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The author or concept searched is found in the following 2 entries.

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

All | Russell | I 80
"All properties of" is illegitimate! -> Reducibility axiom: only "property 2nd order of ..." - of the whole of predicates, not of Napoleon! - Reducibility-axioms are necessary for identity (GoedelVs). I 81f
All properties of a great emperor/Principia Mathematica/Russell: Solution: (j): f (! j ^ z) implies j! (Example of Napoleon) - because the reference to a set of predicates is not itself a predicate of Napoleon. I 143f
Principia Mathematica, 2nd Edition: "Napoleon had all the properties": New: variable function! |
Russell I B. Russell/A.N. Whitehead Principia Mathematica Frankfurt 1986 Russell II B. Russell The ABC of Relativity, London 1958, 1969 German Edition: Das ABC der Relativitätstheorie Frankfurt 1989 Russell IV B. Russell The Problems of Philosophy, Oxford 1912 German Edition: Probleme der Philosophie Frankfurt 1967 Russell VI B. Russell "The Philosophy of Logical Atomism", in: B. Russell, Logic and KNowledge, ed. R. Ch. Marsh, London 1956, pp. 200-202 German Edition: Die Philosophie des logischen Atomismus InEigennamen, U. Wolf (Hg) Frankfurt 1993 Russell VII B. Russell On the Nature of Truth and Falsehood, in: B. Russell, The Problems of Philosophy, Oxford 1912 - Dt. "Wahrheit und Falschheit" InWahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996 |

Constructivism | Russell | I XX
Constructivist attitude/Constructivism/Russell/Gödel: was abandoned in the first edition, since the reducibility axiom for higher types makes it necessary that basic predicates of an infinitely high type exist - of constructivism only remains:
1) Classes as facon de parler 2) The definition of ~, v, etc. as valid for propositions that contain quantifiers 3) Gradual construction of functions of orders higher than 1 (superfluous of course, because of the reducibility-axiom) 4) Interpretation of definitions as mere typographical shortcuts. GoedelVs: because of reducibility axiom: there always exist real objects in the form of basic predicates corresponding to each defined symbol. I XX
Constructivist attitude/Constructivism/Principia Mathematica/Gödel: is taken in again in the second edition and the reducibility axiom is dropped. - It is determined that all basic predicates belong to the lowest type. |
Russell I B. Russell/A.N. Whitehead Principia Mathematica Frankfurt 1986 Russell II B. Russell The ABC of Relativity, London 1958, 1969 German Edition: Das ABC der Relativitätstheorie Frankfurt 1989 Russell IV B. Russell The Problems of Philosophy, Oxford 1912 German Edition: Probleme der Philosophie Frankfurt 1967 Russell VI B. Russell "The Philosophy of Logical Atomism", in: B. Russell, Logic and KNowledge, ed. R. Ch. Marsh, London 1956, pp. 200-202 German Edition: Die Philosophie des logischen Atomismus InEigennamen, U. Wolf (Hg) Frankfurt 1993 Russell VII B. Russell On the Nature of Truth and Falsehood, in: B. Russell, The Problems of Philosophy, Oxford 1912 - Dt. "Wahrheit und Falschheit" InWahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996 |

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

Conceptualism | Quine Vs Conceptualism | VII (f) 126
Classes/Conceptualism/Quine: does not require classes to exist beyond expressible conditions of membership of elements.
((s) VsPlatonism: Quasi requires that there should also be classes without such conditions, as classes should be independent of speakers.)Cantor's proof: would lead to something else: He namely appeals to a class h of those members of the class k that are not elements of the subclasses of k to which they refer. VII (f) 127
But thus the class h is specified impredicatively! h is in fact itself part of the subclass of k.
Thus a theorem of classical mathematics goes overboard in conceptualism.The same fate also applies to Cantor's proof of the existence of hyper-countable infinities. QuineVsConceptualism: which is indeed a welcome relief, but there are problems with much more fundamental and desirable theorems of mathematics: Ex proof that every limited sequence of numbers has an upper limit. ConceptualismVsReducibility Axiom: because it reintroduces the entire Platonist class logic. |
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 InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality InFrom a Logical Point of View, Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference InFrom 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 InZur 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 |

Constructivism | Russell Vs Constructivism | Quine IX 184
VsConstructivism/Construction/QuineVsRussell: we have seen how Russell's constructivist access to the real numbers failed (least upper bound (Kos), see above). He gave up the constructivism and took refuge in the reducibility axiom (RA).
---IX 184/185
The way he gave it up, had something perverse in it:
Reducibility axiom/QuineVsRussell: the reducibility axiom implies that all the distinctions that gave rise to its creation, are superfluous.When Russell's system is consistent with reducibility axioms, then no contradictions will arise if we ignore all orders except the predicative. We can determine that the order of each attribute is always the next highest in comparison to the order of things that have this attribute, according to intensional relations. If somehow an attribute of the order n + k is referred to, which is an attribute of objects of the order n, so we need this name only as such, which is based on a systematic reinterpretation that refers to an attribute of the order n + 1 with the same extension. According to intensional relations. Reducibility Axiom: tells us that an equal-extensional attribute or equal-extensional intensional relation of the desired order, and namely in predicative execution, always exists. Is the axiom planned from the outset, so you should avoid its necessity in that we speak in the beginning only of types of attributes instead of orders of any distinctive sense. Orders are only excusable if one wants to maintain a weak constructive theory without reducibility axiom. ((s)Axiom/Quine/(s): should not be taken as necessary) |
Russell I B. Russell/A.N. Whitehead Principia Mathematica Frankfurt 1986 Russell II B. Russell The ABC of Relativity, London 1958, 1969 German Edition: Das ABC der Relativitätstheorie Frankfurt 1989 Russell IV B. Russell The Problems of Philosophy, Oxford 1912 German Edition: Probleme der Philosophie Frankfurt 1967 Russell VI B. Russell "The Philosophy of Logical Atomism", in: B. Russell, Logic and KNowledge, ed. R. Ch. Marsh, London 1956, pp. 200-202 German Edition: Die Philosophie des logischen Atomismus InEigennamen, U. Wolf (Hg) Frankfurt 1993 Russell VII B. Russell On the Nature of Truth and Falsehood, in: B. Russell, The Problems of Philosophy, Oxford 1912 - Dt. "Wahrheit und Falschheit" InWahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |

Principia Mathematica | Gödel Vs Principia Mathematica | Russell I XIV
Circular Error Principle/VsPrincipia Mathematica/PM/Russell/Gödel: thus seems to apply only to constructivist assumptions: when a term is understood as a symbol, together with a rule to translate sentences containing the symbol into sentences not containing it.
Classes/concepts/Gödel: can also be understood as real objects, namely as "multiplicities of things" and concepts as properties or relations of things that exist independently of our definitions and constructions!This is just as legitimate as the assumption of physical bodies. They are also necessary for mathematics, as they are for physics. Concept/Terminology/Gödel: I will use "concept" from now on exclusively in this objective sense. A formal difference between these two conceptions of concepts would be: that of two different definitions of the form α(x) = φ(x) it can be assumed that they define two different concepts α in the constructivist sense. (Nominalistic: since two such definitions give different translations for propositions containing α.) For concepts (terms) this is by no means the case, because the same thing can be described in different ways. For example, "Two is the term under which all pairs fall and nothing else. There is certainly more than one term in the constructivist sense that satisfies this condition, but there could be a common "form" or "nature" of all pairs. All/Carnap: the proposal to understand "all" as a necessity would not help if "provability" were introduced in a constructivist manner (..+...). Def Intensionality Axiom/Russell/Gödel: different terms belong to different definitions. This axiom holds for terms in the circular error principle: constructivist sense. Concepts/Russell/Gödel: (unequal terms!) should exist objectively. (So not constructed). (Realistic point of view). When only talking about concepts, the question gets a completely different meaning: then there seems to be no objection to talking about all of them, nor to describing some of them with reference to all of them. Properties/GödelVsRussell: one could surely speak of the totality of all properties (or all of a certain type) without this leading to an "absurdity"! ((s) > Example "All properties of a great commander". Gödel: this simply makes it impossible to construe their meaning (i.e. as an assertion about sense perception or any other non-conceptual entities), which is not an objection to someone taking the realistic point of view. Part/whole/Mereology/GödelVsRussell: neither is it contradictory that a part should be identical (not just the same) with the whole, as can be seen in the case of structures in the abstract sense. Example: the structure of the series of integers contains itself as a special part. I XVI/XVII
Even within the realm of constructivist logic there are certain approximations to this self-reflectivity (self-reflexivity/today: self-similarity) of impredicative qualities, namely e.g. propositions, which as parts of their meaning do not contain themselves, but their own formal provability.
There are also sentences that refer to a totality of sentences to which they themselves belong: Example: "Each sentence of a (given) language contains at least one relational word".This makes it necessary to look for other solutions to the paradoxes, according to which the fallacy does not consist in the assumption of certain self-reflectivities of the basic terms, but in other assumptions about them! The solution may have been found for the time being in simple type theory. Of course, all this refers only to concepts. Classes: one should think that they are also not created by their definitions, but only described! Then the circular error principle does not apply again. Zermelo splits classes into "levels", so that only sets of lower levels can be elements of sets of higher levels. Reducibility Axiom/Russell/Gödel: (later dropped) is now taken by the class axiom (Zermelo's "axiom of choice"): that for each level, for any propositional function φ(x) the set of those x of this level exists for which φ(x) is true. This seems to be implied by the concept of classes as multiplicities. I XVIII
Extensionality/Classes: Russell: two reasons against the extensional view of classes: 1. the existence of the zero class, which cannot be well a collection, 2. the single classes, which should be identical with their only elements.
GödelVsRussell: this could only prove that the zero classes and the single classes (as distinguished from their only element) are fictions to simplify the calculation, and do not prove that all classes are fictions!Russell: tries to get by as far as possible without assuming the objective existence of classes. According to this, classes are only a facon de parler. Gödel: but also "idealistic" propositions that contain universals could lead to the same paradoxes. Russell: creates rules of translation according to which sentences containing class names or the term "class" are translated into sentences not containing them. Class Name/Russell: eliminate by translation rules. Classes/Principia Mathematica/Russell/Gödel: the Principia Mathematica can do without classes, but only if you assume the existence of a concept whenever you want to construct a class. First, some of them, the basic predicates and relations like "red", "colder" must be apparently considered real objects. The higher terms then appear as something constructed (i.e. something that does not belong to the "inventory of the world"). I XIX
Ramsey: said that one can form propositions of infinite length and considers the difference finite/infinite as not so decisive.
Gödel: Like physics, logic and mathematics are based on real content and cannot be "explained away".Existence/Ontology/Gödel: it does not behave as if the universe of things is divided into orders and one is forbidden to speak of all orders, but on the contrary: it is possible to speak of all existing things. But classes and concepts are not among them. But when they are introduced as a facon de parler, it turns out that the extension of symbolism opens the possibility of introducing them in a more comprehensive way, and so on, to infinity. To maintain this scheme, however, one must presuppose arithmetics (or something equivalent), which only proves that not even this limited logic can be built on nothing. I XX
Constructivist posture/constructivism/Russell/Gödel: was abandoned in the first edition, since the reducibility axiom for higher types makes it necessary that basic predicates of arbitrarily high type exist.
From constructivism remains only1. Classes as facon de parler 2. The definition of ~, v, etc. as valid for propositions containing quantifiers, 3. The stepwise construction of functions of orders higher than 1 (of course superfluous because of the R-Axiom) 4. the interpretation of definitions as mere typographical abbreviations (all incomplete symbols, not those that name an object described by the definition!). Reducibility Axiom/GödelVsRussell: this last point is an illusion, because of the reducibility axiom there are always real objects in the form of basic predicates or combinations of such according to each defined symbol. Constructivist posture/constructivism/Principia Mathematica/Gödel: is taken again in the second edition and the reducibility axiom is dropped. It is determined that all basic predicates belong to the lowest type. Variables/Russell/Gödel: their purpose is to enable the assertions of more complicated truth functions of atomistic propositions. (i.e. that the higher types are only a facon de parler.). The basis of the theory should therefore consist of truth functions of atomistic propositions. This is not a problem if the number of individuals and basic predicates is finite. Ramsey: Problem of the inability to form infinite propositions is a "mere secondary matter". I XXI
Finite/infinite/Gödel: with this circumvention of the problem by disregarding the difference between finite and infinite a simpler and at the same time more far-reaching interpretation of set theory exists:
Then Russell's Apercu that propositions about classes can be interpreted as propositions about their elements becomes literally true, provided n is the number of (finite) individuals in the world and provided we neglect the zero class. (..) + I XXITheory of integers: the second edition claims that it can be achieved. Problem: that in the definition "those cardinals belonging to each class that contains 0 and contains x + 1 if it contains x" the phrase "each class" must refer to a given order. I XXII
Thus whole numbers of different orders are obtained, and complete induction can be applied to whole numbers of order n only for properties of n!
(...) The question of the theory of integers based on ramified type theory is still unsolved. I XXIII
Theory of Order/Gödel: is more fruitful if it is considered from a mathematical point of view, not a philosophical one, i.e. independent of the question of whether impredicative definitions are permissible.
(...) impredicative totalities are assumed by a function of order α and ω .Set/Class/Principia Mathematica/Russell/Type Theory/Gödel: the existence of a well-ordered set of the order type ω is sufficient for the theory of real numbers. Def Continuum Hypothesis/Gödel: (generalized): no cardinal number exists between the power of any arbitrary set and the power of the set of its subsets. Type Theory/VsType Theory/GödelVsRussell: mixed types (individuals together with predications about individuals etc.) obviously do not contradict the circular error principle at all! I XXIV
Russell based his theory on quite different reasons, similar to those Frege had already adopted for the theory of simpler types for functions.
Propositional functions/statement function/Russell/Gödel: always have something ambiguous because of the variables. (Frege: something unsaturated).Propositional function/p.f./Russell/Gödel: is so to speak a fragment of a proposition. It is only possible to combine them if they "fit together" i.e. are of a suitable type. GödelVsRussell: Concepts (terms) as real objects: then the theory of simple types is not plausible, because what one would expect (like "transitivity" or the number two) to be a concept would then seem to be something that stands behind all its different "realizations" on the different levels and therefore does not exist according to type theory. I XXV
Paradoxes in the intensional form/Gödel: here type theory brings a new idea: namely to blame the paradoxes not on the axiom that every propositional function defines a concept or a class, but on the assumption that every concept results in a meaningful proposition if it is claimed for any object as an argument.
The objection that any concept can be extended to all arguments by defining another one that gives a false proposition whenever the original one was meaningless can easily be invalidated by pointing out that the concept "meaningfully applicable" does not always have to be meaningfully applicable itself. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 |

Principia Mathematica | Wittgenstein Vs Principia Mathematica | II 338
Identity/Relation/Notation/WittgensteinVsRussell: Russell's notation triggers confusion, because it gives the impression that the identity is a relationship between two things. This use of the equal sign, we have to differentiate from its use in arithmetics, where we may think of it as part of a replacement rule.
WittgensteinVsRussell: its spelling gives erroneously the impression that there is a sentence like x = y or x = x. But one can abolish the signs of identity.--- II 352
Definition number/Russell/Wittgenstein: Russell's definition of number as a property of a class is not unnecessary, because it states a method on how to find out if a set of objects had the same number as the paradigm.
Now Russell has said, however, that they are associated with the paradigm, not that they can be assigned.--- II 353
The finding that two classes are associated with one another, means, that it makes sense to say so.
WittgensteinVsRussell: But how do you know that they are associated with one another? One cannot know and hence, one cannot know, if they are assigned to the same number, unless you carry out the assignment, that is, to write it down.--- II 402
Acquaintance/description/WittgensteinVsRussell: misleading claim that, although we have no direct acquaintance with an infinite series, but knowledge by description.
---II 415
Number/definition/WittgensteinVsRussell: the definition of the number as the predicate of a predicate: there are all sorts of predicates, and two is not an attribute of a physical complex, but a predicate.
What Russell says about the number, is inadequate because no criteria of identity are named in Principia and because the spelling of generality is confusing.The "x" in "(Ex)fx" stands for a thing, a substrate. Number/Russell/Wittgenstein: has claimed, 3 is the property that is common to all triads. WittgensteinVsRussell: what is meant by the claim that the number is a property of a class? --- II 416
It makes no sense to say that ABC was three; this is a tautology and says nothing when the class is given extensionally. By contrast, it makes sense to claim that in this room there are three people.
Definition number/WittgensteinVsRussell: the number is an attribute of a function which defines a class, not a property of the extension.WittgensteinVsRussell: he wanted to get ,next to the list, another "entity", so he provided a function that uses the identity to define this entity. --- II 418
Definition number/WittgensteinVsRussell: a difficulty in Russell's definition is the concept of the clear correspondence.
Equal sign/Russell/Wittgenstein: in Principia Mathematica, there are two meanings of identity. 1. by definition as 1 + 1 = 2 Df. ("Primary equations")2. the formula "a = a" uses the "=" in a special way, because one would not say that a can be replaced by a. The use of "=" is limited to cases in which a bound variable occurs. WittgensteinVsRussell: instead of (Ex):fx . (y).fy > (x=y), one writes (Ex)fx: ~ (Ex,y).fx.fy, (sic) which states that there are no two things, but only one. --- IV 47/48
So you cannot introduce objects of a formal concept and the formal concept itself, as primitive concepts.
WittgensteinVsRussell: one cannot introduce the concept of function and special functions as primitive concepts, or e.g. the concept of number and definite numbers.--- IV 73
WittgensteinVsRussell/Tractatus: 5.452 in Principia Mathematica (PM) definitions and basic laws occur in words. Why suddenly words here? There is no justification, and it is also forbidden.
Logic/Tractatus: 5.453 All numbers in logic must be capable of justification. Or rather, it must prove that there are no numbers in logic.5.454 In logic there is no side by side and there can be no classification. There can be nothing more universal and more special here. 5.4541 The solutions of logical problems must be simple, because they set the standard of simplicity. People have always guessed that there must be a field of questions whose answers are - a priori - symmetrical, and --- IV 74
lie combined in a closed, regular structure.
In an area in which the following applies: simplex sigillum veri. ((s) Simplicity is the mark (seal) of the truth).Primitive signs/Tractatus: 5:46 the real primitive signs are not "pvq" or "(Ex).fx", etc. but the most general form of their combinations. --- IV 84
Axiom of infinity/Russell/Wittgenstein/Tractatus: 5.534 would be expressed in the language by the fact that there are infinitely many names with different meanings.
Apparant sentences/Tractatus: 5.5351 There are certain cases where there is a temptation to use expressions of the form"a = a" or "p > p": this happens when one wants to talk of archetype, sentence, or thing. WittgensteinVsRussell: (Principia Mathematica, PM) nonsense "p is a sentence" is to be reproduced in symbols by "p > p" and to put as a hypothesis before certain sentences, so that their places for arguments could only be occupied by sentences. That alone is enough nonsense, because it does not get wrong for a non-sentence as an argument, but nonsensical. 5.5352 identity/WittgensteinVsRussell: likewise, one wanted to express "there are no things" by "~ (Ex).x = x" But even if this was a sentence, it would not be true if there IV 85
would be things but these were not identical with themselves?
---IV 85/86
Judgment/sense/Tractatus: 5.5422 the correct explanation of the sentence "A judges p" must show that it is impossible to judge a nonsense. (WittgensteinVsRussell: his theory does not exclude this).
---IV 87
Relations/WittgensteinVsRussell/Tractatus: 5.553 he said there were simple relations between different numbers of particulars (ED, individuals). But between what numbers? How should this be decided? Through the experience?
There is no marked number.--- IV 98
Type theory/principle of contradiction/WittgensteinVsRussell/Tractatus: 6.123 there is not for every "type" a special law of contradiction, but one is enough, since it is applied to itself.
---IV 99
Reducibility axiom/WittgensteinVsRussell/Tractatus: (61232) no logical sentence, if true, then only accidentally true.
6.1233 One can think of a possible world in which it does not apply. But the logic has nothing to do with that. (It is a condition of the world). |
W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 W III L. Wittgenstein The Blue and Brown Books (BB), Oxford 1958 German Edition: Das Blaue Buch - Eine Philosophische Betrachtung Frankfurt 1984 W IV L. Wittgenstein Tractatus Logico-Philosophicus (TLP), 1922, C.K. Ogden (trans.), London: Routledge & Kegan Paul. Originally published as “Logisch-Philosophische Abhandlung”, in Annalen der Naturphilosophische, XIV (3/4), 1921. German Edition: Tractatus logico-philosophicus Frankfurt/M 1960 |

Quine, W.V.O. | Russell Vs Quine, W.V.O. | Prior I 39
Ramified type theory/rTT/Prior: first edition Principia Mathematica: here it does not say yet that quantification on non-nouns (non nominal) is illegitimate, or that they are only apparently not nominal. (Not on names?) But only that you have to treat them carefully.
---I 40
The ramified type theory was incorporated in the first edition.
(The "simple type theory" is, on the other hand, little more than a certain sensitivity to the syntax.)Predicate: makes a sentence out of a noun. E.g. "φ" is a verb that forms the phrase "φx". But it will not form a sentence when a verb is added to another verb. "φφ". Branch: comes into play when expressions form a sentence from a single name. Here we must distinguish whether quantified expressions of the same kind occur. E.g. "__ has all the characteristics of a great commander." logical form: "For all φ if (for all x, if x is a great commander, then φx) then φ__". ΠφΠxCψxφx" (C: conditional, ψ: commander, Π: for all applies). Easier example: "__ has the one or the other property" logical form: "For a φ, φ __" "Σφφ". (Σ: there is a) Order/Type: here one can say, although the predicate is of the same type, it is of a different order. Because this "φ" has an internal quantification of "φ's". Ramified type theory: not only different types, but also various "orders" should be represented by different symbols. That is, if we, for example, have introduced "F" for a predicative function on individuals" (i.e. as a one-digit predicate), we must not insert non-predicative functions for "f" in theorems. E.g. "If there are no facts about a particular individual ..." "If for all φ, not φx, then there is not this fact about x: that there are no facts about x that is, if it is true that there are no facts about x, then it cannot be true. I.e. if it is true that there are no facts about x, then it is wrong, that there is this fact. Symbolically: 1. CΠφNφxNψx.
--- I 41
"If for all φ not φ, then not ψx" (whereby "ψ" can stand for any predicate). Therefore, by inserting "∏φφ" for "ψ":
2. CΠφNφxNΠφNφxTherefore, by inserting and reductio ad absurdum: CCpNpNp (what implies its own falsehood, is wrong) 3. CΠφNφx. The step of 1 to 2 is an impermissible substitution according to the ramified type theory. Sentence/ramified type theory/Prior: the same restriction must be made for phrases (i.e. "zero-digit predicates", propositions). Thus, the well-known old argument is prevented: E.g. if everything is wrong, then one of the wrong things would be this: that everything is wrong. Therefore, it may not be the case that everything is wrong. logical form: 1. CΠpNpNq
by inserting:
2. CΠpNpNPpNpand so by CCpNpNp (reductio ad absurdum?) 3. NΠpNp, Ramified type theory: that is now blocked by the consideration that "ΠpNp" is no proposition of the "same order" as the "p" which exists in itself. And thus not of the same order as the "q" which follows from it by instantiation, so it cannot be used for "q" to go from 1 to 2. RussellVsQuine/Prior: here propositions and predicates of "higher order" are not entirely excluded, as with Quine. They are merely treated as of another "order". VsBranched type theory: there were problems with some basic mathematical forms that could not be formed anymore, and thus Russell and Whitehead introduce the reducibility axiom. By contrast, a simplified type theory was proposed in the 20s again. Type Theory/Ramsey: was one of the early advocates of a simplification. Wittgenstein/Tractatus/Ramsey: Thesis: universal quantification and existential quantification are both long conjunctions or disjunctions of individual sentences (singular statements). E.g. "For some p, p": Either grass is green or the sky is pink, or 2 + 2 = 4, etc.". (> Wessel: CNF, ANF, conjunctive and adjunctive normal form) Propositions/Wittgenstein/Ramsey: no matter of what "order" are always truth functions of indiviual sentences. Ramified Type TheoryVsRamsey/VsWittgenstein: such conjunctions and disjunctions would not only be infinitely long, but the ones of higher order would also need to contain themselves. E.g. "For some p.p" it must be written as a disjunction of which "for some p, p" is a part itself, which in turn would have to contain a part, ... etc. RamseyVsVs: the different levels that occur here, are only differences of character: not only between "for some p,p" and "for some φ, φ" but also between "p and p" and "p, or p", and even the simple "p" are only different characters. Therefore, the expressed proposition must not contain itself. |
Russell I B. Russell/A.N. Whitehead Principia Mathematica Frankfurt 1986 Russell VII B. Russell On the Nature of Truth and Falsehood, in: B. Russell, The Problems of Philosophy, Oxford 1912 - Dt. "Wahrheit und Falschheit" InWahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996 Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 |

Reducibility Axiom | Berka Vs Reducibility Axiom | Berka I 373
Def Reduzibilitätsaxiom/Russell/Berka: besagt, daß es zu jeder Aussagenfunktion (AF) höherer Ordnung eine entsprechende AF erster Ordnung (d.h. eine prädikative AF) gibt, die mit ihr formal äquivalent ist. (> RA wird durch Forderung nach Prädikativität bedingt).
VsReduzibilitätsaxiom: "Einfache Typentheorie" ( Chwistek, (1921) Ramsey (1926) |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |

Type Theory | Gödel Vs Type Theory | Russell I XXV
Type Theory/Gödel: in the realistic (intensional) interpretation there is an additional assumption: "Whenever an object x can replace another object y in a meaningful proposition, it can do so in every meaningful proposition".
The consequence of this is that the objects are divided into mutually exclusive areas of meaning. GödelVsRussell: suspect that his assumption itself makes his formulation as a meaningful principle impossible: because x and y then have to be narrowed down to definite realms of meaning that are either the same or different and in both cases the statement does not express the principle or even part of it. Other consequence: the fact that an object x is of a given type (or not) cannot be expressed by a meaningful proposition either. I XXVI
A solution is not impossible. It might turn out that any concept is meaningful everywhere except for certain "singular points" or "boundary points" so that the paradoxes appeared as something like the "division by zero". I XXVI
Axioms/Russell/Gödel: Question: are they analytical (as Russell claims here?).
Analyticity/Gödel: can mean two things: 1. purely formal, eliminable. In this sense, even the theory of integers is non-analytical, provided one requires the elimination to be carried out in a finite number of steps. ((s) Otherwise e.g. for each number individually). But the whole of mathematics as applied to propositions of infinite length must be assumed to prove this analyticity, e.g. the axiom of choice can only prove that it is analytical if it is assumed to be true! I XXXIV
Analyticity in the 2. sense: "Due to the sense of the terms occurring in it". Thereby this "sense" is perhaps indefinable (i.e. irreducible to something more fundamental). For example, if one defined "class" and "" as "the concepts (terms) which satisfy the axioms", one would not be able to prove their existence.
"Concept" could perhaps be defined in terms of "proposition", but then certain axioms about propositions become necessary, which can only be justified by reference to the undefined sense of this term. This view of analyticity in turn makes it possible that perhaps any mathematical proposition could be reduced to a special case of a = a. I XXVII
Russell: went the way of seeing both classes and concepts (except for the logically uninteresting basic predicates) as non-existent and replacing them with our own constructions.
Russell/Gödel/(s): constructivist.Reducibility Axiom: is provably wrong in the case of infinitely many individuals, unless one assumes the existence of classes or infinitely many "qualitates occultae". The actual development of mathematical logic has gone the way of the existence of classes and concepts, and Russell himself was later forced to go that way. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 |

Various Authors | Hintikka Vs Various Authors | Hintikka II 136
Picture Theory/Image Theory/Tractatus/Wittgenstein/Hintikka Thesis: Wittgenstein’s image concept is little more than a particularly vivid formulation of the same idea that underlies the usual truth-condition for atomic sentences.
Hintikka: structural equality is a truth condition for elementary propositions.II 137
False: in the most comments it is assumed that the image theory is a complete theory of language understanding.
The difference lies in the completeness thesis that is attributed to Wittgenstein.What needs to be presumed so that the isomorphic relation appears as a complete explanation of language understanding? 1) No separate explanation required 2) The domain of all permissible compounds of names must match the entire domain of possible configurations of their objects. Otherwise one has to decide first whether the connection really exists. Wittgenstein makes both conditions in the Tractatus. HintikkaVsOther Authors: it is unfortunate if all three ideas are mixed up with each other. If the three were made to resemble each other, as many commentators do, the question of whether Wittgenstein abandons the picture theory later would lose its charm. Wittgenstein II 131
Hypothesis/Wittgenstein: Whenever a hypothesis is "always true", so that there is no falsification or verification, this hypothesis is meaningless:
Eddington said: every time a light beam falls upon an electron, it disappears. Then you could also say, there is a white rabbit sitting on the couch sits, and every time I look it’s gone. WittgensteinVsEddington.Hintikka II 59
HintikkaVsCopi: Wittgenstein’s remarks on
Def Reducibility Axiom (Russell)/Hintikka: the axiom states: that for any given property or relation of a certain type (higher lever) there is an equivalent predicative property or relation. It is not about the absolute existence or non-existence, but by the configurations.Therefore, the reducibility axiom cannot belong to the logic! II 60ff
Character/Relation/Denote/Tractatus/Wittgenstein: Not the complex sign "aRb" says that a is in a certain relationship to b, but the fact that "a" is in a certain relation to "b", says that aRb. (3.1432) quotation marks sic!)
But Wittgenstein is getting at something else: The number of names that occur in the elementary proposition must be the same, according to the Tractatus, as that of the objects in the situation represented by the sentence. What situation that is, however, is not determined solely by the names a and b.Copi: thinks (falsely) that Wittgenstein basically abstracts from the relation sign by using the phrase "in certain respects" and undertakes an existential generalization. (HintikkaVsCopi). |
Hintikka I Jaakko Hintikka Merrill B. Hintikka Investigating Wittgenstein German Edition: Untersuchungen zu Wittgenstein Frankfurt 1996 Hintikka II Jaakko Hintikka Merrill B. Hintikka The Logic of Epistemology and the Epistemology of Logic Dordrecht 1989 W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 W IV L. Wittgenstein Tractatus Logico-Philosophicus (TLP), 1922, C.K. Ogden (trans.), London: Routledge & Kegan Paul. Originally published as “Logisch-Philosophische Abhandlung”, in Annalen der Naturphilosophische, XIV (3/4), 1921. German Edition: Tractatus logico-philosophicus Frankfurt/M 1960 |