38,868 research outputs found
Frequentist statistics as a theory of inductive inference
After some general remarks about the interrelation between philosophical and
statistical thinking, the discussion centres largely on significance tests.
These are defined as the calculation of -values rather than as formal
procedures for ``acceptance'' and ``rejection.'' A number of types of null
hypothesis are described and a principle for evidential interpretation set out
governing the implications of -values in the specific circumstances of each
application, as contrasted with a long-run interpretation. A variety of more
complicated situations are discussed in which modification of the simple
-value may be essential.Comment: Published at http://dx.doi.org/10.1214/074921706000000400 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Literal Perceptual Inference
In this paper, I argue that theories of perception that appeal to Helmholtz’s idea of unconscious inference (“Helmholtzian” theories) should be taken literally, i.e. that the inferences appealed to in such theories are inferences in the full sense of the term, as employed elsewhere in philosophy and in ordinary discourse.
In the course of the argument, I consider constraints on inference based on the idea that inference is a deliberate acton, and on the idea that inferences depend on the syntactic structure of representations. I argue that inference is a personal-level but sometimes unconscious process that cannot in general be distinguished from association on the basis of the structures of the representations over which it’s defined. I also critique arguments against representationalist interpretations of Helmholtzian theories, and argue against the view that perceptual inference is encapsulated in a module
Cut Elimination for a Logic with Induction and Co-induction
Proof search has been used to specify a wide range of computation systems. In
order to build a framework for reasoning about such specifications, we make use
of a sequent calculus involving induction and co-induction. These proof
principles are based on a proof theoretic (rather than set-theoretic) notion of
definition. Definitions are akin to logic programs, where the left and right
rules for defined atoms allow one to view theories as "closed" or defining
fixed points. The use of definitions and free equality makes it possible to
reason intentionally about syntax. We add in a consistent way rules for pre and
post fixed points, thus allowing the user to reason inductively and
co-inductively about properties of computational system making full use of
higher-order abstract syntax. Consistency is guaranteed via cut-elimination,
where we give the first, to our knowledge, cut-elimination procedure in the
presence of general inductive and co-inductive definitions.Comment: 42 pages, submitted to the Journal of Applied Logi
Homotopy Type Theory in Lean
We discuss the homotopy type theory library in the Lean proof assistant. The
library is especially geared toward synthetic homotopy theory. Of particular
interest is the use of just a few primitive notions of higher inductive types,
namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201
Rules and derivations in an elementary logic course
When teaching an elementary logic course to students who have a general
scientific background but have never been exposed to logic, we have to face the
problem that the notions of deduction rule and of derivation are completely new
to them, and are related to nothing they already know, unlike, for instance,
the notion of model, that can be seen as a generalization of the notion of
algebraic structure. In this note, we defend the idea that one strategy to
introduce these notions is to start with the notion of inductive definition
[1]. Then, the notion of derivation comes naturally. We also defend the idea
that derivations are pervasive in logic and that defining precisely this notion
at an early stage is a good investment to later define other notions in proof
theory, computability theory, automata theory, ... Finally, we defend the idea
that to define the notion of derivation precisely, we need to distinguish two
notions of derivation: labeled with elements and labeled with rule names. This
approach has been taken in [2]
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