. This reduces the problem of independence to that of consistency. This means that if we place an interpretation on the system, all the axioms except x become true, since x is independent.
2. If A and B are wf (A D B) is wf. and no other formula is wf This sort of definition is one that supplies defining provides a limitation that other than those that follow from these rules. a recursive one, where a recursive definition is characteristics such as in 1 and 2 above, and then this is all that is a defining characteristic. We shall now consider only wff in our discussion, and one of our main aims is how to distinguish those wff that are theorems (or true statements of the system) from those that are not.
Let us explain our dot convention. p D (q D p) is written P D • Q D P since p D q D p would be interpreted as (P D q ) 3 P Similarly P D Q D P D P is read as ( ( ( p D q ) D P) D P) Precision, Language and Logic 40 whereas if we wanted (p D (q D (p D P))) would be written P D • Q D : P D P where : takes precedence over . and, of course, :. so on. In other words dots are effectively brackets. would precede : , and Look again at:- P D • q D'P we insert a left bracket for the dot and then need its right bracket to properly mate it:P D (q DP) But there is also the association to the left rule which means (p D q) DP is the same as P D q DP Similarly, combining the two rules we read P D P D q D - P D q as ((p DP) D q ) D (p Dq) and P D - p D q D : p D q D q D- P D q as ((p D(pD q)) D ( ( ( p D q ) Dq) D(p Dq) Sometimes a compromise notation would lead to the writing of this last example as:( p D - P D q ) D .