Changes

As before, all of the boolean functions of fewer variables are subsumed in this Table, though under a set of alternative names and possibly different interpretations.  Just to acknowledge a few of the more notable pseudonyms:

As before, all of the boolean functions of fewer variables are subsumed in this Table, though under a set of alternative names and possibly different interpretations.  Just to acknowledge a few of the more notable pseudonyms:

: The constant function <math>\underline{0} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> appears under the name <math>F_{0}^{(2)}.</math>

The constant function %0% : %B%^2 -> %B% appears under the name of F^2_00.

The constant function %1% : %B%^2 -> %B% appears under the name of F^2_15.

: The constant function <math>\underline{1} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> appears under the name <math>F_{15}^{(2)}.</math>

The negation and identity of the first variable are F^2_03 and F^2_12, resp.

: The negation and identity of the first variable are <math>F_{3}^{(2)}</math> and <math>F_{12}^{(2)},</math> respectively.

The negation and identity of the other variable are F^2_05 and F^2_10, resp.

: The negation and identity of the second variable are <math>F_{5}^{(2)}</math> and <math>F_{10}^{(2)},</math> respectively.

The logical conjunction is given by the function F^2_08 (x, y) = x · y.

: The logical conjunction is given by the function <math>F_{8}^{(2)} (x, y) = x \cdot y.</math>

The logical disjunction is given by the function F^2_14 (x, y) = ((x)(y)).

: The logical disjunction is given by the function <math>F_{14}^{(2)} (x, y) = \underline{((} ~x~ \underline{)(} ~y~ \underline{))}.</math>

Functions expressing the "conditionals", "implications",

Functions expressing the ''conditionals'', ''implications'', or ''if-then'' statements are given in the following ways:

or "if-then" statements are given in the following ways:

[x => y] = F^2_11 (x, y) = (x (y)) = [not x without y].

: <math>[x \Rightarrow y] = F_{11}^{(2)} (x, y) = \underline{(} ~x~ \underline{(} ~y~ \underline{))} = [\operatorname{not}~ x ~\operatorname{without}~ y].</math>

[x <= y] = F^2_13 (x, y) = ((x) y) = [not y without x].

: <math>[x \Leftarrow y] = F_{13}^{(2)} (x, y) = \underline{((} ~x~ \underline{)} ~y~ \underline{)} = [\operatorname{not}~ y ~\operatorname{without}~ x].</math>

The function that corresponds to the "biconditional",

The function that corresponds to the ''biconditional'', the ''equivalence'', or the ''if and only'' statement is exhibited in the following fashion:

the "equivalence", or the "if and only" statement is

exhibited in the following fashion:

[x <=> y] = [x = y] = F^2_09 (x, y) = ((x , y)).

: <math>[x \Leftrightarrow y] = [x = y] = F_{9}^{(2)} (x, y) = \underline{((} ~x~,~y~ \underline{))}.</math>

Finally, there is a boolean function that is logically associated with

Finally, there is a boolean function that is logically associated with the ''exclusive disjunction'', ''inequivalence'', or ''not equals'' statement, algebraically associated with the ''binary sum'' operation, and geometrically associated with the ''symmetric difference'' of sets. This function is given by:

the "exclusive disjunction", "inequivalence", or "not equals" statement,

algebraically associated with the "binary sum" or "bitsum" operation,

and geometrically associated with the "symmetric difference" of sets.

This function is given by:

[x =/= y] = [x + y] = F^2_06 (x, y) = (x , y).

: <math>[x \neq y] = [x + y] = F_{6}^{(2)} (x, y) = \underline{(} ~x~,~y~ \underline{)}.</math>

Let me now address one last question that may have occurred to some.

Let me now address one last question that may have occurred to some. What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called ''conditionals'' and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question: What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>?  My answer is this:  For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.

What has happened, in this suggested scheme of functional reasoning,

to the distinction that is quite pointedly made by careful logicians

between (1) the connectives called "conditionals" and symbolized by

the signs "->" and "<-", and (2) the assertions called "implications"

and symbolized by the signs "=>" and "<=", and, in a related question:

What has happened to the distinction that is equally insistently made

between (3) the connective called the "biconditional" and signified by

the sign "<->" and (4) the assertion that is called an "equivalence"

and signified by the sign "<=>"?  My answer is this:  For my part,

I am deliberately avoiding making these distinctions at the level

of syntax, preferring to treat them instead as distinctions in

the use of boolean functions, turning on whether the function

is mentioned directly and used to compute values on arguments,

or whether its inverse is being invoked to indicate the fibers

of truth or untruth under the propositional function in question.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

070.  http://suo.ieee.org/ontology/msg03473.html

070.  http://suo.ieee.org/ontology/msg03473.html

071.  http://suo.ieee.org/ontology/msg03479.html

071.  http://suo.ieee.org/ontology/msg03479.html

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