WEBVTT

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The paper assembles established finite Boolean, topological, order and distance constructions in

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one propositional model.

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Its contribution is organizational and explanatory.

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It does not claim a new class of topological spaces merely because

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formulas provide a convenient finite presentation.

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A restricted family of positive tests determines which truth events are certifiable,

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which valuations are indistinguishable and which one-way comparisons preserve positive information.

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Selecting all definable subsets would instead give a discrete topology and remove

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that restriction.

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A signature is the selected truth-answer vector; its support lists the

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true positions.

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A signature fiber is a class of worlds with the same selected

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signature.

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Our information order preserves those positives.

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Opens describe what positive tests can certify.

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Numerical distances also need the indexed test list and its weights.

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Adding tests changes the interface, while accepted premises restrict the surviving worlds.

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Signatures recover topological indistinguishability and the specialization preorder.

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In this finite setting the preorder determines the opens as upper sets.

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These are different descriptions of the same qualitative structure under the chosen

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convention.

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Numerical distances require the indexed formula family and weights.

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Adding a redundant coordinate or changing a weight can change numbers without

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changing the topology.

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A topology by itself does not determine a unique formula list or

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distance table.

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Positive observation generates opens.

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Adding both signs opens original signature fibers.

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Adding tests refines the interface on a fixed carrier.

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Learning premises restricts possible valuations and uses the induced subspace topology.

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Those operations must not be merged into one picture of information gain.

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Classical consequence remains inclusion of truth regions.

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An inclusion map keeps a subspace point as the same ambient point.

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Its continuity says that ambient-open events restrict to open events in

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that subspace.

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The topology and its continuous inclusions organize this reasoning but do not

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create a new validity rule.

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In the optional open-set lessons, negation uses the interior of a

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complement, and implication uses the interior of its classical candidate.

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Each rule is applied at its own connective in the expression.

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Heyting open-set operations were explicitly a different recursive interpretation, not a

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retroactive change to classical evaluation.

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A relation nerve records which indexed tests can be simultaneously satisfied.

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In this diagram, vertices stand for indexed tests.

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An edge means two tests share one satisfying world; a filled triangle

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requires one world satisfying all three.

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Any such compatible group is called a simplex.

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Homology is being used here to discuss holes, such as an unfilled

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loop.

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The diagram’s vertices are test slots, rather than worlds.

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It can distinguish pairwise from joint compatibility.

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But it is not determined by the observation topology, and its holes

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need not describe the homology of the valuation space.

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A tautology is a formula true at every valuation.

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Adding it supplies a constant true coordinate, leaving generated opens and both

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distances unchanged.

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In the test-compatibility complex it adds a new vertex joined to

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every old simplex, forming a cone.

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Any homological claim must therefore identify the presentation and state the intended

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invariance separately.

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The extension lessons discuss connectedness, a compatibility complex, local consistency and observation

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design.

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Some are historical enrichment removed or narrowed in v0.

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9.

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They are labelled accordingly; taking them does not turn suggested directions into

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established new results of the manuscript.

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The notes distinguish complete Boolean truth sets, two-sided indiscernibility partitions and

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positive observation opens.

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For this optional comparison, a conjunctive extent is the set of worlds

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satisfying a chosen group of tests together.

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The formal concept lattice organizes these extents by inclusion.

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The generated opens also allow unions of the conjunction regions.

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A union of conjunctive extents need not itself be an extent, so

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the open-set lattice is not generally the formal concept lattice.

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These are comparisons, not new duality theorems.

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Does the observation topology alone determine numerical weights, the homology of an

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indexed compatibility complex, or a new classical inference rule?

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Explain one concrete reason for each answer before checking the notes.

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Weights are extra quantitative choices.

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A tautology can change the nerve while preserving the topology.

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Classical validity still asks whether every premise survivor satisfies the conclusion.

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None of those stronger claims follows merely from naming the same finite

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topology.

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The useful outcome is knowing which distinctions a chosen interface can support

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and which conclusions require additional data.

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New algorithmic results, new invariants or validated applications would require further work.

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This course has taught the finite constructions and their boundaries, not claimed

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those further achievements.