Leibniz's complete propositional logic

Conclusion I have shown (to my satisfaction) that Leibniz's final attempt at a generalized syllogistico-propositional calculus in the Generales Inquisitiones was pretty successful. The calculus includes the truth-table semantics for the propositional calculus. It contains an unorthodox view of conjunction. It offers a plethora of very important logical principles. These deserve to be called a set of fundamentals of logical form. Aside from some imprecisions and redundancies the system is a good systematization of propositional logic, its semantics, and a correct account of general syllogistics. For 1686 it was quite an accomplishment. It is a pity that Leibniz himself did not fully appreciate what he had achieved. It does seem to me that this was due in part, as the Kneales urge (Note 4), to his having kept the focus of his attention on traditional syllogistics. It is a great pity that he did not polish GI 195–200 for publication. The publication of GI 195, 198, and 200 would have most likely promoted further research.This paper was conceived in a Seminar on the Generales Inquisitiones offered by Professor Klaus Jacobi at the University of Freiburg during the 1987 winter semester. I am grateful to him for having allowed me to participate in that exciting seminar. I am grateful to all the seminar participants, especially to Professor Jacobi, Professor Klaus Erich Kaehler, Doctor Helmut Pape, and Herr Hans-Peter Engelhart for sustained and illuminating discussions of some passages of the GI. Jacobi was extremely kind in reading the second version of this paper with a highly refined comb. I am most grateful to him for having pointed out typos, stylistic infelicities, and conceptual obscurities. He also provided advice on the translation, and, most generously and cooperatively, offered suggestions for improving the exposition and the arguments.     

A pragmatic interpretation of intuitionistic propositional logic

We construct an extension ^P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of ^P;rfs preceded by theassertion sign constituteelementary assertive formulas of ^P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of ^P is endowed with atruth value defined classically, and everyaf is endowed with ajustification value, defined in terms of the intuitive notion of proof and depending on the truth values of its radical subformulas. In this framework, we define the notion ofpragmatic validity in ^P and yield a list of criteria of pragmatic validity which hold under the assumption that only classical metalinguistic procedures of proof be accepted. We translate the classical propositional calculus (CPC) and the intuitionistic propositional calculus (IPC) into the assertive part of ^P and show that this translation allows us to interpret Intuitionistic Logic as an axiomatic theory of the constructive proof concept rather than an alternative to Classical Logic. Finally, we show that our framework provides a suitable background for discussing classical problems in the philosophy of logic.This paper is an enlarged and entirely revised version of the paper by Dalla Pozza (1991) worked out in the framework of C.N.R. project n. 89.02281.08, and published in Italian. The basic ideas in it have been propounded since 1986 by Dalla Pozza in a series of seminars given at the University of Lecce and in other Italian Universities. C. Garola collected the scattered parts of the work, helped in solving some conceptual difficulties and refining the formalism, yielded the proofs of some propositions (in particular, in Section 3) and provided physical examples (see in particular Remark 2.3.1).     

A propositional logic with explicit fixed points

This paper studies a propositional logic which is obtained by interpreting implication as formal provability. It is also the logic of finite irreflexive Kripke Models.A Kripke Model completeness theorem is given and several completeness theorems for interpretations into Provability Logic and Peano Arithmetic.     

Infinitary propositional normal modal logic

A logic with normal modal operators and countable infinite conjunctions and disjunctions is introduced. A Hilbert's style axiomatization is proved complete for this logic, as well as for countable sublogics and subtheories. It is also shown that the logic has the interpolation property.     

A weak intuitionistic propositional logic with purely constructive implication

We introduce subsystems WLJ and SI of the intuitionistic propositional logic LJ, by weakening the intuitionistic implication. These systems are justifiable by purely constructive semantics. Then the intuitionistic implication with full strength is definable in the second order versions of these systems. We give a relationship between SI and a weak modal system WM. In Appendix the Kripke-type model theory for WM is given.This work was partially supported by NSF Grant DCR85-13417     

The method of axiomatic rejection for the intuitionistic propositional logic

We prove that the intuitionistic sentential calculus is -decidable (decidable in the sense of ukasiewicz), i.e. the sets of theses of Int and of rejected formulas are disjoint and their union is equal to all formulas. A formula is rejected iff it is a sentential variable or is obtained from other formulas by means of three rejection rules. One of the rules is original, the remaining two are ukasiewicz's rejection rules: by detachement and by substitution. We extensively use the method of Beth's semantic tableaux.To the memory of Jerzy SupeckiTranslated from the Polish by Jan Zygmunt. Preparation of this paper was supported in part by C.P.B.P. 08-15.     

Model checking propositional dynamic logic with all extras

This paper presents a model checking algorithm for Propositional Dynamic Logic (PDL) with looping, repeat, test, intersection, converse, program complementation as well as context-free programs. The algorithm shows that the model checking problem for PDL remains PTIME-complete in the presence of all these operators, in contrast to the high increase in complexity that they cause for the satisfiability problem.     

Matrix representations for structural strengthenings of a propositional logic

The aim of this paper is to show that the operations of forming direct products and submatrices suffice to construct exhaustive semantics for all structural strengthenings of the consequence determined by a given class of logical matrices.     

The logic of thought experiments

In this paper I argue that (at least many) philosophical thought experiments are unreliable. But I argue that this notion of unreliability has to be understood relative to the goal of thought experiments as knowledge producing. And relative to that goal many thought experiments in science are just as unreliable. But in fact thought experiments in science play a varied role and I will suggest that knowledge production is a goal only under quite limited circumstances. I defend the view that these circumstances can (sometimes) arise in philosophy as well.