Changes

=====3.1.1.1.  Blank and Bound Connectives=====

=====3.1.1.1.  Blank and Bound Connectives=====

Given an alphabet A = {a1,&nbsp;…,&nbsp;an} and a universe U = <A>, we write expressions for the propositions p&nbsp;:&nbsp;U&nbsp;&rarr;&nbsp;B upon the following basis.  The ai&nbsp;:&nbsp;U&nbsp;&rarr;&nbsp;B are interpreted as coordinate functions.  For each natural number k we have two k-ary operations, called the blank or unmarked connective and the bound or marked connective.

Given an alphabet <font face="lucida calligraphy">A</font> = {'''a'''₁,&nbsp;…,&nbsp;'''a'''_''n''} and a universe ''U'' = <font face="symbol">á</font><font face="lucida calligraphy">A</font><font face="symbol">ñ</font>, we write expressions for the propositions ''p''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' upon the following basis.  The '''a'''_''i''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' are interpreted as coordinate functions.  For each natural number ''k'' we have two ''k''-ary operations, called the ''blank'' or ''unmarked'' connective and the ''bound'' or ''marked'' connective.

The blank connectives are written as concatenations of k expressions and interpreted as k-ary conjunctions.  Thus,

The blank connectives are written as concatenations of ''k'' expressions and interpreted as ''k''-ary conjunctions.  Thus,

: e1 e2 e3 means "e1 and e2 and e3".

: ''e''₁ ''e''₂ ''e''₃ means "''e''₁ and ''e''₂ and ''e''₃".

The bound connectives are written as lists of k expressions (e1,&nbsp;…,&nbsp;ek), where the parentheses and commas are considered to be parts of the connective notation.  In text presentations the parentheses will be superscripted, as (e1,&nbsp;…,&nbsp;ek), to avoid confusion with other uses.  The bound connective is interpreted to mean that just one of the k listed expressions is false.  That is, (e1,&nbsp;…,&nbsp;ek) is true if and only if exactly one of the expressions e1,&nbsp;…,&nbsp;ek is false.  In particular, for k = 1 and 2:

The bound connectives are written as lists of k expressions (e1,&nbsp;…,&nbsp;ek), where the parentheses and commas are considered to be parts of the connective notation.  In text presentations the parentheses will be superscripted, as (e1,&nbsp;…,&nbsp;ek), to avoid confusion with other uses.  The bound connective is interpreted to mean that just one of the k listed expressions is false.  That is, (e1,&nbsp;…,&nbsp;ek) is true if and only if exactly one of the expressions e1,&nbsp;…,&nbsp;ek is false.  In particular, for k = 1 and 2: