Date: Feb 21, 2013 8:44 PM
Subject: Re: distinguishability - in context, according to definitions
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
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