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Table 4 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.

Table 4 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.

* A bracketed list of propositional expressions in the form <math>(e_1, e_2, \ldots, e_{k-1}, e_k)</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>(e_1, e_2, \ldots, e_{k-1}, e_k)</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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|+ '''Table 4.  Syntax and Semantics of a Calculus for Propositional Logic'''

|+ '''Table 4.  Syntax and Semantics of a Propositional Calculus'''

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|- style="background:ghostwhite"

! Expression

! Expression

<br>

<br>

All other propositional connectives can be obtained through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracket form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.  The briefest expression for logical truth is the empty word, abstractly denoted <math>\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  It can be given visible expression in this context by means of the logically equivalent expression "<math>((~))</math>", or, especially if operating in an algebraic context, by a simple "<math>1\!</math>".  Also when working in an algebraic mode, the plus sign "<math>+\!</math> may be used for exclusive disjunction.  For example, we have the following paraphrases of algebraic expressions by bracket expressions:

All other propositional connectives can be obtained through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracket form, but it is convenient to maintain it as an abbreviation for more complicated bracket expressions.  The briefest expression for logical truth is the empty word, abstractly denoted <math>\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  It can be given visible expression in this context by means of the logically equivalent expression "<math>((~))</math>", or, especially if operating in an algebraic context, by a simple "<math>1\!</math>".  Also when working in an algebraic mode, the plus sign "<math>+\!</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by bracket expressions:

<center>

<center>

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>

For more information about this syntax for propositional calculus, see the entries on [[minimal negation operator]]s, [[zeroth order logic]], and [[Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus#Table_1|Table 1 in Appendix 1]].

For more information about this syntax for propositional calculus, see the entries on [[minimal negation operator]]s, [[zeroth order logic]], and [[Directory:Jon_Awbrey/Papers/Differential_Propositional_Calculus#Table_A1.__Propositional_Forms_on_Two_Variables|Table A1 in Appendix 1]].

==Formal development==

==Formal development==