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The author or concept searched is found in the following 1 entries.
| Disputed term/author/ism | Author |
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| Element Relation | Lesniewski | Prior I 163 Epsilon/Classes/Individual/LesniewskiVsRussell/Prior: "ε" constant for the relation between classes - Ex "a ε b": "The a is b" or "There is exactly one a and every a is b". In Russell there are of course such forms, but the form "x ε a" has not this meaning! >Principia Mathematica, >"Exactly one". L: "a = b" : "the a is the b" this does not correspond to the Def class identity/Russell: "the a "s coincide with the b "s". >Coextension, >Identity. But identity in Lesniewski is also not quite the same as individual identity in Russell. >Identity/Russell. Prior I 165ff Epsilon/Lesniewski/Prior: also higher-level: "f ε g": e.g. "the unit class-of-classes-of f is contained in the class-of-classes g". >Classes, >Sets, >Set theory, >Inclusion. |
Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 |
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| Disputed term/author/ism | Author Vs Author |
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| Lesniewski, St. | Prior Vs Lesniewski, St. | I 43 Abstracts/Prior: Ontological Commitment/Quine: quantification of non-nominal variables nominalises them and thus forces us to believe in the corresponding abstract objects. Here is a more technical argument which seems to point into Quine's direction at first: Properties/Abstraction Operator/Lambda Notation/Church/Prior: logicians who believe in the real existence of properties sometimes introduce names for them. Abstraction Operator: should form names from corresponding predicates. Or from open sentences. Lambda: λ followed by a variable, followed by the open sentence in question. E.g. if φx is read as "x is red", I 44 then the property of redness is: λxφx. E.g. if Aφxψx: "x is red or x is green" (A: Here adjunction) "Property of being red or green": λx∀φxψx. To say that such a property characterizes an object, we just put the name of the property in front of the name of the object. Lambda Calculus/Prior: usually has a rule that says that an object y has the property of φ-ness iff. y φt. I.e. we can equate: (λy∀φxψx)y = ∀φyψy. ((s) y/x: because "for y applies: something (x) is...") One might think that someone who does not believe in the real existence of properties does not need such a notation. But perhaps we do need it if we want to be free for all types of quantification. E.g. all-quantification of higher order: a) C∏φCφy∑φyCAψyXy∑xAψxXx, i.e. If (1) for all φ, if y φt, then φt is something then (2) if y is either ψt or Xt, then something results in either ψ or X. That's alright. Problem: if we want to formulate the more general principle of which a) is a special case: first: b) C∏φΘφΘ() Where we want to insert in the brackets that which symbolizes the alternation of a pair of verbs "ψ" and "X". AψX does not work, because A must not be followed by two verbs, but only by two sentences. We could introduce a new symbol A', which allows: (A’ φψ)x = Aψxψx this turns the whole thing into: c) C∏φΘφΘA’ψX From this we obtain by instantiation: of Θ d) C∏φCφy∑xφxCA’ψXy∑xA’ψXx. And this, Lesniewski's definition of "A", results in a). This is also Lesniewski's solution to the problem. I 45 PriorVsLesniewski: nevertheless, this is somewhat ad hoc. Lambda Notation: gives us a procedure that can be generalized: For c) gives us e) C∏φΘφΘ(λzAψzXz) which can be instatiated to: f) C∏φCφy∑xφx(λzAψzXz)y∑x(λzAψzXz)y. From this, λ-conversion takes us back to a). Point: λ-conversion does not take us back from e) to a), because in e) the λ-abstraction is not bound to an individual variable. So of some contexts, "abstractions" cannot be eliminated. I 161 Principia Mathematica(1)/PM/Russell/Prior: Theorem 24.52: the universe is not empty The universal class is not empty, the all-class is not empty. Russell himself found this problematic. LesniewskiVsRussell: (Introduction to Principia Mathematica): violation of logical purity: that the universal class is believed to be not empty. Ontology/Model Theory/LesniewskiVsRussell: for him, ontology is compatible with an empty universe. PriorVsLesniewski: his explanation for this is mysterious: Lesniewski: types at the lowest level stand for name (as in Russell). But for him not only for singular names, but equally for general names and empty names! Existence/LesniewskiVsRussell: is then something that can be significantly predicted with an ontological "name" as the subject. E.g. "a exists" is then always a well-formed expression (Russell: pointless!), albeit not always true. Epsilon/LesniewskiVsRussell: does not only connect types of different levels for him, but also the same level! (Same logical types) E.g. "a ε a" is well-formed in Lesniewski, but not in Russell. I 162 Set Theory/Classes/Lesniewski/Prior: what are we to make of it? I suggest that we conceive this ontology generally as Russell's set theory that simply has no variables for the lowest logical types. Names: so-called "names" of ontology are then not individual names like in Russell, but class names. This solves the first of our two problems: while it is pointless to split individual names, it is not so with class names. So we split them into those that are applied to exactly one individual, to several, or to none at all. Ontology/Lesniewski/Russell/Prior: the fact that there should be no empty class still requires an explanation. Names/Lesniewski/Prior: Lesniewski's names may therefore be logically complex! I.e. we can, for example, use to form their logical sum or their logical product! And we can construct a name that is logically empty. E.g. the composite name "a and not-a". Variables/Russell: for him, on the other hand, individual variables are logically structureless. Set Theory/Lesniewski/Prior: the development of Russell's set theory but without variables at the lowest level (individuals) causes problems, because these are not simply dispensable for Russell. On the contrary; for Russell, classes are constructed of individuals. Thus he has, as it were, a primary (for individuals, functors) and a secondary language (for higher-order functors, etc.) Basic sentences are something like "x ε a". I 163 Def Logical Product/Russell: e.g. of the αs and βs: the class of xs is such that x is an α, and x is a β. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 |
| Russell, B. | Lesniewski Vs Russell, B. | Quine VII 81 Classes/Element/Quine: at first glance it looks as if "x ε y" requires y to be a class. VII 82 But we can also allow the case that it says: "x is the single thing y". This works with the postulate P1 (Principa Mathematica): the connection of every single thing with its unit class. This is harmless. ((s) >Prior shows that this is possible with >LesniewskiVsRussell). VII 87 Logical Sum/Abstraction/Quine: (x U y) is z^ ((z ε x) v (z ε y)). ((s) union corresponds to "or"). Universal class: ϑ is x^(x = x) ((s) Then the universal class cannot be empty. But >Prior. - LesniewskiVsRussell: it should be able to be empty). |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is In From a Logical Point of View, Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From a Logical Point of View, Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg) München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
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