Explore Logic in the second year at UCL. Unravel the intricacies of this fundamental field, delving into the principles and frameworks that govern logical reasoning. Please note that these materials are intended for personal use only and should be used in accordance with academic integrity guidelin...
In order to be able to study the structure of mathematical objects and ideas, we will first provide a
framework in which mathematical objects can be defined and analysed.
1.1 First order predicate language
The symbols which we will use to construct our language consist of:
1) A countably infinite set of variable symbols: {x1 , x2 , x3 · · · } or {w, x, y, z, w0 , x0 , y 0 , z 0 , · · · }.
2) For each non-negative integer n, a countably infinite set of predicate symbols {P1 , P2 , P3 · · · }
or {P, Q, R, P 0 , Q0 , R0 , · · · }, each of which has arity n. If P is a predicate symbol of arity n,
then we call P an n-ary predicate symbol.
3) The symbols ¬, ), 8.
The set of all strings of symbols from the above list is denoted by Lstring .
For example, suppose that x, y, z, x1 , x2 , x3 are variable symbols, P is a 1-ary predicate symbol, and
Q is a 2-ary predicate symbol. Then x, P x, P xy, Qx, Qxy, Qxx, xQy, Qx1 x2 x3 , ¬P x, ) P xQyz,
8x8yQxy, 8x, 8x ) ¬P xQxy, 8x ) P ¬xQxy are all in Lstring .
Let us now consider the subset of Lstring that will be especially useful:
3
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