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EXISTENTIAL GENERALIZATION IN TIL
Abstract
The paper deals with the fundamental rule of extensional logics, namely the rule of Existential Generalization. This rule can be applied in the situation when a function f is applied on its argument a to obtain the value of f at a. If the application does not fail, i.e., if the function f is defined at a, then we can existentially quantify, and derive that there is the value f(a). Our system is based on Transparent Intensional Logic (TIL). Since TIL is a hyperintensional, partial, typed lambda calculus, we examine the validity of the rule in TIL, or rather in its computational variant the TIL-Script language. The rule is context sensitive in the sense that depending on a context we should recognize the type of entity to be abstracted over. This is not to say that the rule can be invalid dependently on context; the rule is valid universally. Only that the type of the argument over which we quantity depends on the context. There are three kinds of contexts to be distinguished, namely extensional, intensional and hyperintensonal. We introduce the definition of these three kinds of context and an algorithm that recognizes in which context a particular construction occurs so that the Existential Generalization can be validly applied. The tool navigates users through the correct application of the deduction rules
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Duží, M., Jespersen, B., Materna, P. Procedural Semantics for Hyperintensional Logic. Foundations and Applications of Trasnsparent Intensional Logic. First edition. Berlin: Springer, series Logic, Epistemology, and the Unity of Science, vol. 17, 2010.
Duží, M., Jespersen, B. (2015): Transparent Quantification into Hyperintensional objectual attitudes. Synthese, vol. 192, No. 3, pp. 635-677. DOI: 10.1007/s11229-0140578-z
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Duží, M., Fait, M., Menšík, M. (forthcoming): Context Recognition for a Hyperintensional Inference Machine. To appear in the AIP proceeding of ICNAAM 2016, International Conference of Numerical Analysis and Applied Mathematics.
Duží, M. (2013): Deduction in TIL: From simple to ramified hierarchy of types. Organon F, vol. 20, supplementary issue 2, pp. 5-36.
Duží, M. (2012): Towards an extensional calculus of hyperintensions. Organon F, vol. 19, supplementary issue 1, pp. 20-45.
Duží, Jespersen (2015): Transparent Quantification into Hyperintensional objectual attitudes. Synthese, vol. 192, No. 3, pp. 635-677.
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