Last Update: 27 October 2008
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Abstract: A syntactical phenomenon common to logics of commands, of questions, and some deontic logics is investigated using techniques of algebraic logic. The phenomenon is simple to describe. In terms of questions, the result of combining an indicative sentence (e.g., 'It is raining') with an interrogative sentence (e.g., 'Should I go home?') in (say) a conditional construction is an interrogative ('If it is raining, should I go home?'). Similarly combining an indicative with a command sentence (e.g., 'Go home!') results in a command sentence ('If it is raining, go home!'). In the deontic logic proposed by Hector-Neri Castañeda (in The Structure of Morality and in Thinking and Doing), the result of thus combining an indicative with a "practitive" is a practitive. These syntactical facts are reminiscent of scalar multiplication in vector spaces: The product of a scalar and a vector is a vector. However, neither vector spaces nor modules are general enough to serve as appropriate algebraic analogues of these logics. Taking the sentential (i.e., nonquantificational, nonmodal) fragment C of Castañeda's deontic logic as a paradigm, it is proposed that the relevant algebraic structure is a "dominance algebra" (DA), where <M, R, I, E> is a dominance algebra (over R) iff (i) R is an abstract algebra, (ii) M is not empty, (iii) non-empty I is a subset of M^{M^n} (n \in \omega), and (iv) non-empty E is a subset of M^{(B^n × M^m) \union (M^m × B^n)} (m, n \in \omega). (Modules are special cases of dominance algebras.) It is proved that the Lindenbaum algebra corresponding to C is a "double Boolean DA" (DBDA) (viz., one in which M and R are Boolean algebras), soundness and completeness theorems for C are obtained, and a representation theorem for DBDAs is proved. The paper concludes with some generalizations of DAs and some remarks on their relevance to Montague-style grammars.