Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes, geometric entities which relate to first-order logical theories in much the same way that algebraic schemes relate to commutative rings.
The construction relies on a Grothendieck-style representation theorem which associates every coherent or classical first-order theory with an affine scheme: a topological groupoid (the spectrum of the theory) together with a sheaf of (local) syntactic categories. The groupoid is constructed from the semantics of the theory (models and isomorphisms) and topologized using a Stone-type construction. The sheaf of categories can be regarded as a logical theory varying over the spectrum, and its global sections recover the theory up to semantic equivalence. These affine pieces can be glued together to give more general logical schemes and these are studied using methods from algebraic geometry. The final chapter also presents some connections between schemes and other areas of logic such as model theory, type theory and topos theory.
The fundamental duality theories relating algebra and geometry that were discovered in the mid-20th century can also be applied to logic via its algebraization under categorical logic. They thereby result in known and new completeness theorems. This idea can be taken even further via what is sometimes called ``categorification'' to establish a new connection between logic and geometry, a glimpse of which can also be had in topos theory.
Chevalley's theorem on the images of morphisms of schemes and the principle of quantifier elimination for the theory of algebraically closed fields are widely understood to be two perspectives on the same theorem. In this paper, we demonstrate that both results can easily be proven simultaneously, using a mixture of geometric and logical techniques. In doing so, we give logical meaning to geometric points of schemes and to finitely presented morphisms thereof, in a manner reminiscent of Spencer Breiner's logical schemes.
In Pure Inductive Logic, the principle of Strong Predicate Exchangeability is a rational principle based on symmetry that sits in between the principles of Predicate Exchangeability and Atom Exchangeability. We will show a de Finetti - style representation theorem for probability functions that satisfy this principle in addition to Unary Language Invariance.
Riesz representation theorem, Daniell-Stone theorem for Daniell integrals and Stone's representation theorem for probability and measure algebras are three important classical results in analysis concerning existence of measures with certain properties. Many proofs of these theorems can be found in the literature of analysis, from elementary ones which use ordinary techniques from measure theory, to more sophisticated ones, such as those employing techniques from nonstandard analysis, in particular for Riesz representation theorem. In this paper, as the first goal, we give new proofs for all t…
Moisil in 1941, while constructing the algebraic models of n-valued Łukasiewicz logic defined the set $B^{[n]}$,where $B$ is a Boolean algebra and `n' being a natural number. Further it was proved by Moisil himself the representations of n-valued Łukasiewicz Moisil algebra in terms of $B^{[n]}$. In this article, structural representation results for Stone, dual Stone and double Stone algebras are proved similar to Moisil's work by showing that elements of these algebras can be looked upon as monotone ordered tuple of sets. 3-valued semantics of logic for Stone algebra, dual Stone algebras and …
Using Sheaf duality theory of Comer for cylindric algebras, we give a representation theorem of of distributive bounded lattices expanded by modalities (functions distributing over joins) as the continuous sections of sheaves. Our representation is defined via a contravariant functor. We give applications to many-valued logics logics and various modifications of first order logic and multi-modal logic, set in an algebraic framework.
A structural theorem for Kleene algebras is proved, showing that an element of a Kleene algebra can be looked upon as an ordered pair of sets. Further, we show that negation with the Kleene property (called the `Kleene negation') always arises from the set theoretic complement. The corresponding propositional logic is then studied through a 3-valued and rough set semantics. It is also established that Kleene negation can be considered as a modal operator, and enables giving a perp semantics to the logic. One concludes with the observation that all the semantics for this logic are equivalent.
We continue work of our earlier paper (Lewitzka and Brunner: Minimally generated abstract logics, Logica Universalis 3(2), 2009), where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the cate…
Various topological concepts are often involved in the research of mathematical logic, and almost all of these concepts can be regarded as developing from the Stone representation theorem. In the Stone representation theorem, a Boolean algebra is represented as the algebra of the clopen sets of a Stone space. And based on this, a natural connection is established between the structure of Stone space and the semantics of propositional logic. In other words, models of a propositional theory are represented as points in a Stone space. This enables us to use the concepts of topology to describe ma…
We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{ω_1 ω}$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property {\bf (T)} can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.
The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the…
In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion s…
In this paper, we explain how the connection between higher-order model-checking and linear logic recently exhibited by the authors leads to a new and conceptually enlightening proof of the selection problem originally established by Carayol and Serre using collapsible pushdown automata. The main idea is to start from an infinitary and colored relational semantics of the lambdaY-calculus already formulated, and to replace it by its finitary counterpart based on finite prime-algebraic lattices. Given a higher-order recursion scheme G, the finiteness of its interpretation in the model enables us…
Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. This logic involves Kleene star, axiomatized by an induction scheme. For a stronger system which uses an $ω$-rule instead (infinitary action logic) Buszkowski and Palka (2007) have proved $Π_1^0$-completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by D. Kozen in 1994. In this article, we show that it is undecidable, more precisely, $Σ_1^0$-complete. We also prove the same complexity results for all recursively enumerable logics between action logic and infi…
