By Anthony P. Morse

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Etc. In other words, ( A B ) is the concatenation of A and B. For example, (‘sin’ ‘x’) is ‘sin x’. Remark. We suppose herein that C is (‘x’‘x’‘x’). 0, it is easy to see that C is ‘xxx’. 12 0. Language and Inference On p. 2 we agreed that quotation marks are not among our symbols. 0 alone a contradiction would arise as follows. We also suppose A is ‘ x ’ ; A’ is ‘x’ ‘ x ’ ; B is ‘x’ ‘ x ’ ; B’ is ‘x’. 0 we have on the one hand C is (AA’) and on the other C is (BB’). But (AA’) is ‘xx’ ‘x’ and (BB’) is ‘x’ ‘xx’.

Theorems, of course, are of particular interest to us. Our rules of inference enable us, step by step, to use theorems already known to us to discover new theorems. Formalization describes with care an explicit process for arriving at Roughly speaking, a variable is free in a formula if and only if every occurrence is a free occurrence. 10 0. Language and Inference theorems. Our rules are to be taken for granted, although some of them can be derived from others. Our rules are akin to, but different from, axioms and theorems.

30 a is of type n if and only if AGREEMENT. 7 9. 8 <+’7 ‘wellorders’, ‘orders’, ‘eq’, ‘topologizes’, ‘<’, ‘>’, ‘ s ’ , <>’, ‘metrizes’, ‘simplymetrizes’, ‘measures’; n = 7andais , n = 8 and a is one of ,, n = 9 and u is one of ‘Q’, ‘+’, ‘ I ‘ 7 . 31 ct . ’, ‘O‘t’, ‘as’, ‘*’, ‘n’; - or is -. 9 AGREEMENTS. A symbol is a binarian if and only if it is a symbol of some type. A symbol is a binariate if and only if it is either ‘x’ or one of the primed symbols derived therefrom. 32 AGREEMENT. c is a nexus if and only if c is an expression in which each symbol is a binarian.