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It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>

It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>

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|+ '''Table 1.  Syntax and Semantics of a Calculus for Propositional Logic'''

|+ '''Table 1.  Syntax and Semantics of a Calculus for Propositional Logic'''

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'''Note.''' The usage that one often sees, of a plus sign "+" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts.  Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts.  Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:

<blockquote>

<blockquote>

The expression ''x'' + ''y'' seems indeed uninterpretable, unless it be assumed that the things represented by ''x'' and the things represented by ''y'' are entirely separate;  that they embrace no individuals in common.  (Boole, 66).

The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate;  that they embrace no individuals in common.  (Boole, 66).

</blockquote>

</blockquote>

It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189).  It seems to have been Schroeder who later reassigned the plus sign to inclusive disjunction (Sty, 208).  Additional information, discussion, and references can be found in (Boole) and (Sty, 177-263).  Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic.  For this reason, I am forced to avoid it here.

It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189).  It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208).  Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263).  Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic.  For this reason, it is best to avoid it here.

==A Functional Conception of Propositional Calculus==

==A Functional Conception of Propositional Calculus==