2 research outputs found
Propositional superposition logic
We extend classical Propositional Logic (PL) by adding a new primitive binary
connective , intended to represent the "superposition" of
sentences and , an operation motivated by the corresponding
notion of quantum mechanics, but not intended to capture all aspects of the
latter as they appear in physics. To interpret the new connective, we extend
the classical Boolean semantics by employing models of the form , where is an ordinary two-valued assignment for the sentences
of PL and is a choice function for all pairs of classical sentences. In the
new semantics is strictly interpolated between
and . By imposing several constraints on
the choice functions we obtain corresponding notions of logical consequence
relations and corresponding systems of tautologies, with respect to which
satisfies some natural algebraic properties such as associativity, closedness
under logical equivalence and distributivity over its dual connective. Thus
various systems of Propositional Superposition Logic (PLS) arise as extensions
of PL. Axiomatizations for these systems of tautologies are presented and
soundness is shown for all of them. Completeness is proved for the weakest of
these systems. For the other systems completeness holds if and only if every
consistent set of sentences is extendible to a consistent and complete one, a
condition whose truth is closely related to the validity of the deduction
theorem.Comment: 55 page
Semantics for first-order superposition logic
We investigate how the sentence choice semantics (SCS) for propositional
superposition logic (PLS) developed in \cite{Tz17} could be extended so as to
successfully apply to first-order superposition logic(FOLS). There are two
options for such an extension. The apparently more natural one is the formula
choice semantics (FCS) based on choice functions for pairs of arbitrary
formulas of the basis language. It is proved however that the universal
instantiation scheme of FOL, , is
false, as a scheme of tautologies, with respect to FCS. This causes the total
failure of FCS as a candidate semantics. Then we turn to the other option which
is a variant of SCS, since it uses again choice functions for pairs of
sentences only. This semantics however presupposes that the applicability of
the connective is restricted to quantifier-free sentences, and thus the
class of well-formed formulas and sentences of the language is restricted too.
Granted these syntactic restrictions, the usual axiomatizations of FOLS turn
out to be sound and conditionally complete with respect to this second
semantics, just like the corresponding systems of PLS.Comment: 35 page