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, 15:58, 17 January 2008→3.1.1.1. Blank and Bound Connectives: markup
=====3.1.1.1. Blank and Bound Connectives=====
=====3.1.1.1. Blank and Bound Connectives=====
Given an alphabet A = {a1, …, an} and a universe U = <A>, we write expressions for the propositions p : U → B upon the following basis. The ai : U → 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'''<sub>1</sub>, …, '''a'''<sub>''n''</sub>} 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'' : ''U'' → '''B''' upon the following basis. The '''a'''<sub>''i''</sub> : ''U'' → '''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''<sub>1</sub> ''e''<sub>2</sub> ''e''<sub>3</sub> means "''e''<sub>1</sub> and ''e''<sub>2</sub> and ''e''<sub>3</sub>".
The bound connectives are written as lists of k expressions (e1, …, 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, …, 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, …, ek) is true if and only if exactly one of the expressions e1, …, ek is false. In particular, for k = 1 and 2:
The bound connectives are written as lists of k expressions (e1, …, 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, …, 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, …, ek) is true if and only if exactly one of the expressions e1, …, ek is false. In particular, for k = 1 and 2: