by Corry Shores
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[Griss, “Logic of Negationless Intuitionistic Mathematics”, entry directory]
[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are surely mistaken or inelegantly articulated. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]
Summary of
George François Cornelis Griss
(G.F.C. Griss)
“Logic of Negationless Intuitionistic Mathematics”
3
Ҥ3. Conditions for the existence of the complementary species and the
inter section”
3.2
[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]
Brief summary:
(3.2) We take two notions to be equally fundamental [and primitive]: being identical and distinguishability. We begin with a set u that has at least two distinguishable elements. “a proper subspecies a of u is a subspecies so that at least one element of u is distinguishable from all elements of a.” [So a is a proper subspecies if it is a set that contains just members of u but not all of them.] Then, the complementary species or compliment as those other u elements that are the remainder: “If a is a proper subspecies of u, the complementary species (complement) ¬a is the species of all elements that are distinguishable from the elements of a. Each element of a is distinguishable from each element of ¬a, a and ¬a are disjoint.” In order for two sets to intersect, a ∩ b, they need to share at least one common element, which is called the touch condition, a χ b, and it results from the rejection of there being any empty species.
[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]
Summary
[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]
[We take two notions to be equally fundamental [and primitive]: being identical and distinguishability. We begin with a set u that has at least two distinguishable elements. “a proper subspecies a of u is a subspecies so that at least one element of u is distinguishable from all elements of a.” [So a is a proper subspecies if it is a set that contains just members of u but not all of them.] Then, the complementary species or compliment as those other u elements that are the remainder: “If a is a proper subspecies of u, the complementary species (complement) ¬a is the species of all elements that are distinguishable from the elements of a. Each element of a is distinguishable from each element of ¬a, a and ¬a are disjoint.” In order for two sets to intersect, a ∩ b, they need to share at least one common element, which is called the touch condition, a χ b, and it results from the rejection of empty species.]
[We take the notion of being identical as fundamental. (So we assume that it is a matter of sameness or perhaps as having the traditional properties of reflexivity, symmetry, and transitivity. Perhaps we are just saying it is a primitive notion that is expressed using the = sign.) We also take the notion of distinguishability as equally fundamental. (It seems we would assume that there are things that we can distinguish from one another, meaning that they have some kind of uniqueness in relation to other things, or a separation of some sort from them.) We begin with a set u, and we will assume that it has at least two distinguishable elements. There are no empty subsets (subspecies). We next define proper subspecies: a is a proper subspecies of u if it is a subset of u where some element(s) of u are distinguishable from those of a (and thus lie outside it): “a proper subspecies a of u is a subspecies so that at least one element of u is distinguishable from all elements of a.” So if we have a proper subspecies a of u, that means there is a set of u members outside of a but that in addition to a complete the set u. We define the complementary species or compliment as those other u elements: “If a is a proper subspecies of u, the complementary species (complement) ¬a is the species of all elements that are distinguishable from the elements of a. Each element of a is distinguishable from each element of ¬a, a and ¬a are disjoint.” (Disjoint here might be a non-negational way of dealing with disjunction like we saw in section 1.0.3 of Griss’ “Negationless Intuitionistic Mathematics, II”. For the two subspecies to be disjunct, that means an item is in either one or the other. What is excluded from this conception is a disjunctive synthesis whereby we would say that we know an that an item is in one subspecies on account of it not being in the other.) In order for two sets to intersect, a ∩ b, they need to share at least one common element, which is called the touch condition, a χ b, and this results from the rejection of there being empty species.]
In negationless intuitionistic mathematics the notion of distinguisha- | bility is equally fundamental as the notion of identity. In the following we shall suppose that u contains at least two distinguishable elements. Then we can define: a proper subspecies a of u is a subspecies so that at least one element of u is distinguishable from all elements of a. If a is a proper subspecies of u, the complementary species (complement) ¬a is the species of all elements that are distinguishable from the elements of a. Each element of a is distinguishable from each element of ¬a, a and ¬a are disjoint. a ≠ u, in words: “a is a proper subspecies of u” is the condition that is necessary to form the complement ¬a. There is also a condition for the existence of an intersection a ∩ b, a so-called touch condition, a χ b 4) expressing that a common element of a and b can be indicated. The appearance of these two conditions, a ≠ u and a χ b, is essential in negationless mathematics. It results from the rejection of empty species.
(44-45)
4) “Condition de composabilité” in the papers quoted sub 1).
(45)
1) PAULETTE DESTOUCHES-FÉVRIER; Logique de l'intuitionisme sans négation et logique de l'intuitionisme positif, C. R. de l'Ac. des Sc. Paris, 226 (1948); RENAUD DE BENGY-PUYVALLÉE, Sur les règles de composabilité dans la logique de la mathématique intuitioniste sans négation. C. R. de l'Ac. des Sc. Paris, 226 (1948).
(41)
Griss, G.F.C. “Logic of Negationless Intuitionistic Mathematics.” Indagationes Mathematicae (Proceedings) 54 (1951): 41–49.
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