By Giuseppe Primiero
This publication develops a philosophical and logical interpretation of the concept that of data in the formal constitution of positive sort idea (CTT), in a fashion concurrent with a various diversity of latest views at the philosophy of knowledge. It offers a newly formulated and conceptually constructed presentation of the matter of Analyticity, and a brand new fascinating point of view at the confident interpretation of information processes.
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Extra info for Information and Knowledge: A Constructive Type-theoretical Approach
G. physics. e. objects in terms of the concepts they express, or they are deﬁned by. Thus, a proper object of ontology is a deﬁned object, an object expressed with all its (essential) properties. Ontology in this sense, conceptually near to the Aristotelian way of understanding it, amounts to a study of objects with the concepts they contain, that in the type-theoretical setting means to express objects in terms of the types they belong to. 3), already mentioned in connection to the question of meaning and the problem of impredicativity.
This diﬀerence is of the greatest importance, because judgements being the building elements of inferences, it is then clear that a proof developed through inferences should be considered as an act itself: to make an inference means to construct a proof, and the proof is at the same time a syntactic object and an act which makes a judgement evident, and for this reason known and understood. 76 In this way the distinction between proposition and judgement amounts to an analysis of the respective kinds of proof in the following terms77 : proof of a proposition = veriﬁcation proof of a judgement = demonstration The nature of proofs is the most relevant epistemological issue questioned by the Intuitionistic approach: it is essential to underline the possible deﬁnitions of “proof” as “act of proving” or “object” which counts as a proof,78 introducing the diﬀerence between a proof act and a proof object.
This correspondence can be interpreted as a way to reconsider set-theory in respect to logic86 : the logical laws are reduced back to a set-theoretical interpretation, representing a complete isomorphism, since proposition and set are exactly the same notion. The equivalence can be stated as follows: A : prop is equivalent to A : set a : A is equivalent to a ∈ A On the basis of the constructive explanation of propositions in terms of proof objects, it is possible to give a constructive explanation of all the logical connectives in terms of set theory as follows: – A ∧ B corresponds to the Cartesian product A × B of sets A and B, and to prove it one must obtain an ordered pair < a, b > such that a is a proof of A and b is a proof of B.