We present a semantic game for Gödel logic and its extensions, where the players’ interaction stepwise reduces arbitrary claims about the relative order of truth degrees of complex formulas to atomic ones. The paper builds on a previously developed game for Gödel logic with projection operator in Fermüller et al., Information processing and management of uncertainty in knowledge-based systems, Springer, Cham, 2020, pp. 257–270). This game is extended to cover Gödel logic with involutive negations and constants, and then lifted to (...) a provability game using the concept of disjunctive strategies. Winning strategies in the provability game, with and without constants and involutive negations, turn out to correspond to analytic proofs in a version of \ :1941–1958, 2010) and in a sequent-of-relations calculus, Automated reasoning with analytic tableaux and related methods, Springer, Berlin, 1999, pp. 36–51) respectively. (shrink)

A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads (...) to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics. We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory. (shrink)

We introduce a game for Gödel logic where the players’ interaction stepwise reduces claims about the relative order of truth degrees of complex formulas to atomic truth comparison claims. Using the concept of disjunctive game states this semantic game is lifted to a provability game, where winning strategies correspond to proofs in a sequents-of-relations calculus.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 63.1 (2002) 129-149 [Access article in PDF] Lord Acton and "the Insanity of Nationality" Timothy Lang "I hope I need not warn you against Montalembert's declamation about Poland—He has no idea of the insanity of nationality...." Acton to Richard Simpson, 25 September 1861 The sixty-year period that culminated in the First World War witnessed a momentous transformation in the European state system. Italian (...) and German unification, the expulsion of the Ottomans from southeastern Europe, the destruction of the three remaining empires—the Austrian, German, and Russian—as casualties of the war, their replacement by a band of successor states stretching from the Baltic to the Balkans: these developments completely altered the face of Europe, as dynastic states gave way to nation-states. Looking back on this transformation once the war was over, many Europeans believed the decision to rebuild their continent on the basis of nation-states made perfect sense. To establish the principles of self-determination and democracy would contribute to peace and prosperity by solving the nationalities problem that had caused such mischief in the past. But in the 1860s, as this transformation was just beginning, the question looked very different, especially to the young John Dalberg Acton—the future Lord Acton, famous for the Cambridge Modern History—as he contemplated without benefit of hindsight the rise of the modern nation-state. 1 [End Page 129]Acton's views on the nation-state appeared in an article on "Nationality," which he wrote in 1862 for the Home and Foreign Review. 2 Nationality—today we would call it nationalism—was, Acton pointed out to his colleague Richard Simpson, a "new question in political philosophy" 3 ; and Acton's essay was a remarkably precocious treatment of the subject. In this tour de force he placed nationalism in its historical context, enabling readers to understand its significance for their own time and to see in nationalism the most potent force unleashed by the French Revolution. For Acton the Revolution had given a new direction to social protest. Whereas in the past, people had resisted oppression by calling for a return to traditional ways, now, after 1789, they demanded a new social and political order. The Revolution thus spawned protest movements of a wholly unprecedented kind. These movements were "spontaneous and aggressive," he warned. They were "popular, unreasoning, and almost irresistible." They appealed to the masses, encouraging them to be "arbitrary as well as insubordinate" and to disregard tradition, prescription, and established rights. Nationalism, with Mazzini as its representative figure, was the most formidable of these new mass movements; for it subverted the existing order by attacking the territorial arrangement and sovereignty of legitimate states, and it provided the impetus behind contemporary revolution. 4 In contrast to later generations, who may have regarded nation-states as likely to promote peace and prosperity, Acton believed that to create these states by imposing the principle of nationality would damage irreparably the Europe he knew and valued.Acton clearly appreciated the power of nationalism—and for this he deserves credit—but too much can, and has, been made of his foresight. He was disturbed by nationalism not because he was clairvoyant, not because he saw in nationalism the twentieth century's totalitarian nightmare, but rather because he was preoccupied with very nineteenth-century concerns. 5 When in 1859 Cavour and Louis Napoleon contrived their war against Austria, hoping to drive the [End Page 130] Habsburgs from Italy and capture the provinces of Lombardy and Venetia for Piedmont, they exacerbated the nationalities problem in much of southern Europe. In Italy this problem was solved by the creation of a unified nation-state. The war against Austria was the first step in the process of unification that culminated in 1861 with the founding of the Kingdom of Italy. As a result of unification, inhabitants of the Italian peninsula found themselves subject to a highly centralized state, with an administration modeled on the Napoleonic system that was adverse to regional autonomy. The Italian war also aggravated the nationalities problem in... (shrink)