# Philosophy Lexicon of Arguments

Leibniz’s Principle: Leibniz's law, or identity principle, states that if in the complete descriptions of objects exactly the same properties are attributed, we are concerned with the same object. In the case of identity, it is never a matter of two or more objects, but one, for which there are often different descriptions with different choice of words. Not every description is complete, so identity does not follow from each indistinguishability. See also identity, intensions, extensions, distinguishability, indistinguishability.

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Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

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I 259
Leibniz Principle/Principle/Identity/Indistinguishability/Leibniz/Millikan: Thesis: I treat his principle so that it is an implicit assertion about grammatical categories.
(x)(y)[(F)(Fx equi Fy) > x = y]
Problem: what is the domain of the quantifier "(F)"? ((s) > second level logic).
Here, there cannot simply elements of the domain be paired with grammatical predicates. The set of grammatical predicates may not be of ontological interest. E.g. neither "... exists" nor "... = A" nor "... means red" is paired with something which has the same meaning as "... is green" paired with a variant of a world state.
Quantification/properties/2nd level logic/Millikan: perhaps we can say that the quantifier (F) is about all properties, but we must characterize this set differently than by pairing with grammatical predicates.
False: For example, the attempt of Baruch Brody's thesis: "to be identical with x" should be understood as a property of x "in the domain of the quantifier (F)" is quite wrong! ((s) "be identical with oneself" as a property).
If so, then every thing that has all the properties of x would be identical with x. ((s) Even if it had additional properties).
Problem: under this interpretation, property is not a coherent ontological category.
How can we treat the Leibniz principle, and keep the notion of "property" so that it is ontologically coherent?
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I 260
Leibniz principle/Principle/Identity/Indistinguishability/Millikan: the Leibniz principle is usually regarded as a claim about the identity of individual substances. Substances in which it is useful to attribute to them place and time. That is, "x" and "y" go over individuals.
Quantifier: (F) is generally understood in the way that it only goes via "general properties". Or via "purely qualitative properties".
Purely qualitative properties: i.e. that they are not defined with regard to certain individuals: e.g. the property "to be higher than Mt. Washington"
N.B.: but: "the property of being higher than something that has these and these properties and which are the properties of Mt. Washington".
Individual related properties/Millikan: are normally excluded because they would allow properties like "to be identical to x". That would lead to an empty reading of the Leibniz principle.
MillikanVs: but it is not at all the case that "is identical to x" would not correspond to any reasonable property.
Leibniz principle/Millikan: however, the principle is mostly examined in the context of the domain of general properties in relation to...
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I 261
...the domain of things that have these properties. Thus question: do we have to postulate a domain of such things beyond the domain of these general properties, or can we define the self-identity of an individual in purely qualitative expressions?
Leibniz principle/Millikan: in this context, the relation to a particular individual ((s) and thus of the thing to itself) appears to be an impure or mixed ontological category.
VsLeibniz/VsLeibniz principle/Principle/Identity/Indistinguishability/Indistinguishable/Millikan: the classic objection VsLeibniz is to point out the possibility that the universe could be perfectly symmetrical, whereby then a perfectly identical (indistinguishable) individual would be in another place.
((s) That is, there is something of x that is indistinguishable, which nevertheless is not identical with x, against the Leibniz principle). (See also Adams).
Variants: For example, a temporal repetitive universe, etc. e.g. two identical water drops, two identical billiard balls at different locations. ((s) Why then identical? Because the location (the coordinates) does not have influence on the identity!)
Property/Leibniz: Thesis: a relation to space and time leads to a property which is not purely qualitative.
Millikan: if one ignores such "impure" properties ((s) thus does not refer to space and time), the two billiard balls have the same properties!
VsLeibniz Principle/Law/R. M. Adams/Millikan: Thesis: the principle that is used when such symmetrical worlds are constructed, is the principle that an individual cannot be distinguished (separated) from itself, so the two world halfs of the world cannot be one and the same half.
Leibniz principle/VsVs/Hacking/Millikan: (recent defense of hacking): the objections do not consider that this could be a curved space instead of a doubling.
Curved Space/Hacking/Millikan: here the same thing emerges again, it is not a doubling as in the Euclidean geometry.
MillikanVsHacking: but that would not answer the question.
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I 262
But there are still two interesting possibilities: > indistinguishability.
Leibniz Principle/Principle/Identity/Indistinguishability/Millikan:
1. symmetrical world: one could argue that there is simply no fact here that decides whether the space is curved or doubled. ((s) > nonfactualism).
N.B.: this would imply that the Leibniz principle is neither metaphysical nor logically necessary, and that its validity is only a matter of convention.
2. Symmetrical world: one could say that the example does not offer a general solution, but the assumption of a certain given symmetrical world: here, there would very well be a fact whether the space is curved or not. A certain given space cannot be both!
N.B.: then the Leibniz principle is neither metaphysical nor logically necessary.
N.B.: but in this case this is not a question of convention, but a real fact!
Curved space/Millikan: here, what is identical is necessarily identical ((s) because it is only mirrored). Here the counterfactual conditional would apply: if the one half were different, then also the other. Here the space seems to be only double.
Doubling/Millikan: if the space (in Euclidean geometry) is mirrored, the identity is a random, not a necessary one. Here one half could change without changing the other half. ((s) No counterfactual conditional).
Identity: is then given when the objects are not indistinguishable because a law applies in situ, but a natural law, a natural necessity.
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I 263
Then, in the second option, identity from causality applies.
(x) (y) {[NN (F) Fx equi Fy] equi x = y}
Natural necessary/Notation: natural necessary under natural possible circumstances.
Millikan: this is quite an extreme view, for it asserts that if there were two sets of equivalent laws that explain all events, one of these sets, but not the other, would be true, even if there was no possibility to find out which of the two sets it is that would be true.
This would correspond to the fact that a seemingly symmetrical world was inhabited. Either the one or the other would be true, but one would never find out which one.

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Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.

Millk I
R. G. Millikan
Language, Thought, and Other Biological Categories: New Foundations for Realism Cambridge 1987

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> Counter arguments in relation to Leibniz Principle

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Ed. Martin Schulz, access date 2018-05-21