Intermediate Logics

With help of a compact Prolog-based theorem prover for Intuitionistic Propositional Logic, we synthesize minimal assumptions under which a given formula formula becomes a theorem. After applying our synthesis algorithm to cover basic abductive reasoning mechanisms, we synthesize conjunctions of literals that mimic rows of truth tables in classical or intermediate logics and we abduce conditional hypotheses that turn the theorems of classical or intermediate logics into theorems in intuitionistic logic. One step further, we generalize our abductive reasoning mechanism to synthesize more expressive sequent premises using a minimal set of canonical formulas, to which arbitrary formulas in the calculus can be reduced while preserving their provability. Organized as a self-contained literate Prolog program, the paper supports interactive exploration of its content and ensures full replicability of our results.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,\rho)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.

Chapter 7 - On some extensions and expansions of the basic logics

The purpose of the research presented in this paper is to extend the author's results on sequent forms of Herbrand theorems for classical and intuitionistic logics onto classical and intuitionistic modal sequent logics. It was found that the technique reported at the SYNASC 2008 symposium and based on the original notions of admissibility and compatibility can satisfactorily be applied for proving Herbrand theorems for the logics under consideration in the sequent form.

We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic $\mathbb{VA}$ .

Prawitz suggested expanding a natural deduction system for intuitionistic logic to include rules for classical logic constructors, allowing both intuitionistic and classical elements to coexist without losing their inherent characteristics. Looking at the added rules from the point of view of the Gödel-Gentzen translation, led us to propose a general method for the coexistent combination of two logics when a conservative translation exists from one logic (the source) to another (the host). Then we prove that the combined logic is a conservative extension of the original logics, thereby preserving the unique characteristics of each component logic. In this way there is no collapse of one logic into the other in the combination. We also demonstrate that a Gentzen calculus for the combined logic can be induced from a Gentzen calculus for the host logic by considering the translation. This approach applies to semantics as well. We then establish a general sufficient condition for ensuring that the combined logic is both sound and complete. We apply these principles by combining classical and intuitionistic logics capitalizing on the Gödel-Gentzen conservative translation, intuitionistic and S4 modal logics relying on the Gödel-McKinsey-Tarski conservative translation, and classical and Jaśkowski’s paraconsistent logics taking into account the existence of a conservative translation.

Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated by L. E. J. Brouwer in 1908. The first Hilbert style formalization of the intuitionistic logic, formulated as a proof system, is due to A. Heyting (1930). In this chapter we present a Hilbert style proof system I that is equivalent to the Heyting’s original formalization and discuss the relationship between intuitionistic and classical logic.

I propose a new semantics for intuitionistic logic, which is a cross between the construction-oriented semantics of Brouwer-Heyting-Kolmogorov and the condition-oriented semantics of Kripke. The new semantics shows how there might be a common semantical underpinning for intuitionistic and classical logic and how intuitionistic logic might thereby be tied to a realist conception of the relationship between language and the world.

We prove that any infinite sequence of countable series-parallel orders contains an increasing (with respect to embedding) infinite subsequence. This result generalizes Laver's and Corominas' theorems concerning betterquasi-order of the classes of countable chains and trees. Let C be a class of structures and < an order on C. This class is well-quasiordered with respect to < if for any infinite sequence C1,, C2... , Ck, ... in C, there exist i < j such that Ci < Cj. An equivalent definition is: any subset of C has finitely many minimal elements with respect to <. We are mainly concerned here with binary relations, the order < being the embedding order. Thus, 1? < R' when the relation R is embedded in the relation R' (in other words, R is an induced relation of R'). One of the very first results concerning well-quasi-order is Higman's theorem ([5]): the class of finite linear orderings (or chains) labeled by a finite set is well-quasi-ordered with respect to embedding. More precisely, the objects of the class are finite chains whose elements are labeled by a finite set, and embedding means order-preserving and label-preserving injection. For example aabcc embeds into abcabcabc but not into cbacbacba. This result was extended by Kruskal to the class of finite trees [8]. Then Nash-Williams introduced two fundamental tools of the theory: the 'minimal bad sequence' which shortened greatly the proofs concerning well-quasi-order (wqo), and the 'better-quasi-order' (bqo), a strengthening of wqo, which provides a tool to deal with countable structures [11]. Indeed, wqo is no longer an appropriate tool in the infinite case, since one can construct, from Rado's counterexample [13], an infinite set of pairwise incomparable countable subsets of a wqo. Laver proved in [9] that the class of countable chains is better-quasi-ordered (and thus well-quasi-ordered) with respect to einbedding. Later, Corominas [1] extended the result of Kruskal to countable trees: the class of countable trees labeled by a better-quasi-order is better-quasi-ordered with respect to embedding. The proof is in two parts, one devoted to the construction of countable trees, the other concerned with the preservation of better-quasi-order under certain operations. This latter aspect can be found in Milner [10] and Pouzet [12]. In this paper, Pouzet poses the problem of another class of orders, the N-free or series-parallel orders. This class contains the class of trees, and Pouzet was able to prove the wqo Received by the editors October 4, 1995 and, in revised form, January 6, 1998. 1991 Mathematics Subject Classification. Primary 05C20; Secondary 05C05,08A65,05C75.

Full classical S5 in natural deduction with weak normalization

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.

Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally $\vee $ complemented Heyting algebra (c$\vee $cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension ${\textrm {ILM}}$-${\vee }$ for c$\vee $cHas are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce’s logic and Vakarelov’s logic MIN. With a focus on properties of the two negations, relational semantics for ILM and ${\textrm {ILM}}$-${\vee }$ are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and ${\textrm {ILM}}$-${\vee }$ are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn’s logical framework for the purpose, two logics $K_{im}$ and $K_{im-{\vee }}$ are discussed, where the language does not include implication. The $K_{im}$-algebras are reducts of ccHas and are different from relevant algebraic structures having two negations. The negations in the $K_{im}$-algebras and $K_{im-{\vee }}$-algebras are shown to occupy distinct positions in an enhanced form of Dunn’s kite of negations. Relational semantics for $K_{im}$ and $K_{im-{\vee }}$ is provided by a class of frames that are based on Dunn’s compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and ${\textrm {ILM}}$-${\vee }$.

Intuitionistic Propositional Logic is complete w.r.t. Kripke semantics: if a formula is not intuitionistically valid, then there exists a finite Kripke model falsifying it. The problem of obtaining concise models has been scarcely investigated in the literature. We present a procedure to generate minimal models in the number of worlds relying on Answer Set Programming (ASP).

We say that a Kripke model is a GL-model (Gödel and Löb model) if the accessibility relation $\prec $ is transitive and converse well-founded. We say that a Kripke model is a D-model if it is obtained by attaching infinitely many worlds $t_1, t_2, \ldots $ , and $t_\omega $ to a world $t_0$ of a GL-model so that $t_0 \succ t_1 \succ t_2 \succ \cdots \succ t_\omega $ . A non-normal modal logic $\mathbf {D}$ , which was studied by Beklemishev [3], is characterized as follows. A formula $\varphi $ is a theorem of $\mathbf {D}$ if and only if $\varphi $ is true at $t_\omega $ in any D-model. $\mathbf {D}$ is an intermediate logic between the provability logics $\mathbf {GL}$ and $\mathbf {S}$ . A Hilbert-style proof system for $\mathbf {D}$ is known, but there has been no sequent calculus. In this paper, we establish two sequent calculi for $\mathbf {D}$ , and show the cut-elimination theorem. We also introduce new Hilbert-style systems for $\mathbf {D}$ by interpreting the sequent calculi. Moreover, we show that D-models can be defined using an arbitrary limit ordinal as well as $\omega $ . Finally, we show a general result as follows. Let X and $X^+$ be arbitrary modal logics. If the relationship between semantics of X and semantics of $X^+$ is equal to that of $\mathbf {GL}$ and $\mathbf {D}$ , then $X^+$ can be axiomatized based on X in the same way as the new axiomatization of $\mathbf {D}$ based on $\mathbf {GL}$ .