On 2/21/2013 2:08 AM, Shmuel (Seymour J.) Metz wrote: > > You have to start with a topology to have a quotient topology. There > is no need to model theories on topological spaces. >
All Boolean-valued forcing models are based on topologies constructed from regular open sets in a complete Boolean algebra. These topologies are semi-regular.
My point is that the very syntax of a first-order language can be recognized as a minimal Hausdorff topology as soon as one places a mutually exclusive bivalent truth functionality onto its symbols. Minimal Hausdorff topologies are semi-regular.
Moreover, you cannot divorce this structure from a logic intended as a deductive calculus because what makes it interpretable as a deductive calculus is its relationship to the truth-conditions of interpretations.
Deductive calculi are neither arithmetical calculi nor computer languages.