Induction and abduction are well known non-deductive inferences. We shall propose a view that design is also another form of non-deductive inference, and give a mathematical model of deductive and non-deductive inferences based on Barwise and Seligman's mathematical theory of information flow. In our model, inferences are classified into three categories, and we can show that deduction and abduction are in the same category, although induction is different. Furthermore, we shall show also that non-deductive inferences are interpretable mutually, and investigate also mathematical properties of the model. In particular, we shall prove a generalized version of the Abstract Completeness Theorem by Barwise and Seligman.
We can find in several places an assertion that Gödel's second incompleteness theorem defeated Hilbert's program. But, (as M. Detlefsen argued in his book) in order to establish this assertion, we need to address additional issues. First we formulate Hilbert's program. Second we reconstruct a standard argument for the claim that Gödel's second incompleteness theorem defeated Hilbert's program. In doing so, we formulate a critical, and problematic assumption which we call "DCT" (Derivability Conditions Thesis). Finally we examine three arguments whose aims are to justify DCT. We show that the first and the second argument are not valid, and discuss the third argument, which is based on Kreisel's idea. We identify a difficulty in this argument as well. After examining the difficulty, we conclude that we cannot claim that Gödel's second incompleteness theorem defeats Hilbert's program. Moreover we clarify what is essentially needed for such an argument to succeed.
Based on classical mechanical picture of physical objects, quantum-mechanical measurements are formulated and analyzed in an abstract way. We interpret a dynamical flow generated by a Hamiltonian vector field as a description of the effect on the objects caused by a device or an environment. By expressing contexts of measurement explicitly, a matrix representation of observables on a finite dimensional Hilbert space is constructed. A quantum-mechanical state vector is introduced as something on which a dynamical flow generated by a Hamiltonian vector field acts. We find conditions that enable us to obtain the Dirac's quantization rule for the mean values. At the same time, several characteristics (i.e., rectifiability, a generator of symmetry, an invariant mean) of quantum-mechanical measurements are formulated in our formalism.
Semirealism divides the properties of unobservables into two types: detection properties and auxiliary properties. Detection properties represent the causal properties of an unobservable entities by which its existence can be detected. On the other hand, auxiliary properties are not involved in detection. Semirealism holds that science gives us knowledge of detection properties but not auxiliary properties. In the present paper, I interpret algebraic quantum field theory from a semirealistic point of view. Especially I investigate localized states, which was introduced by Knight (1961) and Licht (1963). Roughly speaking, a localized state represents a localized excitation of the vacuum. I formulate a detection of a localized excitation, and show that this detection does not exist in algebraic quantum field theory.