The typing approach to Church-Fitch’s knowability paradox and its revenge form

Williamson, Linsky, Paseau and others proposed a solution to Church- Fitch’s knowability paradox that is based on typing knowledge; however, it received some criticism. Carrara and Fassio objected that the approach has no paradox-independent motivation, it is thus ad hoc. In the first part of the paper, I dismiss such criticism by carefully stating typing approach principles that are based on non-circular formation of propositions and intensional operators operating on them. In the second part of the paper, I demonstrate that the firm foundation of the approach prevents the variants of the paradox by Florio, Murzi and Jago that were developed as allegedly unresolvable by typing knowledge. The revenge form of Church-Fitch’s knowability paradox, which had been proposed by Williamson, Hart, Carrara and Fassio, fares badly as well, since it is likewise based on violation of reasonable typing rules

Pravdivost mezi syntaxí a sémantikou

Sir s m c lem t eto pr ace je vyjasnit vztah mezi syntax a s emantikou, zejm ena pokud jde o jazyky s p resn e speci kovanou strukturou. Hlavn ot azky, kter ymi se zab yv ame, jsou: Co cin s emantick y pojem s emantick ym? Co zp usobuje, ze je pouh a s emantick a anal yza takov eho pojmu nedostate cn a? Co je t m rozhoduj c m krokem, kter y mus me u cinit, abychom pronikli k v yznamov e str ance jazyka? T emito ot azkami se nezab yv ame p r mo, ale prost rednictv m anal yzy typick eho s emantick eho pojmu, a sice pravdivosti. Na s hlavn ot azkou tedy je: Jak e pojmov e prost redky jsou nezbytn e pro uspokojivou de nici pravdivosti? Ke zkoum an pojmu pravdivosti a jednotliv ych zp usob u, jak jej lze de- novat, jsme si vybrali t ri konkr etn syst emy: kumulativn verzi Russellovy rozv etven e teorie typ u, Zermelovu druho r adovou teorii mno zin a Carnapovu logickou syntax. Ka zd y syst em je podroben d ukladn emu studiu. P redkl adan a pr ace je tedy souborem t r v ce m en e samostatn ych studi , je z popisuj mo znosti explicitn de nice pravdivosti a nezbytn eho pojmov eho z azem . Poznamenejme, ze na s m c lem nen historicky v ern a prezentace uveden ych syst em u, n ybr z snaha o dal s rozvinut toho cenn eho, co nab zej , ve sv etle sou casn ych poznatk u. Obecn ym z av erem, k n emu z dosp ejeme na z...The broad aim of this thesis is to clarify the relationship between syntax and semantics, mainly in connection with languages with exactly speci ed structure. The main questions we raise are: What is it that makes a semantic concept genuinely semantic? What exactly makes a merely semantic characterization of such a concept inadequate? What is the decisive step we have to make if we want to start speaking about the meaning-side of language? We approach these questions indirectly: via an analysis of a typically semantic concept, namely that of truth. Our principal question then becomes: What conceptual resources are required for a satisfactory de nition of truth? To investigate the concept of truth and di erent ways in which it can be de ned, we have chosen three individual systems: (a cumulative version of) Russell's rami ed theory of types, Zermelo's second-order set theory and Carnap's logical syntax. Each of the systems is studied in considerable detail. The presented thesis is, in e ect, a collection of three case-studies into the ways in which the concept of truth is explicitly de nable and into the requisite conceptual background, each study forming a more or less closed unity. It should be noted that we are not interested in a historically faithful representation of these systems; our goal is to get...Institute of Philosophy and Religious StudiesÚstav filosofie a religionistikyFilozofická fakultaFaculty of Art

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