On Philosophical Sense of Metamathical Limitative Theorems
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چکیده
Logic was always considered as closely related to philosophy, even as an essential part of the latter. For example, Russell used to say that logic is the heart of philosophy. The connection of two fields can be illustrated by some views of Aristotle. According to the Stagirite the world consists of individual substances. Every substance has two elements, its matter and its form. Although the former particularizes substances, the latter brings generality. Due to the form, all human beings belong to the same kind (genus), because they share the property of being human. On the other hand, particular human exemplars differ with respect to their form. Now, one can investigate how this metaphysical or ontological idea is related to Aristotelian logic. As we know Aristotle constructed logic of categorical sentences of the form '… S is (are) … P', where the places marked by dots should be filled by 'All', 'No', some' or 'not'. As a result we have four forms 'All S are P', 'No S are P', 'Some S are P' and 'Some S are not P'. Take 'All S are P' illustrated as (a) 'All humans are mor-tal', for example. Presumably, the subject of (a), that is, the word 'humans' refers to particular human beings, but the predicate 'mortal' points out their essential property. Even if we disregard various controversial questions, like the nature of form or how it is bonded with matter in substances, we can still discuss whether Aristotle's ontological view determined his logic in the sense that taking categorical sentences as the basic logical forms was motivated by the doctrine that every substance consists of the matter and the form. However, the opposite direction is always open, because one can maintain that Aristotle passed from grammar to logic. The same problem appears in modern logic. In first-order logic, the form Pa, where the letter P functions as a predicate and the letter a is an individual name, that is, a proper name of an object. Now, one can say that whereas the name a refers to the matter, the component P expresses the form. And controversial issues appear once again. One can argue that so-called bare particulars, that is, pieces of pure matter (without properties) do not exist, although the opposite view is also present in philosophy. One will look for the route from logic to ontology, but others would like to insist that …
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