307 research outputs found
FraenkelâCarnap Questions for Equivalence Relations
An equivalence is a binary relational system A = (A,ϱA) where ϱA is an equivalence relation on A. A simple expansion of an equivalence is a system of the form (Aa1âŠan) were A is an equivalence and a1,âŠ,an are members of A. It is shown that the Fraenkel-Carnap question when restricted to the class of equivalences or to the class of simple expansions of equivalences has a positive answer: that the complete second-order theory of such a system is categorical, if it is finitely axiomatizable
On the Axiom of Canonicity
The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox (without reference to the L\"{o}wenheim-Skolem theorem) 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
Natorpâs Neo-Kantian Mathematical Philosophy of Science
This paper deals with Natorpâs version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorpâs mathematical philosophy of science is contained in his âknowledge equationâ that may be considered as a mathematical model of the âtranscendental methodâ conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the Aufbau, contended that this endeavor can be divided into two distinct parts, namely, a finite âconstitutionâ of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this âAufgabeâ as an infinite dialectical process, in which relative âdataâ and âconceptual constructsâ determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorpâs defense of Cohenâs âinfinitesimalâ metaphysics
FraenkelâCarnap Questions for Equivalence Relations
An equivalence is a binary relational system A = (A,ϱA) where ϱA is an equivalence relation on A. A simple expansion of an equivalence is a system of the form (Aa1âŠan) were A is an equivalence and a1,âŠ,an are members of A. It is shown that the Fraenkel-Carnap question when restricted to the class of equivalences or to the class of simple expansions of equivalences has a positive answer: that the complete second-order theory of such a system is categorical, if it is finitely axiomatizable
Interactive and common knowledge in the state-space model
This paper deals with the prevailing formal model for knowledge in contemporary economics, namely the state-space model introduced by Robert Aumann in 1976. In particular, the paper addresses the following question arising in this formalism: in order to state that an event is interactively or commonly known among a group of agents, do we need to assume that each of them knows how the information is imparted to the others? Aumann answered in the negative, but his arguments apply only to canonical, i.e., completely specified state spaces, while in most applications the state space is not canonical. This paper addresses the same question along original lines, demonstrating that the answer is negative for both canonical and not-canonical state spaces. Further, it shows that this result ensues from two counterintuitive properties held by knowledge in the state-space model, namely Substitutivity and Monotonicity.
Mathematics and language
This essay considers the special character of mathematical reasoning, and
draws on observations from interactive theorem proving and the history of
mathematics to clarify the nature of formal and informal mathematical language.
It proposes that we view mathematics as a system of conventions and norms that
is designed to help us make sense of the world and reason efficiently. Like any
designed system, it can perform well or poorly, and the philosophy of
mathematics has a role to play in helping us understand the general principles
by which it serves its purposes well
To be a realist about quantum theory
I look at the distinction between between realist and antirealist views of the quantum state. I argue that this binary classification should be reconceived as a continuum of different views about which properties of the quantum state are representationally significant. What's more, the extreme cases -- all or none --- are simply absurd, and should be rejected by all parties. In other words, no sane person should advocate extreme realism or antirealism about the quantum state. And if we focus on the reasonable views, it's no longer clear who counts as a realist, and who counts as an antirealist. Among those taking a more reasonable intermediate view, we find figures such as Bohr and Carnap -- in stark opposition to the stories we've been told
Roman Suszko's works in logic at the PoznaĆ University (1946 - 1953)
We discuss the scientific achievements of one of the most prominent Polish logicians of the 20th century - ROMAN SUSZKO in the period when he was active at the University in PoznaĆ (1946 - 1953), i.e. at the very beginning of his academic career. We discuss the scientific achievements of one of the most prominent Polish logicians of the 20th century - ROMAN SUSZKO in the period when he was active at the University in PoznaĆ (1946 - 1953), i.e. at the very beginning of his academic career
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