Quantum computer is considered as a generalization of Turing machine. The bits are substituted by qubits. In turn, a "qubit" is the generalization of "bit" referring to infinite sets or series. It extends the consept of calculation from finite processes and algorithms to infinite ones, impossible as to any Turing machines (such as our computers). However, the concept of quantum computer mets all paradoxes of infinity such as Gödel's incompletness theorems (1931), etc. A philosophical reflection on how quantum computer might (...) implement the idea of "infinite calculation" is the main subject. (shrink)

The book is devoted to the contemporary stage of quantum mechanics – quantum information, and especially to its philosophical interpretation and comprehension: the first one of a series monographs about the philosophy of quantum information. The second will consider Be l l ’ s inequalities, their modified variants and similar to them relations. The beginning of quantum information was in the thirties of the last century. Its speed development has started over the last two decades. The main phenomenon is entanglement. (...) The subareas are quantum computer, quantum communication (and teleportation), and quantum cryptography. The book offers the following main conceptions, theses and hypotheses: – dualistic Phythagoreanism as a new kind among the interpretations of quantum mechanics and information: arithmetical, logical, and metamathematical one; – Gödel ’ s first incompleteness theorem is an undecidable proposition, and consequently the second one,too. – a partial rehabilitation of Hilbert ’ s program for the self-foundation of mathematics; – the dual-foundation of mathematics; – Skolemian relativity between: Cantor ’s kinds of infinity, finiteness and infinity, discreteness and continuity, completeness and incompleteness, etc.; – information is a physical quantity representing the non-reducibility of a system to its parts, particularly nonaddtivity; – there exist pure relations «by itself», which cannot be reduced to predications; – energy conservation can and should be generalized; – Einstein’ s «general covariance» or «principle of relativity» can and should be generalized to cover discrete morphisms where the notion of velocity does not make sense. (shrink)

The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications.

This paper discusses the principle of recombination for possible worlds. It argues that arguments against unrestricted recombination offered by Forrest and Armstrong and by David Lewis fail, but a related argument is a challenge, and recommends that we accept an unrestricted principle of recombination and the conclusion that possible worlds form a proper class.

This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order-Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of (...) modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplares (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which Firs-Order-Logic matured will be seen to have played a primary role in its shaping. (shrink)

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...) set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functionsin theory. However, the approach to partial functions traditionally employed by mathematicians is quite adequatein practice. This paper shows how the traditional approach to partial functions (...) can be formalized in a range of formalisms that includes first-order logic, simple type theory, and Von-Neumann—Bernays—Gödel set theory. It argues that these new formalisms allow one to directly reason about partial functions; are based on natural, well-understood, familiar principles; and can be effectively implemented in mechanized mathematics systems. (shrink)

This paper builds on the system of David Lewis’s “Parts of Classes” to provide a foundation for mathematics that arguably requires not only no distinctively mathematical ideological commitments (in the sense of Quine), but also no distinctively mathematical ontological commitments. Provided only that there are enough individual atoms, the devices of plural quantification and mereology can be employed to simulate quantification over classes, while at the same time allowing all of the atoms (and most of their fusions with which we (...) are concerned) to be individuals (that is, urelements of classes). The final section of the paper canvasses some reasons to be committed to the required ontology for other than mathematical reasons. (shrink)

Languages involving modalities and languages involving vagueness have each been thoroughly studied. On the other hand, virtually nothing has been said about the interaction of modality and vagueness. This paper aims to start filling that gap. Section 1 is a discussion of various possible sources of vague modality. Section 2 puts forward a model theory for a quantified language with operators for modality and vagueness. The model theory is followed by a discussion of the resulting logic. In Section 3, the (...) framework will permit us to address a puzzle raised by Elizabeth Barnes and Robert Williams. (shrink)

The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following Williamson and Russell, we distinguish between logical and scientific strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientific strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories.

Many contributors of the debate about knowledge by testimony concentrate on the problem of justification. In my paper I will stress a different point – the concept of testimony itself. As a starting point I will use the definitional proposal of Jennifer Lackey. She holds that the concept of testimony should be regarded as entailing two aspects – one corresponding to the speaker, the other one to the hearer. I will adopt the assumption that we need to deal with both (...) aspects. Nevertheless, I will show that her concept – which suggests regarding testimony as an act of communication conveying information – is too broad and, therefore, I will end up with a different twofold definition. (shrink)