According to the analysis of concessive conditionals suggested by Crupi and Iacona, a concessive conditional \(p{{\,\mathrm{\hookrightarrow }\,}}q\) is adequately formalized as a conjunction of conditionals. This paper presents a sound and complete axiomatic system for concessive conditionals so understood. The soundness and completeness proofs that will be provided rely on a method that has been employed by Raidl, Iacona, and Crupi to prove the soundness and completeness of an analogous system for evidential conditionals.

We formulate a Hilbert-style axiomatic system and a tableau calculus for the STIT-based logic of imagination recently proposed in Wansing. Completeness of the axiom system is shown by the method of canonical models; completeness of the tableau system is also shown by using standard methods.

According to the analysis of concessive conditionals suggested by Crupi and Iacona, a concessive conditional $$p{{\,\mathrm{\hookrightarrow }\,}}q$$ p ↪ q is adequately formalized as a conjunction of conditionals. This paper presents a sound and complete axiomatic system for concessive conditionals so understood. The soundness and completeness proofs that will be provided rely on a method that has been employed by Raidl, Iacona, and Crupi to prove the soundness and completeness of an analogous system for evidential conditionals.

This paper sketches the antilogism of Christine Ladd-Franklin and historical advancement about antilogism, mainly constructs an axiomatic system Atl based on first-order logic with equality and the wholly-exclusion and not-wholly-exclusion relations abstracted from the algebra of Ladd-Franklin, with soundness and completeness of Atl proved, providing a simple and convenient tool on syllogistic reasoning. Atl depicts the empty class and the whole class differently from normal set theories, e.g. ZFC, revealing another perspective on sets and set theories. Two series (...) of Dotterer and the theorem proved by Russinoff are re-characterized in antilogistic formulas corresponding to the square of opposition and twenty-four moods of syllogism. Then the restricted equivalence between two forms of antilogism and minimal inconsistency is built up, implying the internal connection between antilogisms and minimal inconsistency and leaving two related conjectures not proved. Besides, this paper provides a new explanation for the reason why contrariety, subcontrariety, and subalternation and the nine special moods require the non-empty assumption. (shrink)

In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to (...) Bourbaki’s system the axiom of universes for the purpose of considering the theory of categories. In this regard, we make some historical and epistemological remarks that could explain the conservative attitude of the Group. (shrink)

Lately, an increased interest in formal devices has led to an attempt on the part of some mathematicians to do without those aspects of mathematics which require intuition. One consequence of this movement has been a new conception of pure mathematics as a science of axiomatic systems. According to this conception, there is no reality beyond an axiomatic system which the statements of mathematics are about; the fact that a statement is a theorem in the system (...) is all that is of interest. This is a sort of nominalist position, since the contrary position seems to be committed to a belief in the existence of a domain of mathematical entities, which are just as suspect as the universale commonly discussed in metaphysical treatises. More often it is called formalism. (shrink)

The paper contains an overview of the most important results presented in the monograph of the author "Teorie Językow Syntaktycznie-Kategorialnych" ("Theories of Syntactically-Categorial Languages" (in Polish), PWN, Warszawa-Wrocław 1985. In the monograph four axiomatic systems of syntactically-categorial languages are presented. The first two refer to languages of expression-tokens. The others also takes into consideration languages of expression-types. Generally, syntactically-categorial languages are languages built in accordance with principles of the theory of syntactic categories introduced by S. Leśniewski [1929,1930]; they are (...) connected with- the Ajdukiewicz’s work [1935] which was a continuation of Leśniewski’s idea and further developed and popularized in the research on categorial grammars, by Y. Bar-Hillel [1950,1953,1964]. To assign a suitable syntactic category to each word of the vocabulary is the main idea of syntactically-categorial approach to language. Compound expressions are built from the words of the vocabulary and then a suitable syntactic-category is assigned to each of them. A language built in this way should be decidable, which means that there should exist an algorithm for deciding about each expression of it, whether it is well-formed or is syntactically connected . The traditional, originating from Husserl, understanding of the syntactic category confronts some difficulties. This notion is defined by abstraction using the concept of affiliation of two expressions to the same syntactic category. (shrink)

The Constitution is considered as an informal axiomatic system. The strategy proposed by the authors rests on the following propositions: axioms are considered as contextual definitions of those concepts by means of which they are formulated; and the main requirement for this type of system is internal consistency. The first proposition is necessary for considering the Constitution as an informal axiomatic system, while the second is sufficient, because the approach proposed, apart from consistency, must certainly (...) consider the requirements for formal axiomatic systems, such as independence and completeness. The authors argue that the Constitution can be compared to axiomatic constructions in modern science in the sense that is given in the research on the logic and methodology of deductive sciences. This analogy is appropriate to the extent to which constitutional provisions are interpreted as the basic elements of the legal system, just as in the formal sciences axioms are regarded as basic principles that define the main features of the formal system. This means that the Constitution itself is seen as coherent, consistent discourse that contextually defines the meaning of the basic terms of the legal system. (shrink)

The logic FBCI given by linearly ordered BCI-matrices is known not to be an axiomatic extension of the well-known BCI logic. In this paper we axiomatize FBCI by adding a recursively enumerable set of schemes of inference rules to BCI and show that there is no finite axiomatization for FBCI.

In a language for arithmetic with a predicate T, intended to mean “ x is the Gödel number of a true sentence”, a set S of axioms and rules of inference has the truth disjunction property if whenever S ⊢ T ∨ T, either S ⊢ T or S ⊢ T. Similarly, S has the truth existence property if whenever S ⊢ ∃χ T ), there is some n such that S ⊢ T ). Continuing previous work, we establish whether (...) these properties hold or fail for a large collection of possible axiomatic systems. (shrink)

Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable (...) in the logical calculus of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.The theory is not set up as a pure formalism, but rather in the usual manner of elementary axiom theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell). (shrink)

We present an axiomatization of non-relativistic Quantum Mechanics for a system with an arbitrary number of components. The interpretation of our system of axioms is realistic and objective. The EPR paradox and its relation with realism is discussed in this framework. It is shown that there is no contradiction between realism and recent experimental results.

Classic of Changes is a Chinese cultural classic born more than 3000 years ago. Its profound philosophical thoughts and the use of divination have brought Classic of Changes to a strong oriental mysticism. The view of the heaven and man of yin and yang and the five elements states of Classic of Changes are completely different from the Western elemental theory of ancient Greece. The latter gave birth to classical and modern scientific theories, and the yin and yang and the (...) eight trigrams symbol has become synonymous with oriental mysticism. In fact, the cosmology of the Holism of Classic of Changes is a precious scientific heritage of mankind. The axiomatization of the symbolic formal system of Classic of Changes aims to unveil the veil of oriental mysticism and provide oriental wisdom for the development of modern science. Transforming the symbolic system of Classic of Changes into a formal axiom system is the crystallization of the fusion of wisdom between the East and the West. The axiomatization of the symbolic formal system of Classic of Changes shows that the oriental and the western scientific tradition harmony but not sameness and there is no conflict. Classic of Changes can also be interpreted by the axiomatic system like Euclid’s Elements. The main contribution of this paper is that the author skillfully uses mathematical language to formulate the system of Classic of Changes, reconstructs the ideological system of Classic of Changes with the axiomatic method and realizes the scientificalization of Chinese classical natural philosophy. The formal axiom system of Classic of Changes may give us such a revelation: the gap between the oriental and western scientific traditions is mainly in the axiomatization, but the lack of oriental scientific tradition may be the bottleneck of the development of modern science. (shrink)

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of (...) infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined. (shrink)