Two Applications of Topology to the Study of Non-Classical Logics

<p>This thesis consists of two separate chapters, each exploring a different connection between logic and topology.</p> <p>Chapter 1 concerns product constructions within the continuous-logic framework of Ben Yaacov, Berenstein, Henson, and Usvyatsov. Continuous-logic analogues are presented for the direct product, direct sum, countably direct sum, almost everywhere direct product, cardinal sum, ordinal sum, and ordinal product analyzed in the work of Feferman and Vaught. We show that these constructions possess a number of preservation properties analogous to those enjoyed by their classical counterparts in ordinary first-order logic. For example, each of the above constructions preserves elementary equivalence in the following sense. Given a nonempty index set <em>I</em> and collections of metric structures <em>M_i</em> and <em>N_i</em> indexed by the elements of <em>I</em>: if <em>M_i</em> ≡ <em>N_i</em> for all <em>i</em> ∈ <em>I</em>, then ∏<em>_ieIM_i</em> ≡ ∏<em>_ieIN_i</em>.</p> <p>We also analyze several preservation properties of the classical direct product, direct sum, countably direct sum, almost everywhere direct product, and cardinal sum which follow (in ordinary model theory) from the Feferman-Vaught Theorem. Although the techniques of Feferman and Vaught do not carry over directly to the continuous-logic context, appropriate analogues of these results are established using other methods. For example, we show that given a collection of metric structures Mi indexed by the natural numbers: if a sentence <em>Θ</em> is true in ∏<em>^k</em>_<em>i</em>=0<em>M_i</em> for every <em>k</em> ≡ N, then <em>Θ</em> is true in ∏_{<em>i</em>∈ℕ}<em>Mi</em>.</p> <p>In Chapter 2, the focus is on Scott's topological model for intuitionistic analysis and the truth conditions for certain kinds of formula within this context. Attention is I Ii restricted to a certain class of real-algebraic predicates which behave, in Scott's model, much as they do when classically interpreted. Given predicates <em>M</em> and <em>N</em> belonging to this class, we extend prior work of Scowcroft to obtain decision procedures for sentences of the form ∀<em>x̄</em>(<em>M</em>(<em>x̄</em>)→¬¬∃<em>ȳN</em>(<em>x̄</em>,<em>ȳ</em>) and ∀<em>x̄</em>(<em>M</em>(<em>x̄</em>)→∃<em>y¬¬N</em>(<em>x̄</em>,<em>y</em>)). Sentences of the first form are shown to hold in Scott's model just in case they are true classically. Given a sentence of the second form, we obtain (effectively) a new sentence in the same language whose classical truth is equivalent to the truth of the original statement in Scott's model.</p>