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We introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model-theoretic constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this structure is that its elements can be seen both as truth values and as (generalized) realizers, thus blurring the frontier between proofs and types. We show that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Relating this construction with the corresponding constructions in forcing and realizability, we conclude that the class of implicative triposes encompasses all forcing triposes (both intuitionistic and classical), all classical realizability triposes (in the sense of Krivine), and all intuitionistic realizability triposes built from partial combinatory algebras.
We propose the new concept of Krivine ordered combinatory algebra (
) as foundation for the categorical study of Krivine's classical realizability, as initiated by Streicher (2013).
We show that
's are equivalent to Streicher's abstract Krivine structures for the purpose of modeling higher-order logic, in the precise sense that they give rise to the same class of triposes. The difference between the two representations is that the elements of a
play both the role of truth values and realizers, whereas truth values are sets of realizers in
To conclude, we give a direct presentation of the realizability interpretation of a higher order language in a
, which showcases the dual role that is played by the elements of the
This paper deals with the specification problem in classical realizability (such as introduced by Krivine (2009 Panoramas et synthéses27)), which is to characterize the universal realizers of a given formula by their computational behaviour. After recalling the framework of classical realizability, we present the problem in the general case and illustrate it with some examples. In the rest of the paper, we focus on Peirce's law, and present two game-theoretic characterizations of its universal realizers. First, we consider the particular case where the language of realizers contains no extra instruction such as ‘quote’ (Krivine 2003 Theoretical Computer Science308 259–276). We present a first game
0 and show that the universal realizers of Peirce's law can be characterized as the uniform winning strategies for
0, using the technique of interaction constants. Then we show that in the presence of extra instructions such as ‘quote’, winning strategies for the game
0 are still adequate but no more complete. For that, we exhibit an example of a wild realizer of Peirce's law, that introduces a purely game-theoretic form of backtrack that is not captured by
0. We finally propose a more sophisticated game
1, and show that winning strategies for the game
1 are both adequate and complete in the general case, without any further assumption about the instruction set used by the language of classical realizers.
Building upon Deci's and Ryan (1985) Self-determination theory as well as the sportive behavioral correlates of the model of Commitment (Scanlan et al., 1976), this study tries to establish the relationship between motivation and commitment in youth sport. For this purpose 454 young competitive soccer players answered the Sport Motivation Scale (SMS) and the Sport Commitment Questionnaire (SCQ) during the regular season.
The SMS measures the three dimensions of the Motivational continuum (the Amotivation, the Extrinsic Motivation and the Intrinsic Motivation). The SCQ measures the Sportive Commitment and its composing factors such as the Enjoyment, the Alternatives to the sport, and the Social Pressure. Our findings provided a clear pattern of the influence of motivation in sport enjoyment and commitment, outlining the positive contribution of intrinsic and extrinsic motivation to enjoyment and commitment. Amotivation, contributes positively to alternatives to sport and negatively to enjoyment and commitment. It should be noted that extrinsic motivation has a higher contribution to enjoyment whereas intrinsic motivation has a higher contribution to commitment.
We present an extension of the λ(η)-calculus with a case construct that propagates through functions like a head linear substitution, and show that this construction permits to recover the expressiveness of ML-style pattern matching. We then prove that this system enjoys the Church–Rosser property using a semi-automatic ‘divide and conquer’ technique by which we determine all the pairs of commuting subsystems of the formalism (considering all the possible combinations of the nine primitive reduction rules). Finally, we prove a separation theorem similar to Böhm's theorem for the whole formalism.
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