# Does a Logical possibility exist necessarily or not-necessarily?

First off: What is the definition of a logical possibility?

My personal suggestion is this: A logical possibility is a proposition in which multiple terms are connected in such a way that the proposition does not contain a contradiction. (A blue square, or a non-blue square, where blue is a term and is connected to the term square. This proposition does not include a contradiction and thus is a logical possibility. Contrary to square hexagon.) Would this be a correct definition of a logical possibility? (I.e. does it contain too much or eliminate too much?) To say that something is possible in this view would just mean that it is non-contradictory.

Secondly: Is it possible that a term does not necessarily exist? For example is it necessary that the idea, which constitutes a term, of a "white house" "is/exists" in some way? Or is the "being/existence" of the idea contingent?

Thirdly: If a logical possibility is indeed defined as the connection of terms that does not lead to a contradiction. And there "exists/are" terms which are not necessary but instead contingent. (And not eternal) Would the existence of some logical possibilities then not also be contingent? (And not eternal)

## 2 Answers

1) Your personal suggestion is seconded by Wikipedia.

2) It is possible that a concept does not "exist" in any way (even in "possible worlds"), if there is a contradiction in its definition, e.g. Russell's barber who shaves all those and only those who do not shave themselves. Trying to answer if he shaves himself leads to a contradiction. However, if a concept is free of contradiction then per your, and Wikipedia's, definition it does "exist" 'necessarily'. This use of 'necessarily' is colloquial however, in the technical sense of modal logic "possibility exists necessarily" is not a sentence allowed by the syntax, so your question can not be asked.

3) You are trying to apply necessity and contingency to "existence" or "possibility" of objects, which is not a first order predicate, so in the standard semantic of possible worlds such application can not be interpreted. What you need is something like second order modal logic with meta-possible worlds filled with collections of possible worlds, so that you can interpret modal operators applied to second order predicates. I have seen higher order modal logics, see e.g. Muskens, but not with meta-extensional interpretation like that.

• 2: I like the way you put it. What I mean is this: Could there "be" concepts which at a time did not exist, such that prior to their invention, the concept and the proposition were not only not-contradictory, but simply "non-existent". After the conception (of the concept) of course the proposition "comes to be", and if it is possible, it is necessarily possible, while the proposition itself did not have to come into existence. (And thus the necessity of the possibility is dependent upon the existence of the proposition which itself would be contingent) (S5) – St.Clair Bij Jun 19 '15 at 0:37
• @ St.Clair Bij Traditional modal logic does not entertain "invention" of concepts, but there is temporal modal logic of Arthur Prior with temporal operators "in the future" and "in the past" en.wikipedia.org/wiki/Temporal_logic#History, so I suppose you could get an evolving collection of possible worlds with concepts blinking into "possibility", and then becoming "necessarily possible". I think I saw philosophers try to interpret evolution by natural selection like that, but can't recall where. – Conifold Jun 19 '15 at 1:26
• Thanks Conifold! I think you understand what I mean. Indeed in traditional logic there is no concept of the conception (contrasted with perception) of ideas. I know about temporal logic, but not in modal logic. I have seen the proof of the position contrary to mine: 1. It states: 2+2=4 is necessarily true. 2. If it was not necessarily true, there might have been a time in which where 2+2=4 is untrue/impossible. Since 2. is clearly untrue, therefore 2+2=4 is eternally true. But what I state = there might have been a time when 2 did not exist. such that it was not untrue that 2+2=4. – St.Clair Bij Jun 19 '15 at 17:44
• But that it simply not existed at a time. Have I overlooked reasons for the eternal/timeless existence of concepts? – St.Clair Bij Jun 19 '15 at 17:44
1. What you're speaking of is contingency. Possibility is a different concept. It can be defined using possible words or - in FOL - with respect to constraints.
2. Being isn't existence in most metaphysical theories. Being depends first of all on your domain of discourse (universe). An individual from your domain of discourse can have different modes of existence (possibility, necessity, belief, fictitious existence...).
• Thank you. Contingency is deeply related to possibility. Contingency is the state of being possible while not necessary. While necessary means possible, but not contingently. 2. I know, I meant existence in its broadest sense, where I think a form of being is more fitting. Everything which is not nothing "is" something, and in that sense I mean the "is/exists" The question however is about the contingency of possibility: Are possibilities necessary (i.e. do they necessarily exist) or are possibilities contingent and not necessarily eternal – St.Clair Bij Jun 18 '15 at 15:24
• ad 1. You seem to confuse the terminology. Contingency has nothing to do with modal possibility. Maybe you should rephrase your question. – Atamiri Jun 18 '15 at 17:52
• I'm sorry i think you are wrong on this matter: Contingency is a modal operator. You have the necessity operator indicated by a square and possibility operator indicated by a diamond. (in modal logic). I know most of the time contingency is not connected to possibility in the sense that I connect it, but it is not confused. I'm talking about whether a possibility is necessary, or contingent. – St.Clair Bij Jun 18 '15 at 20:45
• @St.ClairBij The existence of a contingent formula simply means that a theory isn't complete. It's not linked to modality. Possibility and necessity as modal operators are defined by means of the relation of accessibility. Maybe you could give a model that captures your understanding of "contingency." – Atamiri Jun 18 '15 at 21:36
• Here is the reference: en.wikipedia.org/wiki/Modal_logic Under for example alethic logic: contingent= if and only if it is not necessarily false and not necessarily true (i.e. possible but not necessarily true). The above describes contingency as I mean it. Only alethic logic applies it to truth, and I want to apply it to "existence/being" and not truth in that sense. (I.e. could exist/be but not necessarily "is/exists") – St.Clair Bij Jun 18 '15 at 21:57