Semantically closed intuitionistic abstract logics

Does Frege have a metatheory for his logic? There is an obvious and uncontroversial sense in which he does. Frege introduces and discusses his new logic in natural language; he argues, in response to criticisms of Begriffsschrift, that his logic is superior to Boole's by discussing formal features of both systems. In so far as the enterprise of using natural language to introduce, discuss, and argue about features of a formal system is metatheoretic, there can be no doubt: Frege has a metatheory. There is also an obvious and uncontroversial sense in which Frege does not have a metatheory for his logic. The model theoretic semantics with which we are familiar today are a post-Fregean development. The question I address in this paper is, does Frege have a metatheory in the following sense: do his justifications of his basic laws and rules of inference employ, or even require, ineliminable use of a truth predicate and metalinguistic variables? My answer is ‘no’ on both counts. I argue that Frege neither uses, nor has any need to use, a truth predicate or metalinguistic variables in his justifications of his basic laws and rules of inference. Quine's famous explanation of the need for semantic ascent simply does not apply to Frege's logic. The purpose of the discussions that are typically understood as constituting Frege's metatheory is, rather, elucidatory. And once we see what the aim of these particular elucidations is, we can explain Frege's otherwise puzzling eschewal of the truth predicate in his discussions of the justification of the laws and rules of inference.

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].

The paper is devoted to the contributions of Helena Rasiowa to the theory of non-classical negation. The main results of Rasiowa in this area concerns –constructive logic with strong (Nelson) negation, –intuitionistic negation and some of its generalizations: minimal negation of Johansson and semi-negation. We discuss also the impact of Rasiowa works on the theory of non-classical negation.

Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of what (in Section 2) we call a contra-classical modal logic, an unusual logic boasting a connective (“demi-negation”) whose double application is equivalent to a single application of the negation connective. Pondering the example points the way to a general characterization of contra-classicality (Theorems 3.3 and 4.6). In an Appendix (Section 5), we look at one alternative to classical logic as the target for such translational assimilation, intuitionistic logic, calling logics which resist the assimilation, in this case, contraintuitionistic. We will show that one such logic is classical logic itself, thereby strengthening a result of Wojcicki’s to the effect that the consequence relation of classical logic cannot be faithfully embedded by any connective-by-connective translation into that of intuitionistic logic. (What the “faithfully” means here is that not only is the translation of anything provable in the ‘source’ logic provable in the ‘target’ logic, but that also anything whose translation is provable in the target logic is provable in the source logic.)

Full classical S5 in natural deduction with weak normalization

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 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.

Deflationists about truth seek to undermine debates about the nature of truth by arguing that the truth predicate is merely a device that allows us to express a certain kind of generality. I argue that a parallel approach is available in the case of logical consequence. Just as deflationism about truth offers an alternative to accounts of truth's nature in terms of correspondence or justification, deflationism about consequence promises an alternative to model‐theoretic or proof‐theoretic accounts of consequence's nature. I then argue, against considerations put forward by Field and Beall, that Curry's paradox no more rules out deflationism about consequence than the liar paradox rules out deflationism about truth.

The present article is a brief informal survey o$\textit {computability logic}$ (CoL). This relatively young and still evolving nonclassical logic can be characterized as a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader sense, being conceived semantically rather than proof-theoretically, CoL is not just a particular theory but an ambitious and challenging long-term project for redeveloping logic. 
 
 In CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their "truth" is seen as algorithmic solvability. In turn, computational problems – understood in their most general, interactive sense – are defined as games played by a machine against its environment, with "algorithmic solvability" meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question "What can be computed?", just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases "What can be computed" always allows itself to be replaced by "How can be computed", which makes CoL a problem-solving tool. 
 
 CoL is a conservative extension of classical first order logic but is otherwise much more expressive than the latter, opening a wide range of new application areas. It relates to intuitionistic and linear logics in a similar fashion, which allows us to say that CoL reconciles and unifies the three traditions of logical thought (and beyond) on the basis of its natural and "universal" game semantics. 

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.

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.

Formal logic can be used as a tool for representing complex and heterogeneous data such as beliefs, knowledge and preferences. This study proposes an approach for defining clustering methods that deal with bases of propositional formulas in classical logic, i.e., methods for dividing formula bases into meaningful groups. We first use a postulate-based approach for introducing an intuitive framework for formula clustering. Then, in order to characterize interesting clustering forms, we introduce additional properties that take into consideration different notions, such us logical consequence, overlapping, and consistent partition. Finally, we describe our approach that shows how the inconsistency measures can be involved in improving the task of formula clustering. The main idea consists in using the measures for quantifying the quality of the inconsistent clusters. In this context, we propose further properties that allow characterizing interesting aspects related to the amount of inconsistency.

Intermediate logics are logics stronger than intuitionistic logic I but weaker than classical logic C. Most of them are motivated by their semantical characterization. In this chapter we see how the goal-directed approach can be extended to this area by analysing two case-studies. We have seen in the previous chapter that intuitionistic logic is complete with respect to the class of finite Kripke models. One can refine the completeness theorem and show that intuitionistic logic is complete with respect to Kripke models whose possible-worlds structure form a finite tree. Given a Kripke model M = (W, ≤, w 0, V), we can concentrate on the structure (W, ≤, w 0), which is a finite tree, and forget about the evaluation function V for atoms. We write λ(E) for the set of labels occurring in E. Changing the terminology a bit, we will speak about models based on a given finite tree (W, ≤, w 0), since varying V we will have several models based on it. The completeness result can then be re-phrased to assert that intuitionistic logic is complete with respect to the class of finite trees, that is to say, with respect to Kripke models based on finite trees. This change of terminology matters as we are naturally lead to consider subclasses of finite trees and ask what axioms we can add to obtain a characterization of valid formulas in these subclasses. For instance here are two natural subclasses: (1) for any n, the class of finite trees of height ≤ n; a finite tree T = (W, ≤, w 0) is in this class if there are not n + 1 different elements w 0, w 1, ... , w n , such that $${w_o} \leqslant {w_1} \leqslant ... \leqslant {w_n}hold.$$ This means that all chains are of length ≤ n. Valid formulas in these subclasses are axiomatized by the axioms BH n of the next section. (2) for any n, the class of finite trees of width ≤ n; a finite tree T = (W, ≤, w 0) is in this class if there are not n different elements which are pairwise incomparable. Valid formulas in these subclasses are axiomatized by taking the axiom schema $$V_{i = 1}^n({A_i} \to {V_{i \ne j}}{A_j})$$ for finite width trees of width ≤ n. Notice that for n = 2, we have the axiom $$({A_1} \to {A_2}) \vee ({A_2} \to {A_1})$$ which gives the well known logic LC introduced independently by Dummett [1959] and Gödel [1932] that we have already mentioned in the previous chapter. This axiom corresponds to the property of linearity: the models are finite ordered sequence of worlds (i.e. there are not two incomparable points). KeywordsInduction HypothesisClassical LogicProof SystemIntuitionistic LogicKripke ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.