This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

Die Grundlagen gehören zu den klassischen Texten der Sprachphilosophie, Logik und Mathematik. Frege stützt sein Programm einer Begründung von Arithmetik und Analysis auf reine Logik, indem er die natürlichen Zahlen als bestimmte Begriffsumfänge definiert. Die philosophische Fundierung des Fregeschen Ansatzes bilden erkenntnistheoretische und sprachphilosophische Analysen und Begriffserklärungen. Studienausgabe aufgrund der textkritisch herausgegebenen Jubiläumsausgabe (Centenarausgabe). Mit Einleitung, Anmerkungen, Literaturverzeichnis und Namenregister.

Dieser Band enthält die vier Arbeiten Freges: Begriffsschrift, eine der arithmetischen nachgebildeten Formelsprache, 1879; Anwendungen der Begriffsschrift, 1879; Über den Briefwechsel Leibnizens und Huggens mit Papin, 1881; Über den Zweck der Begriffsschrift, 1883; Über die wissenschaftliche Berechtigung einer Begriffsschrift, 1882. Frege's research work in the field of mathematical logic is of great importance for the present-day analytic philosophy. We actually owe to Frege a great amount of basical insight and exemplary research, which set up a new standard also in other (...) fields of knowledge. As the founder of mathematical logic he severely examindes the syllogisms on which arithmetic is built up. In doing so, Frege recognized that our colloquial language is inadequate to define logic structures. His notional language corresponded to the artaivicial logical language demandes by Leibniz. Frege's achievement in the field of logic were so important, that they radiated into the domain of philosophy and influenced the development of mathematical logic decisively. (shrink)

Modern modal logic originated as a branch of philosophical logic in which the concepts of necessity and possibility were investigated by means of a pair of dual operators that are added to a propositional or first-order language. The field owes much of its flavor and success to the introduction in the 1950s of the “possible-worlds” semantics in which the modal operators are interpreted via some “accessibility relation” connecting possible worlds. In subsequent years, modal logic has received attention as an attractive (...) approach towards formalizing such diverse notions as time, knowledge, or action. Nowadays, modal logics are applied in various disciplines, ranging from economics to linguistics and computer science. Consequently, there is by now a large variety of modal languages, with an even greater wealth of interpretations. For instance, many applications require a poly-modal framework consisting of a language with a family of modal operators and a semantics in which the corresponding accessibility relations are connected somehow. (shrink)

The paper studies the relation between systems of modal logic and systems of consequential implication, a non-material form of implication satisfying "Aristotle's Thesis" (p does not imply not p) and "Weak Boethius' Thesis" (if p implies q, then p does not imply not q). Definitions are given of consequential implication in terms of modal operators and of modal operators in terms of consequential implication. The modal equivalent of "Strong Boethius' Thesis" (that p implies q implies that p does not imply (...) not q) is identified. (shrink)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These (...) practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

The title reflects my conviction that, viewed semantically,modal logic is fundamentally dialogical; this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior (notably the use of nominals, special proposition symbols which ‘name’ worlds) I will show how to lift the dialogical (...) conception to modal proof theory. I argue that this shift to hybrid logic has consequences for both modal and dialogical logic, and I discuss these in detail. (shrink)

Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this (...) context was called. This theory gave a justification for the validity of what we todayexpress through the formulae ¬ and ¬. Hugh MacColl and, more recently, Storrs McCall searched for a formal system in which the validity ofthese formulae could be expressed. Unfortunately neither of the resulting systems is very satisfactory. In this paper we introduce dialogical games with the help of a new connexive If-Then (), the structural rules of which allow the Proponent to develop winning strategies not only for the above-mentioned connexive theses but also for ¬ and ¬. Further on, we developthe corresponding tableau systems and conclude with some remarks on possibleperspectives and consequences of the dialogical approach to connexivity including the loss of uniform substitution leading to a new concept of logical form. (shrink)

This collection is the first anthology of works on Boole.

George Boole (1815-1864) is well known to mathematicians for his research and textbooks on the calculus, but his name has spread world-wide for his innovations in symbolic logic and the development and applications made since his day. The utility of "Boolean algebra" in computing has greatly increased curiosity in the nature and extent of his achievements. His work is most accessible in his two books on logic, "A mathematical analysis of logic" (1947) and "An investigation of the laws of thought" (...) (1954). But at various times he wrote manuscript essays, especially after the publication of the second book; several were intended for a non-technical work, "The Philosophy of logic", which he was not able to complete. This volume contains an edited selection which not only relates them to Boole's publications and the historical context of his time, but also describes their strange history of family, followers and scholars have treid to confect an edition. The book will appeal to logicians, mathematicians and philosophers, and those interested in the histories of the corresponding subjects; and also students of the early Victorian Britain in which they were written. (shrink)

In this paper I consider three mathematicians who allowed some role for menial processes in the foundations of their logical or mathematical theories. Boole regarded his Boolean algebra as a theory of mental acts; Cantor permitted processes of abstraction to play a role in his set theory; Brouwer took perception in time as a cornerstone of his intuitionist mathematics. Three appendices consider related topics.

The core ideas of the dialogicalapproach to modal propositional logic are explainedby means of an elementary example. Subsequently,ways of extending this approach to the system G ofso-called provability logic are checked, therebyraising the question whether the dialogician is inneed of shaping his Nichtverzögerungsregel(non-delay-rule), in order to get it sufficiently precise,in different ways for different modal systems.

Kapitel XV von Über Annahmen (1. Aufl.) beginnt mit einigen bitteren Bemerkungen über den Mangel an Einsicht unter Logikern und Philosophen über das Wesen logischen Schließens. Meinong beklagt, daß man sich mit der Klassifizierung von Schlußformen aufgehalten habe, ohne den Versuch einer Analyse des Folgerungsbegriffs zu unternehmen. Es überrascht wenig, daß Meinong auch bei der Analyse des Folgerungsbegriffs der Annahme eine Schlüsselrolle zuweist. Seine Diskussion führt ihn dabei zu einer UntersuChung des Verhältnisses von Schlüssen aus Urteilen, hypothetischen Urteilen und „Annahmeschlüssen", (...) d.h. Argumenten, deren Prämissen Annahmen sind.Meinongs These, daß es so etwas wie Annahmeschlüsse gibt, erscheint selbstverständlich genug. Die Einführung von Annahmen als Prämissen ist heutzutage nicht nur in Kalkülen natürlichen Schließens eine Standardprozedur, die keiner Rechtfertigung bedarf. Umso mehr muß es daher überraschen, daß Russell in seiner ansonsten enthusiastischen Rezension in Mind (\904) vorgibt, für Schlüsse aus Annahmen nichts übrig zu haben: „I cannot understand how the notion of assumptive inference... can be maintained." Daß es hier in der Tat nicht um einen bloßen Streit um Worte oder ein Mißverständnis geht, wird spätestens dadurch offenkundig, daß auch Frege in diesem Punkt mit Russell vollkommen übereinstimmt. Wie sich jedoch herausstellt, ist der eigentliche Streitpunkt das korrekte Verständnis hypothetischer Urteile: Nach Meinong sind hypothetische Urteile keine Urteile, sondern verkappte Annahmeschlüsse, während Russell (und Frege) genau umgekehrt Annahmeschlüsse als verkappte hypothetische Urteile auffassen. (shrink)

Kapitel XV von Über Annahmen (1. Aufl.) beginnt mit einigen bitteren Bemerkungen über den Mangel an Einsicht unter Logikern und Philosophen über das Wesen logischen Schließens. Meinong beklagt, daß man sich mit der Klassifizierung von Schlußformen aufgehalten habe, ohne den Versuch einer Analyse des Folgerungsbegriffs zu unternehmen. Es überrascht wenig, daß Meinong auch bei der Analyse des Folgerungsbegriffs der Annahme eine Schlüsselrolle zuweist. Seine Diskussion führt ihn dabei zu einer UntersuChung des Verhältnisses von Schlüssen aus Urteilen, hypothetischen Urteilen und „Annahmeschlüssen", (...) d.h. Argumenten, deren Prämissen Annahmen sind.Meinongs These, daß es so etwas wie Annahmeschlüsse gibt, erscheint selbstverständlich genug. Die Einführung von Annahmen als Prämissen ist heutzutage nicht nur in Kalkülen natürlichen Schließens eine Standardprozedur, die keiner Rechtfertigung bedarf. Umso mehr muß es daher überraschen, daß Russell in seiner ansonsten enthusiastischen Rezension in Mind (\904) vorgibt, für Schlüsse aus Annahmen nichts übrig zu haben: „I cannot understand how the notion of assumptive inference... can be maintained." Daß es hier in der Tat nicht um einen bloßen Streit um Worte oder ein Mißverständnis geht, wird spätestens dadurch offenkundig, daß auch Frege in diesem Punkt mit Russell vollkommen übereinstimmt. Wie sich jedoch herausstellt, ist der eigentliche Streitpunkt das korrekte Verständnis hypothetischer Urteile: Nach Meinong sind hypothetische Urteile keine Urteile, sondern verkappte Annahmeschlüsse, während Russell (und Frege) genau umgekehrt Annahmeschlüsse als verkappte hypothetische Urteile auffassen. (shrink)