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The author or concept searched is found in the following 14 entries.

Disputed term/author/ism | Author |
Entry |
Reference |
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Compositionality | Montague | Cresswell I 149
Compositionality/Frege-Principle/Montague/Cresswell: Authors using higher order entities (Montague and Cresswell) do not see themselves as deniers of the Frege principle. This seems to be acknowledged by Hintikka (1982^{(1)}, p. 231).
>Levels, >Levels of description, >Second Order Logic, >Frege principle, >Quantification over properties, >Sentence meaning, >M. J. Cresswell, >J. Hintikka. 1. Jaakko Hintikka. Comments and replies. Philosophia 11 (1-2):105-119 (1982) |
Cr I M. J. Cresswell Semantical Essays (Possible worlds and their rivals) Dordrecht Boston 1988 Cr II M. J. Cresswell Structured Meanings Cambridge Mass. 1984 |

Equality | Logic Texts | Menne I 62 Identity: one thing.
Equality: two things.
Equality is expressed in relation to a characteristic.Menne: one cannot meaningfully speak of the identity of properties. >Properties, >Schematic letters, >Quantification over properties. Principia Mathematica ^{(1)}/Russell/Whitehead/Menne: Identity of individuals: expressed by "x = x". Equivalence of statements, i.e. the equality of the truth value of two statements: by an Identity sign: "p ≡ p". classes: are called "identical". Identity: here a thing appears under two names. Equality: is expressed by reference to a property. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. Hoyningen-Huene II 60
Equality/Form/Hoyningen-Huene: Equality and difference belong to the logical form and not to the content.
>Cf. >Identity. |
Logic Texts Me I Albert Menne Folgerichtig Denken Darmstadt 1988 HH II Hoyningen-Huene Formale Logik, Stuttgart 1998 Re III Stephen Read Philosophie der Logik Hamburg 1997 Sal IV Wesley C. Salmon Logic, Englewood Cliffs, New Jersey 1973 - German: Logik Stuttgart 1983 Sai V R.M.Sainsbury Paradoxes, Cambridge/New York/Melbourne 1995 - German: Paradoxien Stuttgart 2001 Me I A. Menne Folgerichtig Denken Darmstadt 1997 |

Everyday Language | Montague | Hacking I 180
Everyday language/Montague: thesis: the everyday language primarily uses quantifiers of the second order.
>Quantifiers, >Quantification, >Comparisons, >Comparability.((s) For relations, comparisons etc. - Logical quantifiers in first and second order logic are "all", "at least one". >Logic, >Second order logic, >Properties, >Quantification over properties. Second order quantifiers in normal language are expressions like "the bigger ones", "two of her sisters" etc. other examples are "someone", "nobody". (Second order quantifiers go ovor properties.) >Every/each/all, >Nobody. |
Hacking I I. Hacking Representing and Intervening. Introductory Topics in the Philosophy of Natural Science, Cambridge/New York/Oakleigh 1983 German Edition: Einführung in die Philosophie der Naturwissenschaften Stuttgart 1996 |

Identity | Logic Texts | Menne I 62 Identity: one thing.
Equality: two things.
Equality is expressed in relation to a characteristic.Menne: one cannot meaningfully speak of the identity of properties. >Properties, >Schematic letters, >Quantification over properties. Read III 127
Identity/Leibniz: indistinguishability of identical: here there is no problem.
The inversion is problematic: the identity of indiscernibles: no two different things can have all their properties in common. - This is controversial.>Indistinguishability, >Leibniz principle. |
Logic Texts Me I Albert Menne Folgerichtig Denken Darmstadt 1988 HH II Hoyningen-Huene Formale Logik, Stuttgart 1998 Re III Stephen Read Philosophie der Logik Hamburg 1997 Sal IV Wesley C. Salmon Logic, Englewood Cliffs, New Jersey 1973 - German: Logik Stuttgart 1983 Sai V R.M.Sainsbury Paradoxes, Cambridge/New York/Melbourne 1995 - German: Paradoxien Stuttgart 2001 Me I A. Menne Folgerichtig Denken Darmstadt 1997 Re III St. Read Thinking About Logic: An Introduction to the Philosophy of Logic. 1995 Oxford University Press German Edition: Philosophie der Logik Hamburg 1997 |

Ontological Commitment | Prior | I 43
Ontological commitment/Quine: quantification over non-nominal variables (higher quantification, over properties) nominalised them and thus forces us to believe in corresponding abstract objects.
>Abstract objects, >Quantification over properties, cf. >Mentalism, >Objects of thought, >Objects of belief. |
Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 |

Predicates | Field | I 176
Use/quantification/Field: use of predicates does not imply quantification over properties.
Cf. >Properties/Quine, >Properties. II 356
Expansion/theory/language/predicate/Field: you cannot just decide to introduce a new predicate for which the indeterminacy of all extensions shall not apply.
>Introduction. |
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 InTheories of Truth, Paul Horwich Aldershot 1994 |

Predicates | Husserl | Tugendhat I 168f
Predicate/Husserl: the meaning of the predicate could be an object or an attribute. TugendhatVsHusserl: it is not real, the meaning of the predicate is not an object. It is simply drawn up linguistically (VsObject Theory). Instead of standing for an object: the function of the predicate is characterization. Predicates are unsaturated, they are only meaningful in connection with singular terms.
>"Unsaturated", >Singular terms, >Predication, >Quantification over properties. |
E. Husserl I Peter Prechtl, Husserl zur Einführung, Hamburg 1991 II "Husserl" in: Eva Picardi et al., Interpretationen - Hauptwerke der Philosophie: 20. Jahrhundert, Stuttgart 1992 Tu I E. Tugendhat Vorlesungen zur Einführung in die Sprachanalytische Philosophie Frankfurt 1976 Tu II E. Tugendhat Philosophische Aufsätze Frankfurt 1992 |

Properties | Property: what can be ascribed to an object in order to distinguish it from other objects. In philosophy, there is debate about whether properties exist or whether "bare particulars" exist. Expressions for properties are predicates. Not every predicate will refer to a property. See also quantification over properties, 2nd order logic, HOL, completeness. |
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Properties | Castaneda | Frank I, 380f
Features /properties/ CastanedaVsChisholm:
1st take properties to be subjects of predication2nd quantifies over them. I 382
This is devastating in deontological contexts. - It is too complicated in the case of cumulative citations.
>Predication, >Attribution, >Properties/Chisholm, >Quantification over properties. |
Cast I H.-N. Castaneda Phenomeno-Logic of the I: Essays on Self-Consciousness Bloomington 1999 |

Properties | Field | I 176
Properties/Field: we need to reduce properties! Instead: predicates - predicates do not imply use of quantification over properties.
>Predicates, Cf. >Schematic letters/Quine, >Prediates/Quine.II 54
Properties/propositions/Field: Example two predicates like "x has a temperature of 210" and "x has a average molecular energy ..." can stand for the same property, although they have different meanings. - Properties are quite different from meanings (propositions).
>Meaning, >Propositions, >Reference. |
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 InTheories of Truth, Paul Horwich Aldershot 1994 |

Properties | Strawson | IV 67/68
properties/Strawson: one could concede that attributes and properties are ontologically of secondary importance.
Reference to characteristics presupposes the reference to objects but not vice versa. >Reference, >Attributes, >Things/Strawson, >Properties. IV 69
VsQuine: quantification over properties: e.g. "there is a property that no thing has: perfection".
>Quantification/Quine, >Schematic letters/Quine, >Properties/Quine, >Second Order Logic.IV 67
Reference/Strawson: particulars are possible without reference to properties.
((s) QuineVsStrawson: when Quine says being means being value of a bound variable, it means there are no properties without objects).>Bound variable. |
Strawson I Peter F. Strawson Individuals: An Essay in Descriptive Metaphysics. London 1959 German Edition: Einzelding und logisches Subjekt Stuttgart 1972 Strawson II Peter F. Strawson "Truth", Proceedings of the Aristotelian Society, Suppl. Vol XXIV, 1950 - dt. P. F. Strawson, "Wahrheit", InWahrheitstheorien, Gunnar Skirbekk Frankfurt/M. 1977 Strawson III Peter F. Strawson "On Understanding the Structure of One’s Language" InTruth and Meaning, G. Evans/J. McDowell Oxford 1976 Strawson IV Peter F. Strawson Analysis and Metaphysics. An Introduction to Philosophy, Oxford 1992 German Edition: Analyse und Metaphysik München 1994 Strawson V P.F. Strawson The Bounds of Sense: An Essay on Kant’s Critique of Pure Reason. London 1966 German Edition: Die Grenzen des Sinns Frankfurt 1981 Strawson VI Peter F Strawson Grammar and Philosophy in: Proceedings of the Aristotelian Society, Vol 70, 1969/70 pp. 1-20 InLinguistik und Philosophie, G. Grewendorf/G. Meggle Frankfurt/M. 1974/1995 Strawson VII Peter F Strawson "On Referring", in: Mind 59 (1950) InEigennamen, Ursula Wolf Frankfurt/M. 1993 |

Second Order Logic, HOL | 2nd order Logic: Predicate logic of the 2nd order goes beyond predicate logic of the 1st level allowing quantification over properties and relations, and not just objects. Thus comparisons of the powerfulness of sets become possible. Problems which are expressed in everyday terms with terms such as "greater", "between", etc., and e.g. the specification of all the properties of an object require predicate logic of the 2nd order. Since the 2nd level logic is not complete (because there are, for example, an infinite number of properties of properties), one often tries to get on with the logic of the 1st order. |
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Second Order Logic, HOL | Cresswell | I 134
Imbroglio/Geach/Cresswell: e.g. Each of two Turks fought against each of two Greeks. - Problem: the following does not work: each of two Greeks was F and each of two Turks was F.
>Quantification over properties.I 135
E.g. most fundamentalists are creationists.
Problem: it is not easy with two predicates F and C - it is not possible in 1st order logic to bring it in an order. I 137
Solution: 2nd order Logic: here we can say that there is a 1:1 function of F-creationists to fundamentalists, but not vice versa.
>Everyday language, >Unambiguity, >Ordering. |
Cr I M. J. Cresswell Semantical Essays (Possible worlds and their rivals) Dordrecht Boston 1988 Cr II M. J. Cresswell Structured Meanings Cambridge Mass. 1984 |

Syntax | Prior | I 46
Syntax/Prior: variables and constants belong to the same syntactic category.
>Variables, >Constants Problem: what is the meaning of the quantifier with quantification over properties? >Quantification over properties, >Quantification, >Quantifiers. Should the following variable (to be bound by the quantifier) belong to it? >Bound Variables.
Solution: if we consider lambda operators as the only operators that may bind the variables, then the quantifier can build the sentence :
∏(λxφx) (which is equivalent to the simple φ) is briefly ∏φ, everything φ-s. The quantifier builts the sentence. >Lambda calculus, >Lambda notation, >Range. Syntactic status of Lambda: symbolic crutch. Problem: e.g. Something is not the case: SN: S builds a sentence out of a one-digit compound or an adverb. >Sets, >Clauses, >Adverbs. |
Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 |

Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
---|---|---|---|

Quine, W.V.O. | Strawson Vs Quine, W.V.O. | NS I 149
Strawson/Newen/Schrenk: pro descriptive metaphysicsVsRevisionist metaphysics.
Definition descriptive metaphysics/Strawson: detects which ontology suggests our every day doing and speaking.Definition revisionists Metaphysics/StrawsonVsQuine: a physicalist ontology. This stands in contrast to the everyday's way of thinking. StrawsonVsQuine: for Strawson it is just about the everyday language, not about the ontology of any language. Ontology/language/Strawson: Thesis: pro-thing-property-ontology. This is necessarily the most elementary. Because of the similarity to the subject-predicate form. --- NS I 150
Space/Time/Strawson: are tools to differentiate different cases.
Transcendental/Kant: are arguments that relate to the conditions of possibility.Strawson/Newen/Schrenk: his arguments are transcendental. --- Strawson I 198
QuineVsGeach/QuineVsFrege: singular expressions (singular term) can occur at the points of quantifiable variables, general expressions cannot.
Singular Term: can be quantified, general term: not quantifiable.StrawsonVsQuine: on closer inspection, these differences of approach seem far less significant. Quine strongly distinguishes between types of non-linguistic objects on one side and the distinction between singular and general terms, on the other side. (Word/object). In Quine "piety" and "wisdom" are singular expressions, namely names of abstract objects like the nouns "Socrates" and "earth" are the names of concrete objects. Abstract Singular Term/Quine: E.g. "piety" (Universal). The distinction between singular and general term is more important for Quine from the logical point of view. The singular term gives the impression, and to name only one object, while the general term does not claimed at all, to name something, although it "may be true of many things." StrawsonVsQuine: this is an unsatisfactory way of explaining that the word "philosopher" should be a general and not a singular term. We would not like to say that this expression is true of many things or people. --- Strawson I 252
Circle/StrawsonVsQuine: regardless of their captivating simplicity of this analysis, I believe that it will be unacceptable by the form in which it is created. The language terms, in which the analysis is drawn up, presuppose the existence of subject expressions of linguistic singular terms.
Other consequence: we are invited, to see the expressions that replace the "Fs" and "Gs" in the quantified sentences as ordinary predicate expressions. That is allright.--- I 253
Circle/StrawsonVsQuine: but again these forms have only their place in normal language because singular terms, subject expressions occupy the place they have there.
Circularity: because we cannot simultaneously regard Fs and Gs as predicate expressions and accept that they all resolve subject expressions totally in the form of quantified sentences.Circle/StrawsonVsQuine: the argument is based on the linguistic forms that require in turn the use of these expressions. StrawsonVsGadamer/StrawsonVsQuine: one could argue against that this is too narrow, one must proceed inventively. In the case one would have to say what a teaching really should say, which is, taken literally, unacceptable. --- Strawson IV 69
StrawsonVsQuine: Suppose we want to manage without quantification over properties. Does it follow that the belief in objects would be justified, but not the belief in properties?
---IV 70
Strawson: we can accept a different kind of existence. A secondary, although a usual sense of existence, which applies to properties and relations.
---IV 71
Vs: E.g. a) "There is at least one property that has no machine, namely perfect efficiency".
b) "no machine is completely efficient." In a) I quantify, in b) I do not. |
Strawson I Peter F. Strawson Individuals: An Essay in Descriptive Metaphysics. London 1959 German Edition: Einzelding und logisches Subjekt Stuttgart 1972 Strawson II Peter F. Strawson "Truth", Proceedings of the Aristotelian Society, Suppl. Vol XXIV, 1950 - dt. P. F. Strawson, "Wahrheit", InWahrheitstheorien, Gunnar Skirbekk Frankfurt/M. 1977 Strawson III Peter F. Strawson "On Understanding the Structure of One’s Language" InTruth and Meaning, G. Evans/J. McDowell Oxford 1976 Strawson IV Peter F. Strawson Analysis and Metaphysics. An Introduction to Philosophy, Oxford 1992 German Edition: Analyse und Metaphysik München 1994 Strawson V P.F. Strawson The Bounds of Sense: An Essay on Kant’s Critique of Pure Reason. London 1966 German Edition: Die Grenzen des Sinns Frankfurt 1981 Strawson VI Peter F Strawson Grammar and Philosophy in: Proceedings of the Aristotelian Society, Vol 70, 1969/70 pp. 1-20 InLinguistik und Philosophie, G. Grewendorf/G. Meggle Frankfurt/M. 1974/1995 Strawson VII Peter F Strawson "On Referring", in: Mind 59 (1950) InEigennamen, Ursula Wolf Frankfurt/M. 1993 |

Quine, W.V.O. | Schiffer Vs Quine, W.V.O. | I 137
Paul and Elmer/SchifferVsQuine: Quine: there are no countable belief objects. E.g. If John believes that snow is white, and Mary believes that snow is white, there must be something that both believe.
Schiffer: this conditional is wrong: I 138
either that or the alleged quantification for belief objects is not what it appears to be in the Quine's eye. I 144
SchifferVsQuine: harmless apparent quantification. I 235
Substitutional Quantification/Schiffer. E.g. (c) There is something that Mother Teresa, (namely modesty) is true because a substitution instance of
"Mother Teresa X" is true, namely (b): Mother Teresa has the property to be modest. ontological commitment: at substitutional quantification: are only those of the true substitution instances. Universals/Quine: (On what there is, 1953, 10): it is misleading to say that red houses, red roses and red sunsets have something in common. SchifferVsQuine: for whom these everyday speech would it misleading? One can therefore say something true, assuming substitutional quantification. Similarly E.g. "there is a chance that you will win". there are/exist/substitutional quantification/substitutional quantification/Lycan: (1979): Allowed e.g. "There are many things that do not exist". E.g. Loch Ness monster, etc. Properties/Schiffer: in most books of Non-Platonists there is quantification over properties. ((s)> Second order Logic). Quine himself gives an e.g. Properties/Attribute/Existence/"There is"/quantification/second order logic/Schiffer: Quine 1966, p 164): "is valid" is a verb that can be appended to the name of a sentence, and expresses an attribute of the designated sentence. I 237
Schiffer: nobody would assume here that Quine hereby makes an ontological commitment to the existence of attributes. Solution: It is "apparent" quantification that is true, if it is understood as a substitutional quantification. |
Schi I St. Schiffer Remnants of Meaning Cambridge 1987 |

Russell, B. | Quine Vs Russell, B. | Chisholm II 75
Predicates/Denote/Russell: denoting expressions: proper names stand for individual things and general expressions for universals. (Probleme d. Phil. p. 82f). In every sentence, at least one word refers to a universal.
QuineVsRussell: confusion!II 108
Theory of Descriptions/VsRussell/Brandl: thus the whole theory is suspected of neglecting the fact that material objects can never be part of propositions. QuineVsRussell: confusion of mention and use.
Quine II 97
Pricipia mathematica, 1903: Here, Russell's ontology is rampant: every word refers to something. If a word is a proper name, then its object is a thing, otherwise it is a concept. He limits the term "existence" to things, but has a liberal conception of things which even includes times and points in empty space!
Then there are, beyond the existent things, other entities: "numbers, the gods of Homer, relationships, fantasies, and four-dimensional space". The word "concept", used by Russell in this manner, has the connotation of "merely a concept". Caution: Gods and fantasies are as real as numbers for Russell!QuineVsRussell: this is an intolerably indiscriminate ontology. Example: Take impossible numbers, e.g. prime numbers that are divisible by 6. It must be wrong in a certain sense that they exist, and that is in a sense in which it is right that there are prime numbers! Do fantasies exist in this sense? II 101
Russell has a preference for the term "propositional function" against "class concept". In P.M. both expressions appear. Here:
Def "Propositional Function": especially based on forms of notation, e.g. open sentences, while concepts are decidedly independent of notation. However, according to Meinong Russell's confidence is in concepts was diminished, and he prefers the more nominalistic sound of the expression "propositional function" which is now carries twice the load (later than Principia Mathematica.)Use/Mention/Quine: if we now tried to deal with the difference between use and mention as carelessly as Russell has managed to do sixty years ago, we can see how he might have felt that his theory of propositional functions was notation based, while a theory of types of real classes would be ontological. Quine: we who pay attention to use and mention can specify when Russell's so-called propositional functions as terms (more specific than properties and relations) must be construed as concepts, and when they may be construed as a mere open sentences or predicates: a) when he quantifies about them, he (unknowingly) reifies them as concepts. For this reason, nothing more be presumed for his elimination of classes than I have stated above: a derivation of the classes from properties or concepts by means of a context definition that is formulated such that it provides the missing extensionality. QuineVsRussell: thinks wrongly that his theory has eliminated classes more thoroughly from the world than in terms of a reduction to properties. II 102
RussellVsFrege: "~ the entire distinction between meaning and designating is wrong. The relationship between "C" and C remains completely mysterious, and where are we to find the designating complex which supposedly designates C?"
QuineVsRussell: Russell's position sometimes seems to stem from a confusion of the expression with its meaning, sometimes from the confusion of the expression with its mention.II 103/104
In other papers Russel used meaning usually in the sense of "referencing" (would correspond to Frege): "Napoleon" particular individual, "human" whole class of such individual things that have proper names.
Russell rarely seems to look for an existing entity under any heading that would be such that we could call it the meaning that goes beyond the existing referent. Russell tends to let this entity melt into the expression itself, a tendency he has in general when it comes to existing entities. QuineVsRussell: for my taste, Russell is too wasteful with existing entities. Precisely because he does not differentiate enough, he lets insignificance and missed reference commingle. Theory of Descriptions: He cannot get rid of the "King of France" without first inventing the description theory: being meaningful would mean: have a meaning and the meaning is the reference. I.e. "King of France" without meaning, and "The King of France is bald" only had a meaning, because it is the short form of a sentence that does not contain the expression "King of France". Quine: actually unnecessary, but enlightening. Russell tends commingle existing entities and expressions. Also on the occasion of his remarks on Propositions: (P.M.): propositions are always expressions, but then he speaks in a manner that does not match this attitude of the "unity of the propositions" (p.50) and of the impossibility of infinite propositions (p.145) II 105
Russell: The proposition is nothing more than a symbol, even later, instead: Apparently, propositions are nothing..." the assumption that there are a huge number of false propositions running around in the real, natural world is outrageous."
Quine: this revocation is astounding. What is now being offered to us instead of existence is nothingness. Basically Russell has ceased to speak of existence.What had once been regarded as existing is now accommodated in one of three ways a) equated with the expression, b) utterly rejected c) elevated to the status of proper existence. II 107
Russell/later: "All there is in the world I call a fact."
QuineVsRussell: Russell's preference for an ontology of facts depends on his confusion of meaning with reference. Otherwise he would probably have finished the facts off quickly.What the reader of "Philosophy of logical atomism" notices would have deterred Russell himself, namely how much the analysis of facts is based on the analysis of language. Russell does not recognize the facts as fundamental in any case. Atomic facts are as atomic as facts can be. Atomic Facts/Quine: but they are composite objects! Russell's atoms are not atomic facts, but sense data! II 183 ff
Russell: Pure mathematics is the class of all sentences of the form "p implies q" where p and q are sentences with one or more variables, and in both sets the same. "We never know what is being discussed, nor if what we say is true." II 184
This misinterpretation of mathematics was a response to non-Euclidean geometry.
Numbers: how about elementary arithmetic? Pure numbers, etc. should be regarded as uninterpreted. Then the application to apples is an accumulation.Numbers/QuineVsRussell: I find this attitude completely wrong. The words "five" and "twelve" are nowhere uninterpreted, they are as much essential components of our interpreted language as apples. >Numbers. They denote two intangible objects, numbers that are the sizes of quantities of apples and the like. The "plus" in addition is also interpreted from start to finish, but it has nothing to do with the accumulation of things. Five plus twelve is: how many apples there are in two separate piles. However, without pouring them together. The numbers "five" and "twelve" differ from apples in that they do not denote a body, that has nothing to do with misinterpretation. The same could be said of "nation" or "species". The ordinary interpreted scientific speech is determined to abstract objects as it is determined to apples and bodies. All these things appear in our world system as values of variables. II 185
It even has nothing to do with purity (e.g. of the set theory). Purity is something other than uninterpretedness. XII 60
Expression/Numbers/Knowledge/Explication/Explanation/Quine: our knowledge of expressions is alone in their laws of interlinking. Therefore, every structure that fulfills these laws can be an explication. XII 61
Knowledge of numbers: consists alone in the laws of arithmetic. Then any lawful construction is an explication of the numbers.
RussellVs: (early): Thesis: arithmetic laws are not sufficient for understanding numbers. We also need to know applications (use) or their embedding in the talk about other things.Number/Russell: is the key concept here: "there are n such and suches". Number/Definition/QuineVsRussell: we can define "there are n such and suches" without ever deciding what numbers are beyond their fulfillment of arithmetic addition. Application/Use/QuineVsRussell: wherever there is structure, the applications set in. E.g. expressions and Gödel numbers: even the mention of an inscription was no definitive proof that we are talking about expressions and not about Gödel numbers. We can always say that our ostension was shifted. VII (e) 80
Principia Mathematica ^{(1)}/PM/Russell/Whitehead/Quine:shows that the whole of mathematics can be translated into logic. Only three concepts need to be clarified: Mathematics, translation and logic. VII (e) 81
QuineVsRussell: the concept of the propositional function is unclear and obscures the entire PM. VII (e) 93
QuineVsRussell: PM must be complemented by the axiom of infinity if certain mathematical principles are to be derived. VII (e) 93/94
Axiom of infinity: ensures the existence of a class with infinitely many elements.
Quine: New Foundations instead makes do with the universal class: θ or x^ (x = x).1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. VII (f) 122
Propositional Functions/QuineVsRussell: ambiguous:
a) open sentencesb) properties. Russell no classes theory uses propositional functions as properties as value-bound variables. IX 15
QuineVsRussell: inexact terminology. "Propositional function", he used this expression both when referring to attributes (real properties) and when referring to statements or predicates.
In truth, he only reduced the theory of classes to an unreduced theory of attributes.IX 93
Rational Numbers/QuineVsRussell: I differ in one point: for me, rational numbers are themselves real numbers, not so for Russell and Whitehead.
Russell: rational numbers are pairwise disjoint for them like those of Peano. (See Chapter 17), while their real numbers are nested. ((s) pairwise disjoint, contrast: nested)Natural Numbers/Quine: for me as for most authors: no rational integers. Rational Numbers/Russell: accordingly, no rational real numbers. They are only "imitated" by the rational real numbers. Rational Numbers/QuineVsRussell: for me, however, the rational numbers are real numbers. This is because I have constructed the real numbers according to Russell's version b) without using the name and the designation of rational numbers. Therefore, I was able to retain name and designation for the rational real numbers IX 181
Type Theory/TT/QuineVsRussell: in the present form our theory is too weak to prove some sentences of classical mathematics. E.g. proof that every limited class of real numbers has a least upper boundary (LUB). IX 182
Suppose the real numbers were developed in Russell's theory similar to Section VI, however, attributes were now to take the place of classes and the alocation to attributes replaces the element relation to classes.
LUB: (Capters 18, 19) of a limited class of real numbers: the class Uz or {x:Ey(x ε y ε z)}.Attribute: in parallel, we might thus expect that the LUB of a limited attribute φ of real numbers in Russell's system is equal to the Attribute Eψ(φψ u ψ^x). Problem: under Russell's order doctrine is this LUB ψ is of a higher order than that of the real numbers ψ which fall under the attribute φ whose LUB is sought. Boundary/LUB/QuineVsRussell: You need LUB for the entire classic technique of calculus, which is based on continuity. However, LUB have no value for these purposes if they are not available as values of the same variables whose value range already includes those numbers whose upper boundary is wanted. An upper boundary (i.e. LUB) of higher order cannot be the value of such variables, and thus misses its purpose. Solution/Russell: Axiom of Reducibility: Def Axiom of Reducibility/RA/Russell/Quine: every propositional function has the same extension as a certain predicative one. I.e. Ey∀x(ψ!x φx), Eψ∀x∀y[ψ!(x,y) φ(x,y)], etc. IX 184
VsConstruktivism/Construction/QuineVsRussell: we have seen Russell's constructivist approach to the real numbers fail (LUB, see above). He gave up on constructivism and took refuge in the RA. IX 184/185
The way he gave it up had something perverse to it:
Axiom of Reducibility/QuineVsRussell: the RA implies that all the distinctions that gave rise to its creation are superfluous! (... + ...)IX 185
Propositional Function/PF/Attribute/Predicate/TT/QuineVsRussell: overlooked the following difference and its analogs:
a) "propositional functions": as attributes (or intentional relations) andb) proposition functions: as expressions, i.e. predicates (and open statements: e.g. "x is mortal") Accordingly: a) attributes b) open statements As expressions they differ visibly in the order if the order is to be assessed on the basis of the indices of bound variables within the expression. For Russell everything is "AF". Since Russell failed to distinguish between formula and object (word/object, mention/use), he did not remember the trick of allowing that an expression of higher order refers straight to an attribute or a relation of lower order. X 95
Context Definition/Properties/Stage 2 Logic/Quine: if you prefer properties as sets, you can introduce quantification over properties, and then introduce quantification over sets through a schematic context definition.
Russell: has taken this path.Quine: but the definition has to ensure that the principle of extensionality applies to sets, but not to properties. That is precisely the difference. Russell/QuineVsRussell: why did he want properties? X 96
He did not notice at which point the unproblematic talk of predicates capsized to speaking about properties. ((s) object language/meta language/mention/use).
Propositional Function/PF: Russell took it over from Frege.QuineVsRussell: he sometimes used PF to refer to predicates, sometimes to properties. |
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