Centro de Lógica, Epistemologia e História da Ciência

Ilya Prigogine (1913–2003) discovered that importation and dissipation of energy into complex systems could reverse the maximization of entropy rule imposed by the second law of thermodynamics. Beginning from a sketch of what a complex system consists of and some epistemological consequences, I will show which instances of process approach are able to give some preliminary ontological basis for a more adequate understanding of nature with its own indeterminacy, surfaced as empirical evidence inner non–linear thermodynamics. The idea is to connect these issues for reinterpreting the verification principle in experimental method, avoiding Popper‟s criticism to it and in agreement with falsification procedure.

The main aim of this work is to evaluate whether Boolos' semantics for second-order languages is model-theoretically equivalent to standard model-theoretic semantics. Such an equivalence result is, actually, directly proved in the Appendix. I argue that Boolos' intent in developing such a semantics is not to avoid set-theoretic notions in favor of pluralities. It is, rather, to prevent that predicates, in the sense of functions, refer to classes of classes. Boolos' formal semantics differs from a semantics of pluralities for Boolos' plural reading of second-order quantifiers, for the notion of plurality is much more general, not only of that set, but also of class. In fact, by showing that a plurality is equivalent to subsets of a power set, the notion of plurality comes to suffer a loss of generality. Despite of this equivalence result, I maintain that Boolos' formal semantics does not committ (directly) second-order languages (theories) to second-order entities (and to set theory), contrary to standard semantics. Further, such an equivalence result provides a rationale for many criticisms to Boolos' formal semantics, in particular those by Resnik and Parsons against its alleged ontological innocence and on its Platonistic presupposition. The key set-theoretic notion involved in the equivalence proof is that of many-valued function. But, first, I will provide a clarification of the philosophical context and theoretical grounds of the genesis of Boolos' formal semantics.

The main aim of this work is to depict the relevance of one of the most interesting formal concepts, i.e., duality, with respect to some foundational issues related to the actual ontological debates arising from the epistemological analysis of physical processes. In particular, it is argued that a formal characterization of emergence is possible in terms of the category theoretic notion of duality. This would constitute a significative shifting from a Platonic to an Aristotelian ontology (of mathematics and, then, of physics). Actually,
providing a contribution to this very shift is what this paper wants to be focused on.

This project is an enquiry in the metaphysics of {\itshape processes} and its semantics. For processes we means here, in the most generic way, dynamic entities such as actions or oriented relations, as opposite to static ones, e.g., substances, things or objects. How can we understand, justify and account for the asymmetry of becoming and the consequent emergence of contingently stable objects (properties) in terms of dynamical entities such as processes? The project is, thus, articulated so as to answer the following questions: (i) Are processes, conceived as basic entities, compatible with the constraints imposed by fundamental natural sciences? (ii) What is the basic mechanism that provides an explication of how emergent objectuality arises from processes? (iii) What are the consequences for any formal and semantic analysis of the emergence of objectuality?

Ontology, as science, must not presuppose the entity as its object, but must define its constitution so as to establish what exists. The connection between logic and ontology, in a formalized theory, is perceived by considering the logical forms of that theory, that is, the ways of combining the logical-grammatical categories of a (formal) language. In this, we do not rely on the descriptive apparatus (theoretical) - both linguistic (non-logical constants) and axiomatic (axioms and non-logical definitions) - to decide which ontological categories should be used to represent entities and the way their behavior is regulated. Rather, it is through distinctions in logical-grammatical categories and through rules and the logical axioms that is possible to control symbolic transformations between such categories. It is precisely the variations of these transformations that corresponds to the various definitions of the alleged ontology. In this chapter we will focus on the differences between realism and nominalism.

La connessione fra logica ed ontologia si puó cogliere in maniera il piú possibile precisa osservando una teoria formalizzata. In tal caso l’ontologia relativa a tale teoria è da intendersi, da un punto di vista sintattico, una teoria delle forme logiche, in quanto ha per oggetto i modi di combinare le categorie logico-grammaticali del linguaggio (formale) adottato da tale teoria. In tal modo, agendo su di esse, si puó definire e osservare il comportamento formale di tali forme e in funzione dell’ontologia che si vuole supporre in una data teoria. La definizione delle forme logiche di una teoria non é affidata all’apparato descrittivo, specifico della teoria formalizzata di cui si è in possesso – che sia l’apparato linguistico (le costanti non-logiche), che sia quello assiomatico (gli assiomi e le definizioni non-logici). Quali sono le categorie ontologiche supposte da una specifica teoria e il modo in cui è regolato il loro comportamento  dipende strettamente dalla struttura logica della teoria: mediante le distinzioni in categorie, appunto, logico-grammaticali: mediante la stipulazione delle regole e degli assiomi logici, si controllano le possibili trasformazioni simboliche fra queste categorie. A variazioni relative a tali trasformazioni corrispondono variazioni dell’ontologia di riferimento per la teoria.
In questo lavoro ci concentreremo sulle differenze fra realismo e nominalismo descritte e controllate all’interno della logica matematica, della logica classica del primo e del secondo ordine per le quali rimandiamo a (Enderton 2001)

In some works about what is currently called plural logic, Oliver and Smiley support the thesis that the so-called many-valued functions are not relations. This thesis is
fundamental for the development of the aforementioned logic. With the present work I will first analyse the reasons for this thesis, which ultimately rest on an extensional approach to logic and mathematics. Subsequently I will argue that in non-extensional contexts it is entirely legitimate to state, otherwise, that many-valued functions are relations, strongly reducing the validity of the former thesis.