The Rasiowa-Sikorski Lemma states that for any Boolean algebra B, any countable set Q of subsets of B, and any non-zero element a of B, there exists an ultrafilter U over B such that a belongs to U and U preserves all existing meets in Q. Rasiowa and Sikorski famously applied the lemma to the Lindenbaum-Tarski algebra of Classical Predicate Logic in order to prove its completeness with respect to Tarskian semantics. However, their proof of the lemma was an application of the Stone representation theorem, which relies on the non-constructive Boolean Prime Ideal Theorem (BPI). In fact, the Rasiowa-Sikorski Lemma is equivalent over ZF to the conjunction of BPI and Tarski's Lemma, a weaker statement that is "semi-constructive" in the sense that it is itself equivalent over ZF to a host of well-known results and principles, including the Baire Category Theorem and the Axiom of Dependent Choices. Moreover, Tarski's Lemma itself can be used to prove the completeness of first-order logic with respect to a broader class of structures. This fact yields an alternative semantics for classical logic, known as possibility semantics. In this presentation, Massas will show how these results generalize to the setting of distributive lattices and Heyting algebras, which are the algebraic counterpart of Boolean algebras for intuitionistic logic. In particular, Massas will present a semi-constructive version of the Rasiowa-Sikorski Lemma for distributive lattices, and show how the main idea behind the proof yields a new semantics for intuitionistic logic. Massas will present some of the main algebraic features of this semantics, and discuss its merits in comparison with standard Kripke semantics.
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