Abstract Let $\to $ be a continuous $\protect \operatorname {\mathrm {[0,1]}}$ -valued function defined on the unit square $\protect \operatorname {\mathrm {[0,1]}}^2$ , having the following properties: (i) $x\to (y\to z)= y\to (x\to z)$ and (ii) $x\to y=1 $ iff $x\leq y$ . Let $\neg x=x\to 0$ . Then the algebra $W=(\protect \operatorname {\mathrm {[0,1]}},1,\neg ,\to )$ satisfies the time-honored Łukasiewicz axioms of his infinite-valued calculus. Let $x\to _{\text {\tiny \L }}y=\min (1,1-x+y)$ and $\neg _{\text {\tiny \L }}x=x\to _{\text {\tiny \L }} 0 =1-x.$ Then there is precisely one isomorphism $\phi $ of W onto the standard Wajsberg algebra $W_{\text {\tiny \L }}= (\protect \operatorname {\mathrm {[0,1]}},1,\neg _{\text {\tiny \L }},\to _{\text {\tiny \L }})$ . Thus $x\to y= \phi ^{-1}(\min (1,1-\phi (x)+\phi (y)))$ .

Suppose we are given a set Φ of m Boolean formulas with the information that e of these formulas are unconfirmed, while the actual set of unconfirmed formulas is not disclosed to us. Let us denote by Rest(Φ,e) the family of all subsets of Φ having m−e elements. We are interested in the problem whether a Boolean formula ω is a consequence of Ψ for each Ψ∈Rest(Φ,e)⁠. More generally, given for each i=1,…,h a set Φi of mi Boolean formulas and an integer 0≤ei

We endow the partially ordered set of nonempty faces of the n -cube with a distinguished 0-dimensional face and two operations that naturally extend the Rota-Metropolis partial operations. Although the structures thus obtained turn out to be term-equivalent to Post algebras of order 3, the inclusion order between faces coincides with the De Luca-Termini sharpening order, and yields a compact coNP-complete logic that tolerates a modicum of inconsistency.

Working jointly in the equivalent categories of MV-algebras and lattice-ordered abelian groups with strong order unit (for short, unital ℓ-groups), we prove that isomorphism is a sufficient condition for a separating subalgebra A of a finitely presented algebra F to coincide with F. The separation and isomorphism conditions do not individually imply A = F. Various related problems, like the separation property of A, or A≅F (for A a separating subalgebra of F), are shown to be (Turing-)decidable. We use tools from algebraic topology, category theory, polyhedral geometry and computational algebraic logic.

By an MV-set, we understand a pair (E,X) where X is a set of unit vectors in a Hilbert space E such that the linear span of X is dense in E, and 〈v,w〉 ≥ 0 for all v,w ∈ X. The scalar product 〈v,w〉 ∈ [0,1] is the identity degree of v and w. Building on MV-sets and continuous functions and relations defined on them, we construct a compact [0,1]-valued first-order Łukasiewicz logic, whose set of unsatisfiable formulas is recursively enumerable. In the particular case when X is an orthonormal basis of E we recover classical Skolem first-order logic with identity, constants, functions and relations. Our main tools are the Kolmogorov dilation theorem for positive semidefinite kernels, and the Tarski–Seidenberg decision method for elementary algebra and geometry.

Abstract The inverse of the distance between two structures ≢ of finite typeτis naturally measured by the smallest integerqsuch that a sentence of quantifier rankq− 1 is satisfied by but not by . In this way the space Strτof structures of typeτis equipped with a pseudometric. The induced topology coincides with the elementary topology of Strτ. Using the rudiments of the theory of uniform spaces, in this elementary note we prove the convergence of every Cauchy net of structures, for any typeτ.

Abstract We give a constructive proof of McNaughton's theorem stating that every piecewise linear function with integral coefficients is representable by some sentence in the infinite-valued calculus of Lukasiewicz. For the proof we only use Minkowski's convex body theorem and the rudiments of piecewise linear topology.

Abstract Given an abstract logic , generated by a set of quantifiers Qi, one can construct for each type τ a topological space Sτ, exactly as one constructs the Stone space for τ in first-order logic. Letting T be an arbitrary directed set of types, the set is an inverse topological system whose bonding mappings are naturally determined by the reduct operation on structures. We relate the compactness of to the topological properties of ST. For example, if I is countable then is compact iff for every τ each clopen subset of Sτ is of finite type and Sτ, is homeomorphic to limST, where T is the set of finite subtypes of τ. We finally apply our results to concrete logics.

Abstract We prove the following algebraic characterization of elementary equivalence: ≡ restricted to countable structures of finite type is minimal among the equivalence relations, other than isomorphism, which are preserved under reduct and renaming and which have the Robinson property; the latter is a faithful adaptation for equivalence relations of the familiar model theoretical notion. We apply this result to Friedman's fourth problem by proving that if is an (ω1, ω)-compact logic satisfying both the Robinson consistency theorem on countable structures of finite type and the Löwenheim-Skolem theorem for some λ < ωω for theories having ω1 many sentences, then ≡L = ≡ on such structures.