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.26 Ab-stract facts are needed in Russell s logical realism as the truth makers of thestatements of logic and mathematics that contain no descriptive constants.Thisontology is quite different from an ontology of sets (with or without urelements),i.e., one in which pure mathematics is reduced to set theory as opposed to logic.In set theory there are no set-theoretical facts about pure sets, i.e., sets whosetransitive closures contain no urelements other than the empty set.This is be-24Russell 1959, p.124.25See Russell 1912, p.137.26Ibid., p.105. 116 CHAPTER 5.FORMAL THEORIES OF PREDICATION PART IIcause a set has its being in its members, and a set s being is all that is neededto account for the truth or falsehood of statements about membership in thatset.In other words, the being of a set consists in its having just the membersthat it has, and no fact over and above the being of the set itself is needed toaccount for membership in that set.A property or relation (in intension), onthe other hand, does not have its being in its instances, and therefore its beingcannot of itself account for the truth of statements about objects having thatproperty or relation.Propositions, it should be noted, are not objective truthsor falsehoods in this ontology; rather, according to Russell, there are no propo-sitions other than sentences, i.e., propositions, like propositional functions, arenow only expressions.Russell s 1910 13 logical realist ontology, in other words,is not a simples•"complexes ontology as characterized above, and in that regardthe hyperintensional assumptions (Hyper1) (Hyper3) are inapplicable to thisversion of logical realism.What a complex predicate represents in this ontologyis not a complex property or relation, but only another property or relationthat stands in a certain logical relation between the properties and relationsthat are represented by the component parts of that complex predicate.Thelogical relation, together with these properties and relations, are constituents ofan abstract fact, rather than of a complex property or relation.And abstractfacts, like concrete, physical facts, are extensional entities and not hyperinten-sional entities.The above hyperintensional paradox , accordingly, does notapply to either Russell s or Frege s versions of logical realism.Of course, one might argue that there is still the problem of explaininghyperintensional contexts, i.e., of how a logical realist formal ontology suchFrege s or Russell s can account for the logic of such intentional contexts asbelief, desire, etc.Be that as it may, in any case one cannot reject these formalontologies on the basis of the above hyperintensionality paradox.One might, on the other hand, reject our interpretation of propositional func-tions as expressions, or one might just insist on maintaining a simples•"complexesontology regardless of what Frege s or Russell s own views were.In that case,however, an explanation must be given of how a property or relation can con-tain a logical operation as well other properties or relations as constituents.Itis not enough to simply assume that this is so without giving an ontologicalaccount of how it is possible, i.e., of how there can be such complexes.27 Butthen, assuming such an ontology will bring one back to the problem of how thehyperintensional paradox is to be avoided.Finally, with regard to conceptual realism, it is noteworthy that althoughconcepts are formed, or constructed, on the basis of other concepts, conceptsthemselves, as cognitive capacities, do not contain other concepts or logicaloperations as constituents.In other words, with respect to concepts as cognitivecapacities, conceptual realism is not a simples•"complexes ontology.Of course,with respect to the intensional contents of concepts, including propositions asthe contents of our speech and mental acts, the situation might well be different.27An algebraic or set-theoretical semantics for hyperintensionality, we should note, does notof itself amount to an ontological account. 5.4.HYPERINTENSIONALITY 117In other words, we might well allow intensional objects to be either simpleor complex, though some account will then be needed of the sense in whichthey might contain objects, including the intensional counterparts of logicaloperations, as constituents.But even assuming that a simples•"complexes ontology applies to the inten-sional objects of conceptual realism, nevertheless, the hyperintensional assump-tion (Hyper3) does not apply to conceptual realism as represented by »HST"or HST" for the reason given above.Moreover, none of the above hyperin-»tensional principles (Hyper1) (Hyper3) apply to conceptual realism at thelevel of the logical forms that represent the cognitive structure of our speechand mental acts; nor can most of the steps in the above argument be taken asrepresenting the cognitive structure of a speech or mental act.This is importantbecause it is only on this level of analysis that hyperintensionality has a roleto play.In other words, it is only on this initial level of analysis, as opposedto the second level where deductive transformations are represented, must com-plex predicates be given a fine-grained representation.A speech or mental actin which, e.g., being round and red, i.e., [»x(Round(x) '"Red(x))], is predicatedof an object is not the same as a speech or mental act in which being not eithernot-round or not-red, i.e., [»x¬(¬Round(x) ("¬Red(x))] is predicated of thatobject, even though the predicate expressions representing these concepts arelogically equivalent.Thus, although hyperintensionality does apply on the levelof analysis on which the cognitive structure of our speech and mental acts arerepresented, it does not apply on the level of deductive transformations, suchas those involved in the above paradox.Hyperintensionality, or fine-grained structure, applies only to cognitive struc-ture, which, in conceptual realism is represented only at the initial level of anal-ysis [ Pobierz caÅ‚ość w formacie PDF ]
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.26 Ab-stract facts are needed in Russell s logical realism as the truth makers of thestatements of logic and mathematics that contain no descriptive constants.Thisontology is quite different from an ontology of sets (with or without urelements),i.e., one in which pure mathematics is reduced to set theory as opposed to logic.In set theory there are no set-theoretical facts about pure sets, i.e., sets whosetransitive closures contain no urelements other than the empty set.This is be-24Russell 1959, p.124.25See Russell 1912, p.137.26Ibid., p.105. 116 CHAPTER 5.FORMAL THEORIES OF PREDICATION PART IIcause a set has its being in its members, and a set s being is all that is neededto account for the truth or falsehood of statements about membership in thatset.In other words, the being of a set consists in its having just the membersthat it has, and no fact over and above the being of the set itself is needed toaccount for membership in that set.A property or relation (in intension), onthe other hand, does not have its being in its instances, and therefore its beingcannot of itself account for the truth of statements about objects having thatproperty or relation.Propositions, it should be noted, are not objective truthsor falsehoods in this ontology; rather, according to Russell, there are no propo-sitions other than sentences, i.e., propositions, like propositional functions, arenow only expressions.Russell s 1910 13 logical realist ontology, in other words,is not a simples•"complexes ontology as characterized above, and in that regardthe hyperintensional assumptions (Hyper1) (Hyper3) are inapplicable to thisversion of logical realism.What a complex predicate represents in this ontologyis not a complex property or relation, but only another property or relationthat stands in a certain logical relation between the properties and relationsthat are represented by the component parts of that complex predicate.Thelogical relation, together with these properties and relations, are constituents ofan abstract fact, rather than of a complex property or relation.And abstractfacts, like concrete, physical facts, are extensional entities and not hyperinten-sional entities.The above hyperintensional paradox , accordingly, does notapply to either Russell s or Frege s versions of logical realism.Of course, one might argue that there is still the problem of explaininghyperintensional contexts, i.e., of how a logical realist formal ontology suchFrege s or Russell s can account for the logic of such intentional contexts asbelief, desire, etc.Be that as it may, in any case one cannot reject these formalontologies on the basis of the above hyperintensionality paradox.One might, on the other hand, reject our interpretation of propositional func-tions as expressions, or one might just insist on maintaining a simples•"complexesontology regardless of what Frege s or Russell s own views were.In that case,however, an explanation must be given of how a property or relation can con-tain a logical operation as well other properties or relations as constituents.Itis not enough to simply assume that this is so without giving an ontologicalaccount of how it is possible, i.e., of how there can be such complexes.27 Butthen, assuming such an ontology will bring one back to the problem of how thehyperintensional paradox is to be avoided.Finally, with regard to conceptual realism, it is noteworthy that althoughconcepts are formed, or constructed, on the basis of other concepts, conceptsthemselves, as cognitive capacities, do not contain other concepts or logicaloperations as constituents.In other words, with respect to concepts as cognitivecapacities, conceptual realism is not a simples•"complexes ontology.Of course,with respect to the intensional contents of concepts, including propositions asthe contents of our speech and mental acts, the situation might well be different.27An algebraic or set-theoretical semantics for hyperintensionality, we should note, does notof itself amount to an ontological account. 5.4.HYPERINTENSIONALITY 117In other words, we might well allow intensional objects to be either simpleor complex, though some account will then be needed of the sense in whichthey might contain objects, including the intensional counterparts of logicaloperations, as constituents.But even assuming that a simples•"complexes ontology applies to the inten-sional objects of conceptual realism, nevertheless, the hyperintensional assump-tion (Hyper3) does not apply to conceptual realism as represented by »HST"or HST" for the reason given above.Moreover, none of the above hyperin-»tensional principles (Hyper1) (Hyper3) apply to conceptual realism at thelevel of the logical forms that represent the cognitive structure of our speechand mental acts; nor can most of the steps in the above argument be taken asrepresenting the cognitive structure of a speech or mental act.This is importantbecause it is only on this level of analysis that hyperintensionality has a roleto play.In other words, it is only on this initial level of analysis, as opposedto the second level where deductive transformations are represented, must com-plex predicates be given a fine-grained representation.A speech or mental actin which, e.g., being round and red, i.e., [»x(Round(x) '"Red(x))], is predicatedof an object is not the same as a speech or mental act in which being not eithernot-round or not-red, i.e., [»x¬(¬Round(x) ("¬Red(x))] is predicated of thatobject, even though the predicate expressions representing these concepts arelogically equivalent.Thus, although hyperintensionality does apply on the levelof analysis on which the cognitive structure of our speech and mental acts arerepresented, it does not apply on the level of deductive transformations, suchas those involved in the above paradox.Hyperintensionality, or fine-grained structure, applies only to cognitive struc-ture, which, in conceptual realism is represented only at the initial level of anal-ysis [ Pobierz caÅ‚ość w formacie PDF ]