Connections between Algebraic Logic and (ordinary) Logic. Algebraic counterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections. The class Alg(L) of algebras associated to any logic L. Equivalence theorems stating that L has a certain logical property iff Alg(L) has a certain algebraic property. (E.g. L admits a strongly complete Hilbertstyle inference...

We will use K̂iφ as an abbreviation for ¬Ki¬φ and 〈φ〉ψ as an abbreviation for ¬[φ]¬ψ. Definition 2 (Semantics). An epistemic model (or model for short) is a triple M = (S,∼n, V ) with S a non-empty set of states, ∼i an equivalence relation for i < n, and V a function assigning a subset V (p) ⊆ S with each p ∈ PL. Given a modelM = (S,∼n, V ) and an s ∈ S, we define: M, s |= p iff s ∈ V (p) M, s |...

1. Extensionality (definition of =): ∀x∀y (x = y iff ∀z z ∈ x iff z ∈ y). 2. Pairing (pairs exist): ∀x∀y∃z∀w (w ∈ z iff (w = x or w = y)). 3. Union (unions exist): ∀x∃z∀w (w ∈ z iff ∃y (w ∈ y and y ∈ x)). 4. Power set (power sets exist): ∀x∃z∀w(w ∈ z iff w ⊂ x). 5. Regularity (∀x (x,∈) has a minimal element): ∀x∃y (y ∈ x and y ∩ x = ∅). 6. Infinity (there is an infinite set): ∃x(x 6= ∅ and ∀y (...

The Information Flow Framework (IFF) [1] is a descriptive category metatheory currently under development, which is being offered as the structural aspect of the Standard Upper Ontology (SUO). The architecture of the IFF is composed of metalevels, namespaces and meta-ontologies. The main application of the IFF is institutional: the notion of institutions and their morphisms are being axiomatize...

In this paper, concepts of strongly way below relations, completely precontinuous posets, coprimes and Heyting posets are introduced. The main results are: (1) The strongly way below relations of completely precontinuous posets have the interpolation property; (2) A poset P is a completely precontinuous poset iff its normal completion is a completely distributive lattice; (3) An ω-chain complet...

Solution: Since Λ(y) has finitely many (3) possible values, all values of η between any adjacent pair lead to the same threshold test. Thus, for η > 1, Λ(y) > η, leads to the decision ĥ = 0 if and only if (iff) Λ(y) = 1, i.e., iff −1 ≤ y < 0. For η = 1, the rule is the same, Λ(y) > η iff Λ(y) = 1, but here there is a ‘don’t care’ case Λ(y) = 1 where 0 ≤ y ≤ 1 leads to ĥ = 1 simply because of th...

For 1 1 ( ) ( )∈! m m m x = x ,...,x , y = y ,...,y we put ≤ x y iff i i x y ≤ for each { } 1 2 ∈ i M = , ,...,m ; ≤ x y iff ≤ i i x y for each i M ∈ , with x y; x < y ≠ iff i i x < y for each i M ∈ . We write + ∈! m x iff 0 ≥ x . For an arbitrary vector n x∈! and a subset J of the index set { } 1 2 n , ,..., , we denote by J x the vector with components j x , j J ∈ . Let ( ) μ X , Γ, be a fini...