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Table&nbsp;1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.

Table&nbsp;1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.

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| A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

| A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

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|-

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|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>

|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}</math>

|- style="background:#f0f0ff"

|- style="background:#f0f0ff"

! Expression

| <math>\text{Expression}\!</math>

! Interpretation

| <math>\text{Interpretation}\!</math>

! Other Notations

| <math>\text{Other Notations}\!</math>

|-

|-

| <math>~</math>

| <math>~</math>

'''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:

'''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:

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).

| 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).

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 will be avoided 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 will be avoided here.