RAMIFIED TYPE THEORY AS INTENSIONAL LOGIC

Ovaj doktorski rad sastoji se od dva glavna dijela. Prvi se dio bavi pitanjem što sustava čine funkcije u razgranatoj teoriji tipova Bertranda Russella, kako ju je izložio u filozofijskome uvodu prvoga izdanja Principia Mathematica.U tome se dijelu rada brani eliminativističko tumačenje i pokušava pokazati da Russell sam stavačne funkcije u Principia razumije samo kao izraze, kao tzv. nepotpune simbole, koji ne označavaju nikakve izvanjezične predmete poput pojmova ili atributa.This doctoral thesis consists of two main sections. The first section addresses the background ontology of Bertrand Russell’s ramified type theory as described in Principia Mathematica. More precisely, it deals with the question of the ontological status of propositional functions. The concept of a propositional function is one of the central concepts of Russell’s theory of types, both in the first draft of the theory in “Appendix B” of The Principles of Mathematics andinitsmatureformulationinthefirsteditionofPrincipia.However,howtounderstandwhat Russell meant by “propositional functions” remains controversial. What are propositional functions? Are they some sort of intensional abstract entities, like properties and relations, or just expressionsofthelanguageoftypetheory,i.e.openformulas?Aneliminativistinterpretationis proposedandclaimedthatRussell’spropositionalfunctionsaretobeunderstoodonlyasexpressions,astheso-called“incompletesymbols”,whichdonotdenoteanyextra-linguisticobjects, such as attributes, whether in realist or constructivist sense. It is argued that the ramified type theory of Principia should not be understood as an abandonment of Russell’s earlier substitutional theory, but rather as its continuation. The ramified type hierarchy is a consequence of Russell’s belief that the paradoxes of propositions that plagued the substitutional theory can only be avoided by some kind of a type differentiation of propositions. On the other hand, the elimination of propositional functions (as well as propositions) from the ontology of Principia is a consequence of Russell’s conception of logic as universal science, which must contain only one type of genuine variables – viz., completely unrestricted entity variables, with everything that exists as their values. The doctrine of the unrestricted variable has been formulated by Russell in The Principles of Mathematics and is an inseparable part of his understanding of logic. The theory of denoting phrases he developed in “On Denoting” provided the tool for the elimination of higher-order entities from the background ontology of his logic. This way, Russell managed to retain a complex type hierarchy of expressions needed to avoid the paradoxes and at the same time preserve the doctrine of the unrestricted variable. At the end of the first section, certain advantages of rejecting the doctrine of the unrestricted variable and Russell’s understanding of propositional functions as incomplete symbols are recognized, and suggested that the interpretation of the ramified hierarchy as an ontological hierarchy of concepts might be philosophically justified. Inthesecondsection,aformalsystemofcumulativeintensionalramifiedtypetheory(KIRTT) is presented, guided by a realist interpretation of a ramified type hierarchy and with semantics based on an intensional generalization of Henkin models. The aim was to formalize certain metaphysical intuitions concerning the nature of intensional entities and to sketch one possible formal theory of concept

RAMIFIED TYPE THEORY AS INTENSIONAL LOGIC

Ovaj doktorski rad sastoji se od dva glavna dijela. Prvi se dio bavi pitanjem što sustava čine funkcije u razgranatoj teoriji tipova Bertranda Russella, kako ju je izložio u filozofijskome uvodu prvoga izdanja Principia Mathematica.U tome se dijelu rada brani eliminativističko tumačenje i pokušava pokazati da Russell sam stavačne funkcije u Principia razumije samo kao izraze, kao tzv. nepotpune simbole, koji ne označavaju nikakve izvanjezične predmete poput pojmova ili atributa.This doctoral thesis consists of two main sections. The first section addresses the background ontology of Bertrand Russell’s ramified type theory as described in Principia Mathematica. More precisely, it deals with the question of the ontological status of propositional functions. The concept of a propositional function is one of the central concepts of Russell’s theory of types, both in the first draft of the theory in “Appendix B” of The Principles of Mathematics andinitsmatureformulationinthefirsteditionofPrincipia.However,howtounderstandwhat Russell meant by “propositional functions” remains controversial. What are propositional functions? Are they some sort of intensional abstract entities, like properties and relations, or just expressionsofthelanguageoftypetheory,i.e.openformulas?Aneliminativistinterpretationis proposedandclaimedthatRussell’spropositionalfunctionsaretobeunderstoodonlyasexpressions,astheso-called“incompletesymbols”,whichdonotdenoteanyextra-linguisticobjects, such as attributes, whether in realist or constructivist sense. It is argued that the ramified type theory of Principia should not be understood as an abandonment of Russell’s earlier substitutional theory, but rather as its continuation. The ramified type hierarchy is a consequence of Russell’s belief that the paradoxes of propositions that plagued the substitutional theory can only be avoided by some kind of a type differentiation of propositions. On the other hand, the elimination of propositional functions (as well as propositions) from the ontology of Principia is a consequence of Russell’s conception of logic as universal science, which must contain only one type of genuine variables – viz., completely unrestricted entity variables, with everything that exists as their values. The doctrine of the unrestricted variable has been formulated by Russell in The Principles of Mathematics and is an inseparable part of his understanding of logic. The theory of denoting phrases he developed in “On Denoting” provided the tool for the elimination of higher-order entities from the background ontology of his logic. This way, Russell managed to retain a complex type hierarchy of expressions needed to avoid the paradoxes and at the same time preserve the doctrine of the unrestricted variable. At the end of the first section, certain advantages of rejecting the doctrine of the unrestricted variable and Russell’s understanding of propositional functions as incomplete symbols are recognized, and suggested that the interpretation of the ramified hierarchy as an ontological hierarchy of concepts might be philosophically justified. Inthesecondsection,aformalsystemofcumulativeintensionalramifiedtypetheory(KIRTT) is presented, guided by a realist interpretation of a ramified type hierarchy and with semantics based on an intensional generalization of Henkin models. The aim was to formalize certain metaphysical intuitions concerning the nature of intensional entities and to sketch one possible formal theory of concept