A question concerning formal modal logic
My lack of formal training in this area betrays me. So I'm looking for help.
In S4 or S5, or a derivative therefrom, can an individual exist in every possible world without contradiction?
In predict calculus we might cheat and represent that an individual exists as ?(x)(a=x); is there some way to pars this into modal logic, such that the individual (a) is in every possible world?
I'm pretty sure that any such attempt will be either illformed or lead to contradiction, but the details escape me.
Comments?
In S4 or S5, or a derivative therefrom, can an individual exist in every possible world without contradiction?
In predict calculus we might cheat and represent that an individual exists as ?(x)(a=x); is there some way to pars this into modal logic, such that the individual (a) is in every possible world?
I'm pretty sure that any such attempt will be either illformed or lead to contradiction, but the details escape me.
Comments?
Comments (189)
Quoting Banno
In S5 it seems one can, as I explained to you before. This video is helpful, though a little long (you can skip to 10:50 if you want the short version, but I’d suggest you watch the whole thing):
This kind of logic is founded on philosophy and philosophy can offer very different answers. In my opinion, in philosophy there is nothing contingent or necessary that exists. These are relations of ideas. The world exists neither contingently nor necessarily. It is necessary that the world exist if it is existing but this doesn't mean it had to exist a priori or that it is contingent a priori such that it needs a necessary substance or whatever to back it up. The proper distinction in mind and world situations is "objective vs subjective". Subjective is fooling or lying to oneself. Objective is what is real towards consciousness. As Kant, Hegel, and others point out, the world is there and we don't create it but turn over the coin and we see that we don't know how much we might be contributing to creating the world, and modern psychology backs this up to a great extent
Then how does modal logic apply to reality. It's concepts are necessity and contingency, neither of which can be proven to apply to the world the way they do to ideas
The concepts apply to propositions, or formally - sentences, including propositions about the world.
In mathematics, salient uses of modality include analysis of provability ('The Logic Of Provability' - Boolos) and for semantcs for constructive logic.
The desk across the room is contingent because it can be destroyed and I will of necessity die if hit by a train. That is science, not logic. Modal logic is a waste of time because philosophy easily shows it's bs. Again it's use is for science, not philosophy
"The desk across the room" is a noun phrase, not a proposition. The modal operators are applied to propositions not noun phrases.
Would you please tell me what textbook in modal logic is the basis of whatever familiarity you have with the subject?
Wikipedia and it's application in articles. Modal Logic is a structural thing and when it says a proposition is necessary or contingent, this is not using the word as it is used in traditional philosophy but this doesn't stop people from pretending it does. Why not show us something unique model logic had proven
Wikipedia and those articles don't mention that the operators are applied to sentences and don't refrain from saying that the operators are applied to "ideas" or noun phrases?
You don't like philosophy any
I was explaining how necessity and contingency are used in traditional philosophy and model logic in totally different ways
Do you have a problem with Aristotle's "traditional philosophy"?
Whatever you thought you were explaining, you did it by terribly misunderstanding the modal operators in modal logic, thus giving reason to think you don't know anything about modal logic.
Good be sentences.
What you call classical philosophy was a hodgepodge when it came to modality. The work in modality has given us a syntax within which we can construct a coherent account.
And no, you have not given an account of classical modality. You've done nothing more than make an accusation.
But, hey, you've provided a reason for the thread to appear towards the top of the main page, so that's positive. It might attract someone who can address the OP.
Aristotle is fine but logic is subjective though necessary and doesn't apply in reality. Modal logic is not a philosophically traditional way of thinking. What do you know of Hegel's logic btw
Before you present any model logic you have to prove logic can prove something outside the mind. Can you provide an example? I searched "what had model logic proved" and there was nothing
Quoting Gregory
That depends on a definition of "philosophical traditional way of thinking" and its import depends on whether it is important to adhere to "philosophically traditional ways of thinking".
Quoting Gregory
I haven't made any claims about Hegel's logic.
Quoting Gregory
https://philosophy.stackexchange.com/questions/23929/what-are-the-practical-applications-of-modal-logic
https://www.sciencedirect.com/science/article/pii/S1571066114000905
Quoting Banno
Well, I only quoted your first question, and was waiting for somebody else to answer the second one.
As to that second question:
Quoting Banno
Well, that question just leads to: Does the idea of a being who exists in all possible worlds involve a contradiction? So it just goes back to the first one.
If you could elaborate on what you think entails a contradiction about the idea of such a being (or why it would be illformed) I or somebody else could give you a clear answer, otherwise the question might be a bit too broad.
It's your thread but I'm not your student
Aristotle would not have supported proving anything exists simply from logic structures alone. There are pure Platonic ideas which logic can't touch and the real practical world of reality.
B applies to a theorem, not an individual. I don't think we have an answer yet.
Is ? ?(x)(a=x) well-formed? Is it a theorem of S5?
I didn't say you don't try at philosophy but you've had many discussions with me and to my brain you put things in neat packages. Socrates at the start had a fluid way at coming to truth (as he saw it). Why would you ask logic to tell you if you could exist in every world? To me it sounds strange
And it has advanced greatly since Aristotle.
Ok, well I learn something every day. There might be some of that which would register with me.
What specific theorems of modal logic do you have in mind as "proving something exists from logic structures alone"? And what passages in Aristotle do you take to be predictive that he would not countenance modern modal logic?
Modal logical leads to it's using the ontological argument. That's the best example of how ML can go to far
Ex a=x is a theorem of the pure predicate calculus with identity.
So it seems to me that it is the case that necessarily Ex a=x.
Am I overlooking something? (I admit that I'm rusty in modal logic.)
You said you couldn't find references. I gave you ones.
What system of modal logic and what specific theorem are you referring to?
Those links are about computer science
Aristotle would never take the ontological argument under consideration. But modal theorists struggle with it nevertheless
So is ? ?(x)(a=x) a rendering of "a (that individual) exists in every possible world"?
That seems problematic, since one could specify a possible world in which A did not exist.
You said you couldn't find mention on the Internet of anything modal logic proves. So those are links were modal logic is used for results in computer science.
And one of the links is not just about what is proved but actual use for traffic signal systems.
In
Ex a=x
'a' is merely a variable or constant (depending on your specification). It has no specific referent except by assignment to an individual in the domain of the model.
Quoting Banno
You mean the corollary of axiom B? It does not apply to an individual, but it could apply to a proposition which has an individual as its subject, that is: “Necessarily, God exists”. If this proposition is true in some possible worlds (meaning it does not entail a contradiction), then it's possible that it's necessary that God exists, and using corollary B, the advocate of the modal ontological argument will argue that it implies that God exists, unless one can show that God exists in no possible worlds, which is to show that the proposition: “Necessarily, God exists” entails a contradiction.
Quoting Banno
As TonesInDeepFreeze pointed out, it seems to be well formed, since it follows logically from “the pure predicate calculus with identity”, and truths infered from pure logic are necessarily true if they are deduced from principles which are themselves necessarily true.
But it would mean something like: Necessarily, there exists/is an x (God), such that a (the greatest conceivable being/ subject of all perfections) = x.
I asked what specific system and what specific theorem you have in mind. Apparently you don't have anything specific in formal modal logic in mind.
And I asked you what passages in Aristotle you have in mind. Again, apparently none.
So is the implication of ? ?(x)(a=x) that, in any given model, there must be an individual that exists in every possible world?
I don't want to go there yet - leave the good lord int he background and look at the implication of ? ?(x)(a=x)...
It seems you are wondering how to formalize "there is an individual that exists in every possible world". I don't know at this moment. But I don't think it is Ex a=x.
Ex a=x really doesn't say much. It's so obviously true that translating it into English looks almost silly. Here's one way: "Whatever is named by 'a', there is an object x in the domain such that what is named by 'a' is x." That is obviously true since such an x is whatever is named by 'a'.
I've been clear but you don't follow. Aristotle had philosophy-like arguments for the first mover, something ML doesn't have. I stated right at the start that logic is used in science just as math is. But logic can't say what can exist and what can't. It is plain old logic at the end of the day, and yes for Aristotle metaphysics is not logic. This is all common knowledge
Well, yes, in that I don't think it can be done - it woudl not be well-formed - or it would lead to contradiction.
I guess I'm trying to formalise an apparent contradiction between "there is an individual that exists in every possible world" and "for any given individual, one can specify a possible world in which that individual does not exist".
Again, thanks for your help.
So, still, you've shown nothing specific that Aristotle wrote that would be predictive that he would reject modern modal logic.
The fact alone that Aristotle had certain arguments to prove the existence of God does not predict that he would reject modern modal logic.
Quoting Gregory
It easily says, "There exists an object that has property P if and only if it has property P".
It easily says "There does not exist an object that has property P if and only if it does not have property P".
All that comes from non logic. ML tries to rule over philosophy like logicism tried to do with math
Individuals are members in universes. Now, suppose an individual is a member of a certain universe, of course that individual is not a member of certain other universes. So, yes, there is no individual that is a member of every universe.
On the other hand, whatever individual 'a' names by a model, that individual is a member of the universe for the model that names 'a'. But 'a' may name different individuals according to different models.
And if 'a' is in the language, then, for any model, 'a' must name some individual in the universe for the model. And syntactically in the object language we have: Ex a=x for which the derivation is trivial:
Ax x=x
a=a
Ex a=x
But logic is about contrapositives, inverse minor premises, ect. It's useful to understand how to understand one's own thinking and map the world, but the great questions of philosophy can't be answered thru finding the final modal logic idea that settles everything. Doesn't modal logic implicitly assume it can settle these questions one and for all?
No, it comes from first order logic.
Quoting Gregory
Mathematical logic is a field of study. It has no will such that it could will to rule over anything, including another field of study. Meanwhile people who work in the field of mathematical logic might have a will to rule over people in the field of philosophy. So, yes, we should have that investigated, maybe start with a Presidential commission.
No. That is an hilarious question.
Well, no. Rather it allows you to recognise good and bad arguments.
Name one thing in modal logic literature that proved something in philosophy.
Yep. With first order logic, every model specifies a universe. For modal logic, we specify a set of universes.
First order logic gets its content from non logic
So if I could describe what I think you are saying back to you in my own terms...
To build a model, we set up a bunch of possible worlds. Within that universe, "a" refers to some given individual. In some possible worlds, "a" exists, in others, "a" does not exist - but in those universes "a" refers to that very same individual.
OK so far?
Hmm, but isn't that what the advocates of the modal ontological argument would reject? They would not be convinced with just “of course that individual is not a member of certain other universes” because they argue that God, and God alone, is a member of all “universes” or “possible worlds” without exception.
Or is there some significant difference between “universe” and “possible world” in the case of God?
I have not claimed that modal logic proves anything in philosophy. And that does not entail that modal logic does not prove anything in philosophy nor that I hold that modal logic does not prove anything in philosophy. Still, even though I am not obligated to support that which I have not claimed, one observes that the subjects of necessity and contingency are themselves philosophical topics. And, turning back around, modal logic itself is a subject of philosophy. And modal logic applied to mathematical logic has a role in the philosophy of mathematics. Meanwhile, in your readings about modal logic on the Internet, you really found nothing about application of modal logic to philosophy? You didn't look further beyond the Wikipedia article you looked at (I only guess you actually read it) - on such subjects the article mentions as knowledge, belief, temporality and morality?
It's interesting that people on the Internet have such vociferous opinions on entire fields of study that they know nothing about except skimming a Wikipedia article and a few other dubiously researched web pages and posts.
What I see as the problem with modal logic and the way many posters reason on this forum too is trying to use logic to prove something beyond itself. Proper philosophical intuition rarely considers logic as logic
I'll give the definitions, then I'll address your formulation.
Let 'A' notate the universal quantifier.
We define 'model for a first order language' as a function F on certain symbols of the language, such that
F('A') is a non-empty set. F('A') is called 'the universe for the model F'.
For an n-place (n being 0 or greater) predicate symbol 'R', F('R') is an n-ary relation on the universe.
For an n-place (n being 0 or greater) operation symbol 'g', F('g') is an n-ary function on the universe.
So a model for a first order language specifies a "possible world".
/
We define 'model' for a first order modal language' as a tuple
W is a non-empty set of first order models for the language. W is the set of "possible worlds".
c is a 2-place relation on W. c is the accessibility relation.
/
Quoting Banno
For modal, a model has a non-empty set of possible worlds and an accessibility relation on the set of possible worlds.
Quoting Banno
For modal, there may be more than one universe.
Quoting Banno
For a given possible world, 'a' refers to a member of the universe of that possible world.
Quoting Banno
No, 'a' is a symbol, not an individual. For a given possible world, 'a' names a member of the universe of that possible world. And for any possible world, some member of the universe of that possible world is named by 'a'.
I don't know what you intend to mean with that.
The a first order theory of course includes non-logical symbols and, if not the pure first order theory for that language, the theory includes non-logical theorems. The content expressed is given by models.
Has modal logic always fail or has it proved something which takes logic to prove? After all logic is about proof
So the member of the universe named by "a" exists in some possible worlds but not others.
Must there be an individual named by "a" that exists in every possible world? Can there be an individual that exists in every possible world?
I don't know. I'd have to see the specific argument formalized.
Perhaps such arguments have additional premises other than the mere proof apparatus of whatever given modal system? Of course, with certain premises, we can compel certain semantical results that are not compelled by the logic alone.
Quoting Amalac
That strikes me as being an additional premise. Of course we can't rule out that additional premises have consequences.
Right. For example, suppose one universe is the set of even numbers, and suppose 'a' names 2. Then suppose another universe is the set of odd numbers. Then whatever 'a' names, it can't be 2.
Please provide an example of modern formal modal logic used trying to prove something beyond itself. Of course, a proof may adopt premises that a reader might reject, but I'd like to know what formal modal logic proof you think is not entailed by its premises. (Various systems of modal logic are proven to be sound.)
Quoting TonesInDeepFreeze
Actually, I think they claim that follows from the definition of God, using corollary B or corollary 5. So it's not a premise, but rather something that follows from other premises (they say).
You are right however, in that even if one accepted S5, the modal ontological argument would still have major problems, like dealing with the objection that existence is not a predicate or is a second order predicate, since that premise (that existence is a predicate) is required (it seems to me at least) to hold the claim that the greatest conceivable being or subject of all perfections exists in all possible worlds.
The argument would go something like this:
Quoting Amalac
I can't parse that.
Quoting Gregory
Good for him! Nor does modern logic. indeed, one of the early notable aspects of modern logic is that it extends past syllogisms. And modern logic does not claim that any given logic encompasses all of reasoning. Indeed, that is why, for example, propositional logic is extended to predicate logic, and both are extended to modal logic, and extensions and alternatives to many kinds of mathematical logics and philosophical logics.
I don't understand the question.
Quoting Gregory
One may characterize what logic is "about" in different ways. A common notion is that logic studies entailment and inference. Then proof formulates methods for inference, especially methods that correspond to entailment.
Hegel writes that way. I read and reread hundreds of pages of such. It's a striking style.
Now logic is about the forms of itself and can't comment on the inexperienced. You admit this! So then modal logic is just logic and much closer to programming than philosophy
The answer is "no"?
I'd like to see the logic, the semantics, and the proof explicitly specified.
What is the exact sentence that is proved? Is it of the form?:
Necessarily E!x x has property P [where 'P' stands for the bundle of Godlike properties].
Also, I don't know enough about the theory of definitions in modal logic, but if it is close enough to predicate logic, then a definition of an individual requires first proving an existence and uniqueness theorem. So do the proofs you mention indeed first prove there exists a unique individual with such and such properties that is then named 'God'? One can't prove that God has certain properties without first defining 'God' courtesy of an existence and uniqueness theorem. Especially, one can't just assert without proof that there does exist a unique individual having certain properties and then go on to demonstrate that that individual then has other properties for a QED.
And recall that what I mentioned is based on ordinary specifications for setting up a semantics for modal logic. I don't even know how one would formulate those specifications as formulas in the language itself to use in proof.
I admit that I am very rusty in modal logic so I might need to be corrected or qualified. But as far as I can tell, ordinarily, semantics for modal logic begins with ordinary models for propositional or predicate logic, which includes assignments for constants and variables. And in that ordinary manner, there is no object that is the member of all universes for models.
Indeed. I gave an example. And it's basic set theory. For any set S, and individual d, we have that d is not a member of S\{d}.
I don't know what that means.
Quoting Gregory
How could I admit it when I don't even know what it means?
Quoting Gregory
It depends on a definition of 'philosophy'. Ordinarily philosophy is regarded to include logic and formal logic. If you insist on a definition of 'philosophy' rigged to exclude formal logic, then, of course, you will have prevailed to establish, by definition, that formal logic is not part of philosophy.
Ah, I admit that I am not familiar with a system that has existence as a predicate. For ordinary predicate logic, it's not a formalizable notion. I didn't know that it can be formulated in certain modal logics.
Quoting Amalac
I don't know enough about this. Isn't the above a meta-argument about the semantics for formal logic? That would be okay, but I'd like to be clear what really is afoot.
Quoting Amalac
Quoting Amalac
How does a system of modal logic talk about its own semantics? I'm not saying it can't be done, but I'd like to know how it works.
If you construe existence as a matter of existential quantification over identity, as ?x[x=a], to mean that there is an individual identical to a (the individual you're interested in), then it depends on what your quantifiers range over. In fact, on a classical Kripkean treatment, existence is always necessary existence, since if there is some individual x identical to a in the domain of individuals, then there will be at any world, since the domain of individuals and the domain of worlds are simply separate.
On the other hand, you can make the domain relative to a world, such that at world w, there is an individual x identical to a, but at world w', there is no (because the domain associated with w includes a, while the domain of individuals associated with w' does not). Here, you are not forced to make existence necessary existence, but you can – you can just include a in the individual-domain of every world in your domain of worlds.
A logic that banned the necessary existence of an individual would have to make some special provision for how existence is interpreted, and why you could never have a domain of worlds such that an individual exists at every world.
How do you express that as a modal formula in the object language?
Quoting Snakes Alive
I don't quite follow you. In any domain, there is an individual named by 'a'. But the individual named by 'a' may be different in different domains.
Quoting Snakes Alive
Yes, that is the sense I had been mentioning. If 'a' names a certain individual in one world then it is not required that 'a' names that individual in other worlds. Indeed the universes of the worlds in question could even be disjoint.
Quoting Snakes Alive
Yes. But how do you express that as a formula in the modal logic object language?
Quoting Snakes Alive
What systems have that predicate? Is it definable in typical modal systems? What is the definition?
Quoting Snakes Alive
That I understand. Of course, there is no restriction that demands that the intersection of the set of universes is empty.
You can define any predicate you like, necessary or otherwise. To make a predicate 'P' that is necessary for an individual a at a model, you just posit that the model you're working with is such that for all worlds w in the set of worlds W associated with the frame of the model, P(a) evaluates to true at w.
The standard Kripkean treatment does not, of course, allow for a primitive predicate like this to be true for all individuals (or just some individual) at all models. But necessity, on a Kripkean semantics, is not a matter of logical truth that generalizes over models – it's a matter of truth at all accessible worlds to some particular world, and if we have an accessibility relation on which every world is accessible from every other, then this is equivalent to truth at all worlds in that particular model. There is no impediment to supposing such a model.
Let 'Px' intuitively say "x exists". What is the definition of 'Px' in the form?:
Px <-> Fx where 'Fx' is a formula in a given first order language (or with a modal operator too, if you like) with only 'x' free and not including the symbol 'P'?
Quoting Snakes Alive
I was asking about a predicate "exists".
Anyway, what you mentioned is a semantical. How would we express that as a formula in the modal logic itself?
Quoting Snakes Alive
I do seem to recall those particulars to be correct. But that's all semantical. I don't know how you can express such things in the modal logic itself.
For modal semantics, there are two different methods we may adopt for assigning denotations to the constants:
(1) For each domain of a possible world, we assign the denotations. The assignments for different domains might be different. That is the method I used earlier in this thread.
(2) We make only one assignment, so that each constant is assigned to a member of the union of the domains. Obviously, with this method, my previous remarks don't apply.
But I don't understand how (2) could work with a base of ordinary predicate logic. Suppose individual d is in the union of the domains but not in a particular domain D for a possible world w. Then if constant 'a' is assigned to d, then how could w even be a possible world? It would lack a denotation for 'a'.
I don't know what you're asking.
I don't think so, they define God as having certain properties (perfections or “great making properties”) first, and then through analysis of the concept of “God”, defined as the subject of all perfections or greatest conceivable being, they argue that the proposition “God exists” is analytically true, that is: that God's non existence is as impossible as there being an object that was both round and triangular at the same time and in the same sense.
Quoting TonesInDeepFreeze
That's true, here:
Quoting Amalac
...I should have just said that “there is [not “exists”] an x (God), such that...” (though as I said later in the ontological argument it is argued that God's existence is analytic), since otherwise one would just assume the existence of that “x” right away, which is not what I meant to write.
That way, the existence of x is not assumed, but (supposed to be) proved from analysis of the meanings of the terms involved.
Quoting TonesInDeepFreeze
The corollary of axiom M states that A??A , so systems that have axiom M do consider the actual world as one of the possible worlds, since a possible world is simply a world, real or imagined, that does involve any contradictions, and so the actual world is one of them.
My question was a followup. The previous point is more important:
I know the definition of E!xP, but I would like to know the definition of E!x.
I haven't found a full explication of the argument in a modal logic, with all the terms defined from primitives or previously defined terms.
Quoting Amalac
I don't see how that answers my question.
'x' is a variable. And 'E!' is being used as a 1-place predicate symbol. So either it is a primitive predicate symbol, in which case it would appear in the axioms, or it is a defined predicate symbol.
Then I understand it this way: It's a primitive symbol, but there are no special logical axioms for it, and it can be interpreted differently in different models. But we are particularly interested in those interpretations in which it is interpreted as you described.
But that raises the question: What would be axioms that would entail that any model of the theory evaluates E!a as true only in the way you described?
But, even more basically, in ordinary predicate logic, for any given model for the language, the referent of a variable or constant exists in the domain for that model. That follows from the definition of 'model for the language'. So I don't understand how that can be different with interpretations of a modal language.
PS. I do find discussion about an existence predicate in Hughes & Cresswell. If explanation is too complicated for the confines of posts, I'll try to figure it out from that textbook.
And the semantics for 'E' are fixed (in the manner described by Snakes Alive). (I would say that the semantics for 'E' is fixed in the same sense of 'fixed' when we say the semantics for '=' is fixed.)
And 'E' is not just a predicate symbol with no supporting axioms. Rather Hughes & Cresswell describes a logic system that modifies first order logic and also has axioms mentioning the special primitive predicate symbol 'E'.
Also, Hughes & Cresswell addresses the question I asked about how a model can excuse a constant from having a referent in the domain of the model. I'll understand better as I study more carefully.
Then that opens yet another can of worms. I understand what it means for there to be a unique individual having a certain property: E!xP. And in this thread I'm starting to understand an existence predicate: Ea. And I could understand writing 'E!a' to mean the same as 'Ea'. But I don't know what it would mean to say "individual a has unique existence and not merely existence". What would be the semantics for that? (And then I'd have to see what modifications for the logic would be required.)
Where can I read more about a uniqueness predicate in a second order modal logic?
Quoting Snakes Alive
I understand a uniqueness quantification symbol that is followed by a variable and then a formula: E!xP.
But I am not familiar with a syntax, even in second order logic, that has predicate symbol preceding a variable then a formula. Where can I read about that?
Quoting Snakes Alive
The uniqueness quantifier works that way. But where I can see a specification of the syntax and semantics of a second order modal logic that has a uniqueness predicate?
Where 'v' is a variable and 'p' is a formula, E!v[p] is true at w iff there is exactly one individual x in the domain such that p is true at w on any assignment that maps v to x.
That is the ordinary uniqueness quantifier, not a predicate.
So, thus far, we have:
'E' as a predicate symbol before a term: where 't' is a term, Et.
'E' as the existential quantifier: where 'x' is a variable and P is a formula, ExP.
'E!' as a uniqueness quantifier, where 'x' is a variable and P is a formula, E!xP.
But not yet another thing that is a uniqueness predicate (whatever that would mean).
That's not a usage I have happened to have seen. But that doesn't in itself disqualify it.
But at least now it is clear that 'E' as a predicate is a very different kind of animal from 'E!' as a quantifier, which was not clear, except in stages, in your previous explanations. Anyway, I do appreciate your explanations in general; they have helped. Thank you.
Not to say you couldn't construct such a notion, of unique existence, though – the problem with logic and trying to use it to address philosophical issues is that you can do whatever you want.
Sure, but I wonder what it would be.
Do you have any thoughts on my question: How can we have a method of models in which, for certain models, there are constant symbols such that the model does not assign a member of the domain of the model? It throws off the way we evaluate satisfaction and truth in models.
Well, the point of a non-logical constant is that its value is invariant across worlds. You could, of course, have a modal logic where individual terms like constants have different denotations relative to different worlds. And you could then allow that they refer to 'nothing,' say, at worlds where the relevant individual doesn't exist. How you want to represent this formally is up to you – one old formal trick is to use a dummy object, say *, to which the value of all terms that have nothing satisfying them at the world map to. You would then need to make a semantics that deals with the dummy object – you could assign predications of it to false, or to a third undefined truth value, and so on.
I'm having trouble with that. I can specify world in which Donovan doesn't exist, and speculate about the consequences. Someone in that world might say "Donovan doesn't exist", and state a truth...
I seem to be again on a different page from you.
Quoting Snakes Alive
I'm referring to constant symbols. It is not the case that the point is to have the value for a constant symbol to be invariant across models. A model assigns to each constant some member of the universe of the model. There is no requirement that all models agree on what they assign to the constant symbol. Not all models have the same universe, so it's not even possible that they all agree on what they assign to a constant symbol.
Quoting Snakes Alive
Yes, that is what I imagine is the default.
Quoting Snakes Alive
That I don't understand. As I explained, if we fail to assign a member of the universe to a constant symbol, then the methods of evaluation for satisfaction and truth for formulas and sentences per a model falls apart.
And I don't know what 'the relevant individual' refers to in your remark. For domains with cardinality greater than 1, there is no particular individual that must be mapped to from a constant symbol. But every constant symbol must map to some member of the domain.
Quoting Snakes Alive
That seems to me to be about a separate question.
That seems to be a semantical version of the syntactical Fregean method of "the scapegoat" for failed definite descriptions. I understand that method for handing conditional definitions of constant symbols (and can be expanded to operation symbols). This handles those definite descriptions that fail because either the existence or uniqueness condition fails. The method requires a theory in which at least one constant symbol 's' is either primitive or already defined:
If we want a definition
c = the_unique x P
but have not derived the theorem E!xp
then we revise to
(If E!xP -> c = the_unique x P) & (~E!xP -> c = s).
Quoting Snakes Alive
But the problem of failed definite descriptions needs to be dealt with syntactically first (as I did above) or we cannot ensure that definitions uphold the criteria of eliminability.
Anyway, the question I have is not what happens when a definite description fails, but rather, how do we reconcile ordinary semantics for either predicate logic or modal predicate logic with dangling non-denotating constants?
As I understand, you suggest perhaps marking any formula with such a constant as "not satisfied" or with a third truth value, or some other accommodation we would stipulate. But that would mess up the whole context of addressing the first post in this thread and its corollary questions.
In a standard quantified modal logic, however, you need both a domain of individuals and a domain of worlds. And on a standard reading of the identity relation, it is necessary by definition (Kripke made a big deal about this in NN). Thus, if a = b in w, then in all w', a = b.
Therefore, if you make existence claims as follows:
?x[x = a], to mean 'a exists,'
Then if you have '?' quantifying over the domain of individuals, independent of the domain of worlds, then it will have the same value at any world – either it will be necessarily true, or necessarily false.
If you want to get around this, you have to reinterpret '?' so as only to quantify relative to a world, such that each world is associated with a sub-domain of individuals, and '?x[x = a]' is true in w iff there is an individual x in the sub-domain associated with w that is identical to a. If you read it this way, then the formula can be contingently true, and contingently false – true in worlds whose sub-domains contain a, and false in worlds whose sub-domain does not.
Across worlds, not models. A model has a set of worlds, in its frame.
Quoting TonesInDeepFreeze
What do you mean, 'ordinary?' Obviously in a strict sense you cannot reconcile them, since ordinary predicate logic has no notion of a constant that doesn't refer to anything. There's just a domain, and then the interpretation function maps each constant to a member of that domain.
Quoting TonesInDeepFreeze
It's whoever, intuitively, the constant is supposed to refer to. So if 'b' refers to Bob, then it might refer to Bob in all worlds where Bob exists, but to * in all worlds in which he doesn't exist.
Quoting TonesInDeepFreeze
No, it just requires some distinguished object in the domain, like say *.
I'm not addressing philosophy of language or more advanced modal logic, but in ordinary predicate logic, 'Donovan' is a plain name (essentially, it's a constant symbol). An interpretation of the language assigns a member of the universe to the name.
That is different from a definite description. In some theories there might or not be a theorem "There exists a unique individual that wrote the song "Mellow Yellow"". If we don't have that theorem, then see my post above as to what we can do about the definite description "The unique individual who wrote "Mellow Yellow".
(I'm assuming you mean the Scottish singer and not the main character in the movie 'Donovan's Reef' nor the main character in the TV show 'Ray Donovan'.)
Quoting Snakes Alive
Right, my mistake, we're talking about modal logic. But make the correct substitution, and my remarks still pertain both to domains for models with predicate logic and to the domains for the worlds in models for modal logic.
Quoting Snakes Alive
The method of semantics for first order languages as described in any textbook in mathematical logic (or as they describe methods with inessential differences). And am I wrong that also the most basic methods for modal logic in textbooks follow suit? Doesn't basic modal logic stipulate worlds recursively, based first on the ordinary predicate logic clauses and adding the clause for the modal operator?
Quoting Snakes Alive
Exactly. But isn't that method also used in modal predicate logic too for worlds?
Quoting Snakes Alive
Then there has to be an object that is in the intersection of the domains of all worlds. That seems to be a big requirement. Also, there are domains that might have nothing to do with the intuitive intent.
{0} and {1}. Two domains and empty intersection between them. And a constant symbol 'c' with no intuitive referent. I don't see how what you describe is supposed to work.
[EDIT: The following is okay but perhaps no longer relevant anyway.]
Quoting Snakes Alive
I was referring to the syntactical side of things for definite descriptions, not to your semantical method.
No, I don't.
That seems to imply that Donovan exists in possible world in which he does not exist.
I see now that I face an obstacle in talking about constant symbols and terms with you. That obstacle is that I am using Hughes & Cresswell, but their quantified modal system has no constants, no terms other than the variables themselves, and '=' is introduced only in a later chapter. Their language has only: universal quantifier (with existential quantifier defined), individual variables, sentential connectives, relation symbols, and the necessity operator.
Of course, that is enough, since in theories, constants and operation symbols can be defined from relation symbols.
But it makes it difficult for me to attend to the details vis-a-vis your remarks, because I lack a reference for how terms are dealt with semantically in a definitive textbook.
So, which textbook guides you the most in this subject? I can see whether I can get it cheap enough.
Quoting Snakes Alive
I take that mean that the letters 'a' and 'b' are variables in the meta-language and not constant symbols in the object language. And with that, the above sentence seems to be the way we should understand the semantics.
Quoting Snakes Alive
But there 'a' is used as constant symbol in the object language.
For me, 'Ex x=a' is just a trivial theorem of identity theory. For any term t, we have the theorem:
Ex x=t
Quoting Snakes Alive
As a theorem of identity theory, isn't it true in all models for a language for quantified+identity modal logic?
Quoting Snakes Alive
I don't see how you can use 'a' as a symbol in the object language when you write 'Ex x=a', but also as a symbol in the meta-language when you talk about 'a being in a domain'. Also with 'x', but there I can reword in my own mind to make it work, while, in this particular situation, I can't do that with the way 'a' is being used, and as I don't think this is merely pedantic but rather it confuses me as to what really is being said.
I think I might have had an incorrect premise that modal semantics evaluates truth in these stages: first per world and then per model. Rather, perhaps I can answer my own questions if I dispense with that premise and view semantics as done "top-down" for the model overall. That premise disallowed me from better understanding your posts. I'm going go back to studies I had forgotten a long time ago to see whether I can correct myself now. (If you have a textbook to recommend, then it would be appreciated.)
I don't think you even need S5 for it? Given that you can choose world elements.
W1={egg, bacon}
W2={egg}
The statement E: "At least one entity in this world is an egg"
E's w1 valuation, true.
E's w2 valuation, true.
E is true in all possible worlds, so it's necessary.
If you'd like, call the relation between W1 and W2 the breakfastibility relation. Two worlds are connected iff they consist of only breakfast goods. That accessibility relation is an equivalence relation on the set of possible worlds.
So yes, with that world set and that accessibility relation, I believe the existence of eggs is necessary.
#modal logic isn't metaphysics
The next question is, must there be an individual which exists in every possible world?
It seems not, since we can have
W1={egg, bacon}
W2={toast, coffee}
...and now for breakfast.
In which set of possible worlds?
{egg}, {egg, bacon}
Has "egg must exist" as true, where "eggs must exist" is treated as "necessarily there is at least one egg in every possible world".
But as you noticed:
{egg}, {bacon}
doesn't have this property, so the divine mandate for egg's existence disappears.
Does an entity necessarily exist in all possible worlds? Set out a set of possible worlds and an accessibility relation and we'll talk.
If you don't set out an accessibility relation and set of possible worlds, it looks like quantifying over sets of worlds rather than worlds - ie does there exist a set of worlds in which eggs necessarily exist - yes , I just summoned it vs do eggs necessarily exist in each possible world in all sets of possible worlds - no, I just summoned one where eggs don't -.
...and that's the point. Possible worlds are constructed by fiat, not discovered. Hence arguments which presume to demonstrate a being that exists in every possible world can do so only by assuming there is such a being.
Is logic constructed by fiat? Possible worlds depend upon what is logically possible/ impossible (that's how they are defined), and it seems we don't construct “by fiat” what is and is not logically possible.
That's a different question. A possible world comes about as the result of a "what if..."; then we can see if that "what if..." leads to a consistent story or not. If it is inconsistent, then there can be no such possible world.
That is, logic gives us a grammar with which to judge our statements.
Ok, so we more or less agree then.
But my point was that a statement such as “It is not the case that the sun both was and was not a star, at the same time and in the same sense ” doesn't seem to be merely a truth about “grammar”, but also about the world, and it would seem like that proposition was true before any human being verbally constructed it or thought of it.
I am aware this leads to other difficult questions, related to philosophies like psychologism and the disputes between the advocates of that philosophy and people like Frege and Husserl, so I won't talk about it anymore here to avoid derailing the thread.
Oh, I think it pertinent.
At issue in this thread is whether one can deduce the existence of a being from logic alone. I think that it should be in principle not possible. I think this because logic is about what we can say, and not about the way things are.
Quoting Amalac
If we found a situation in which there was an apparent contradiction, what we would do is to re-think how we set out contradictions. Consider "It is both a wave and a particle"; the prima facie contradiction dissolves in the mathematics. Is Pluto both a planet and not a planet?
Ok then, do you accept that the Law of Contradiction is necessarily true? For there sake of this discussion, I'll maintain that it is.
If so, you must admit that the proposition:
Quoting Banno
... can't be true if you can replace “not a wave” with “a particle” without changing the meaning of the sentence.
If that proposition is true, then particle cannot mean “not a wave”, where both of those words “wave” in that sentence mean exactly the same thing. If they don't, then that proposition does not assert that A and ¬A, since the word wave (in the sentence resulting after you replace the meaning of “a particle”) would not mean “something that is not a particle”. It would have changed meaning in that case, and thus no longer have the same sense.
That's why the Law asserts that you can't assert A and ¬A, unless the second A has a different meaning/sense.
A world = a sub-domain?
Of the set of possible worlds under consideration, I think so.
Within worlds, you still have the base logic and whatever domain of discourse is in that world.
I visualise possible worlds and accessibility relations as a graph whose nodes are sets (worlds) and whose links are the accessibility relation.
Possibility of X in a world is having a neighbour world (which can include itself) in which X evaluates as true.
Necessity of X in a world means having all neighbour worlds of that world evaluate X as true.
Possibility of X is "can I transition in one step to a world where this is true?"
Necessity of X is "if I step one world away, is X true no matter where I step?"
That means you can flesh out statements about possibility and necessity in terms of graph connections.
So a claim like "X is possible at world W" means "W has at least one neighbour where X evaluates as true". If W is a neighbour of a world where X is true, then X is possible at W. If all of W's neighbours have X as true, then X is necessary at W.
For @Banno, what S5 does is make the graph a partition/equivalence relation. It means that for all worlds:
(1) Every world is its own neighbour - that means if X is true in all worlds connected to W (X is necessary), then since W is connected to W (reflexivity), X is true in W. Necessarily A implies A.
(2) If W1 has another world W2 as its neighbour, then W2 has W1 as its neighbour. That means if X is true in W1, X must be possible in W2, so X is necessarily possible in W1 - every time X is true at W1, it will force all of W1's neighbours to have X as a possibility, which means possibly X is true in all of W1's neighbours, which means X => necessarily possibly X.
(3) If W1 is a neighbour of W2, and W2 is a neighbour of W3, then W1 is a neighbour of W3 (transitivity). That means that if X is true at W1, and W3 is a neighbour of W2, and W2 is a neighbour of W1, then X is true at W3 through the chain of links . This is a "walking" condition, it makes being a neighbour the same idea as being connected. In terms of modality, what this means is that if X is necessary at W1, and W1 is a neighbour of W2, then X is necessary at W2. Why? Well in order to be a neighbour of W2, it would also have to be a neighbour of W1, and we know all of W1's neighbours have X evaluate to true since X is necessary.
You can see that this is a fertile ground for ontological arguments - if god is possible and god is possibly necessary, then god is necessary, then god exists...
So, W1 = some world (among others) whose domain is {egg, bacon}?
Yes.
Set of all possible worlds there is:
{W1, W2}
W1 is {egg, bacon}
W2 is {egg}
Not always. There are things that Logic is undeniable cause it comes simply from truth. When logic is based in pure truth facts it is indeed the way things are!
Ah, so not some one among several with the same domain.
So worlds are not in general to be identified by their domain?
That is just a nice thing about your example?
I don't see how I suggested that? Explain it to me please.
Well... equals?
I imagine that equality works on worlds too. I'd say that two worlds are equal when they consist of (all and only) the same elements / when they evaluate the same for all stuff/statements within 'em.
If you have a clock, the set of seconds in a minute form a set of worlds, and the ticking of one second to another forms an accessibility relation - t -> t+1 for all seconds t.
In that situation, two worlds would be equal if they were the same time instant.
Would you please tell me in what book or article I can read the stipulation of semantics for quantified modal logic you use?
Quoting fdrake
What are W1 and W2? I would guess they are domains for two different worlds, since you refer to "world elements".
A world is not ordinarily just a domain, but rather (intuitively, as the full definition is more technical) a domain for the world and a set of relations on the domain for the world. Or, if the semantics stipulates just one overall domain, then a world is just a set of relations on the domain.
So does 'egg' stand for an individual? Or does it stand for the 1-place relation 'is an egg'?
I haven't yet read the rest of the posts, so maybe I'll find out more. I have more questions, but this is a start.
I can't because I'm mostly making it up from SEP and university memories. It is quite possible that what I said was entirely wrong!
I was envisioning the set of worlds:
{W1, W2}
With W1={egg}, and W2={bacon, egg}. The accessibility relation was just R = { (W1,W2), (W2,W1) }. I suppose more formally each of these has a hierarchy of statements regarding eggs, bacon, and the whole underlying logic thrown into them.
Quoting fdrake
So {e b} and {e} are domains. So W1 and W2 are domains. But you say that W1 and W2 are worlds. As far as I can tell, that is conflating 'world' with 'domain for a world'.
Could you explain the difference so it is very clear to me please?
Quantified modal logic is pretty technical. And I am rusty in my brief study of it long ago. So I might not state some things correctly, but I'll do my best. Also, the subject is complicated by the fact that there are various equivalent and pretty technically complicated ways of describing the semantics, while also there are alternatives to choose from. One of the choices is whether there is just one domain for the model or whether there are different domains for different worlds.
I'll say this in an intuitive way (it can be made more formally rigorous):
In predicate logic, a model has a domain, which is a set of individuals, and a set of relations on the domain. So a model is a "state of affairs". In predicate logic, models don't have worlds. Rather, the model is the world. A model (a world, or a state of affairs) is not just a set of individuals, but rather it is a set of individuals and facts about those individuals. The facts are captured as relations (predicates).
In predicate modal logic, models are more complicated. A model for predicate modal logic (I'm leaving out some other stuff here) is a set of worlds and an accessibility relation on the set of worlds. Again, a world is a domain, which is a set of individuals, and a set of relations on the domain. We can choose two different stipulations:
(1) There is only one domain for the model. It is the domain for all the worlds.
(2) Each world has its own domain and that domain may be different from domains for other worlds.
in discussions about existence in worlds, I think there could be a lot riding on which of those two contexts we are in, so we should be clear as to which of the two we mean.
/
By the way, through my conversation with Snakes Alive and looking more closely at some textbooks, I am starting to get a better idea of how an existence predicate works and also how evaluation of truth can work when some of the constants don't map to any member of certain of the domains in (2) above.
/
The textbooks I'm selectively reading now are:
A New Introduction To Modal Logic - Hughes & Cresswell
Logic, Language, And Meaning Volume 2 - Gamuit
:up:
Quoting TonesInDeepFreeze
Thank you, I see. Let me see if I can rephrase the issue.
Imagine a world at which God exists, this world has a domain, and one of the entities in that domain is God. The phrase "God exists" is true in this world, but what set is that existential quantifier quantifying over?
"There exists at least one God" could be quantifying over the domain of the world in particular, in which case "There exists" is a perhaps a merely possible proposition - in a world where there wasn't a God, it would be false.
But it could be that "There exists" is quantifying over the union of the domain of all worlds, in which case if something exists in one world, it exists in all of them - since "exists" is only looking at the shared domain of entities which are distributed over all the worlds. That would make existence necessary existence for everything (not just God).
I think that's a separate issue from the issue with my equivocating between worlds and world domains? The appropriate scope of the existential quantifier "within world" so to speak is distinct from what a world is "made of" - my construal of it as a domain of objects missed out that it's actually a formal language structure within the world as well as there being a domain in the world which the formal language structure takes its (at least actual) truth-value cues from.
But doesn't that mean that the Law of Contradiction reflects some kind of a priori, given structure of the world, such that the world must always follow that Law, and doesn't merely stablish a rule of grammar or a statement about what we must believe?
Do you disagree with any of this?:
[quote= Russell]When we have seen that a tree is a beech, we do not need to look again in order to ascertain whether it is also not a beech; thought alone makes us know that this is impossible. But the conclusion that the law of contradiction is a law of thought is nevertheless erroneous. What we believe, when we believe the law of contradiction, is not that the mind is so made that it must believe the law of contradiction. This belief is a subsequent result of psychological reflection, which presupposes the belief in the law of contradiction. The belief in the law of contradiction is a belief about things, not only about thoughts. It is not, e.g., the belief that if we think a certain tree is a beech, we cannot at the same time think that it is not a beech; it is the belief that if the tree is a beech, it cannot at the same time be not a beech. Thus the law of contradiction is about things, and not merely about thoughts; and although belief in the law of contradiction is a thought, the law of contradiction itself is not a thought, but a fact concerning the things in the world. If this, which we believe when we believe the law of contradiction, were not true of the things in the world, the fact that we were compelled to think it true would not save the law of contradiction from being false; and this shows that the law is not a law of thought.[/quote]
I wish we had a specific formal semantics that together we reference. Otherwise, we risk getting lost in the twists and turns of an analysis bereft of a road map.
In predicate logic, "what does the quantifier range over?" has a simple answer. But in predicate modal logic, it's not immediately clear to me what "the quantifier ranges over" means. The more pertinent question instead might be, "how is the truth value of a quantified statement evaluated?" At least I can say that I don't find mentions of "the quanitifer ranges over" in stipulating the semantics. Instead:
(1) There are two methods: (a) just one domain per model or (b) domains for each world.
(2) Evaluation of truth (per a model) of a sentences with quantifiers.
Quoting fdrake
In the context you set up, I don't like that sentence. First you used 'God' as a name. But here you use 'God' like a predicate ("There exists at least one [g]od" I would take to mean "There exists an x such that x has the property of being a god" ).
But perhaps the best way to proceed would be for you to set out exactly what you think Russell's argument is in the piece you quote; because I don't see an argument there.
Quoting Banno
There is an argument, it's just brief:
1. Either the Law of Contradiction states merely what we must believe, and what we can't believe, or it also asserts how the world necessarily is and must continue to be like.
2. It is impossible for the Law of Contradiction to
ever be false (interpreted as an assertion about the world, and not merely about our thoughts).
3. If the Law of Contradiction merely stated that we can't help believing that a thing cannot have a property X and a property ¬X at the same time and in the same sense, then the Law could be false in spite of the fact that we can't help believing in it, thus it is possible that the Law of Contradiction (in the sense in which it is applicable to the world) is false.
4. 3 contradicts 2.
5. Therefore, the Law of Contradiction must be a fact about the world. ( From 1, 2, 3 and 4)
Well yes, since all proofs assume the Law, it itself cannot be proven, as Aristotle pointed out.
But it's just blindingly obvious, is it not? I mean, if we can't be certain about that, we can't be certain about anything.
Well...
That's the point in question.
Something philosophically quite odd happens when someone uses a name. There's a way in which the use ceases to be words and becomes the thing. If I say here that I am replying to Amalac, it doesn't mean that I am replying to a word, but to a person.
So there is a way of using words that stops being about language and becomes the world.
Perhaps it would help to think of it like this: Russell supposes that either the law of noncontradiction is a fact int he world or a figment of language. But actually, it is both. Even as Amalac is both a word and you.
Not all proofs use the law. Indeed, the law is not even usually one of the logical axioms.
Quoting Amalac
Yes it can. (Here I'm taking the law in the sense of a single instance. For a schema we could adjust):
(1) Trivial. If the law is an axiom then it also provable by the rule of putting an axiom on a line.
(2) If it is not an axiom, then it is still provable from any set of logical axioms for a system that is complete in the sense of proving all validities.
(3) Trivial. It is provable from any inconsistent axiomatization.
Examples for all three cases include all the most common Hilbert style and natural deduction style systems (excepting those that stipulate that all tautologies are axioms, such as Enderton).
Quoting Banno
I don't think so, here he says:
His “only” implies that he holds that the belief in the Law of Contradiction is both about thoughts and about things.
Quoting Banno
A word is not the same thing as that to which the word refers.
So if Amalac, in the following sentence, means “the word Amalac” then I am not Amalac.
If it means “the person writing this right now”, then I am Amalac.
The only way you can assert that “Amalac” is both a person and a word is through either denying the Law of Contradiction, or through the fallacy of equivocation.
Quoting TonesInDeepFreeze
They don't assert the Law of Contradiction explicitly, but they must assume that it is certain implicitly, otherwise it is not even possible to talk meaningfully, as Aristotle pointed out.
Quoting TonesInDeepFreeze
Only if you assume that it is not also the case that the Law is not provable by the rule of putting an axiom on a line, which requires the Law of Contradiction and makes the “proof” circular. It certainly would not convince a LNC sceptic.
And the same applies for the rest of the “proofs”.
They prove it as a theorem. Of course, our motivation for the system would include proving it as a theorem.
Quoting Amalac
"Talk meaningfully" is a large and undefined rubric.
Quoting Amalac
When I resolve the double negative, we get "It is the case that the law is provable by the rule of putting an axiom on a line", which is what I said in the first place.
Quoting Amalac
It is petitio principii. But ordinary logic allows it, otherwise you could never put a premise onto a line.
Quoting Amalac
First, I don't know why you put 'proofs' in what I surmise to be scare quotes.
Second, (1) is not petitio principii, unless you hold that proof from logical axioms is petitio principii. (2) Is basically an example of the principle of explosion itself closely related to LNC.
Yeah, all of that. But that misses the point. There's a difference between "Amalac" and Amalac. But no difference between Amalac and Amalac.
Quoting TonesInDeepFreeze
How do you know that (Edit: sorry, missed adding: “it's not the case that” here, I'm very tired from work) they also don't prove it as a theorem? By accepting the Law of Contradiction as an axiom? Then you are agreeing with me: It cannot be proven so it must be assumed as an axiom. By definition, axioms are accepted as true without proof. So the question is: Why should a LNC sceptic accept that axiom in the first place?
My point is that your proofs only work if someone accepts the Law as an axiom, if they don't, then your “proofs” are circular.
Quoting TonesInDeepFreeze
Without assuming the Law as true without proof, for all I know you may have both asserted and denied everything you've said this far. That's what I mean.
Quoting Banno
Obviously not, but what exactly did you mean then? What point were you trying to get across here? :
Quoting Banno
Ok, but like I said, Russell doesn't disagree and neither do I:
Quoting Amalac
Well, I thought you held that the Law of Contradiction was only a rule of language, but now you are saying that it does reflect how the world is.
What I don't understand is how that's consistent with this earlier statement of yours:
Quoting Banno
I do.
Quoting Amalac
...so does language.
Logic is about the rules of language. Language is about how the world is.
I'm not seeing a problem.
Quoting Banno
The problem is that I don't agree with this statement: I think logic is about what we can say, but also about the way things are (in the sense in which Russell also holds this in that last statement of his that I quoted previously).
So how do you explain this disagreement when we seem to agree about the rest? That's what's got me a little puzzled, unless you changed your mind.
Yeah, alright, take it that I misspoke, and look a the following Sentence:
Quoting Banno
What I was aiming at is that logic is the handmaid of what is the case. One of the things we do with language is that when it doesn't seem to show us what is the case we change what we are saying.
DO you agree with that?
One of the problems here is that an essay is needed but I don't have time for that.
That's a very silly question. I don't know they don't prove it as a theorem, since it is not the case that they don't prove it as a theorem, and I can't know that which is not the case. Are you trolling me?
Or maybe you meant to type: How do you know that they also don't prove it is not a theorem?
And I answered that in a previous post. When we remove the double negative we get: How do you know that they prove is a theorem?
Quoting Amalac
We may agree with LNC and use LNC without LNC being an axiom.
Quoting Amalac
That's one notion. But another definition of 'axiom' is purely syntactical.
If your point is that LNC is endemic in reasoning, then I agree with you on that point. But that doesn't entail that LNC must be an axiom. (And I'm putting aside the matter of paraconsistent logic.)
Quoting Amalac
Now you're just reasserting a claim that I refuted.
Quoting Amalac
It is fine to have it as a logical axiom, since it is logically true. Sceptics should learn that it is logically true.
Quoting Amalac
That's false. The proofs can be mechanically audited whether the auditer knows of LNC as an axiom or not. Indeed, even for everyday reasoning, probably most people haven't even heard of LNC, especially the notion of it is an axiom. And that does not contradict that good reasoning (other than dialethistic) conforms to LNC and sometimes uses it - either as an explicit or implicit principle.
Quoting Amalac
You are reasserting a claim that I refuted.
Quoting Amalac
It is possible for one to assert and deny a proposition. And it is even ubiquitous that people assert propositions that are inconsistent with other propositions. So probably what you mean is that it is not possible to be correct while both asserting and denying a proposition. Then your question seems to be how do we know that contradictions are not the case. But the question of how we know things is different from the question of what axioms we choose. We may know that a statement is true by reasoning from different axioms sets that each yield the statement as a theorem. It is not required that LNC be one of the axioms. If, as we ordinarily do, we require a system that is complete in the sense of proving all validities then it is only required that LNC at least be a theorem even if not an axiom.
Logic is about entailment and inference. Logic concerns both syntax of language and meaning with language. Meaning includes denotation, which concerns individuals in the world. And meaning includes evaluation of truth of sentences, which concerns states-of-affairs in the world. ('the world' may be taken in such senses as the real world, possible worlds, fictional worlds, or mathematical worlds.) Generally speaking, logic doesn't say what is the case in worlds, but logic is not just rules of language unless included in those rules are the means by which we relate language to objects and what is the case in the world. Maybe said this way: Logic does not concern what is the case in the world, but rather logic does include concern of HOW language relates to what is the case in the world.
Quoting TonesInDeepFreeze
Yes, that was my bad, I'm too tired from work, I wrote too quickly and forgot to add that other negation, I'll try writing more slowly now. (Also english is not my mother tongue so please have mercy on me).
Quoting TonesInDeepFreeze
Let's start here, what's this purely syntactical definition of “axiom” you speak of?
Quoting TonesInDeepFreeze
But they must still accept it implicitly without proof (that's what I mean by “accept it as an axiom”). Of course we don't make all our inferences explicit, but we still implicitly infer many things, which require us to accept some propositions as true without proof, among which is the Law of Contradiction. As Aristotle said: It is utterly impossible to prove everything.
Someone who doubted that the Law of Contradiction was true, would not accept any proof that assumed, without proof, that the Law of Contradiction is true, he will demand that you prove it without having it as an axiom or assuming its truth in any way, since he won't accept circular arguments.
In one sense, it is trivially true that if you have the Law as an axiom, then you can prove that the Law is true. Likewise, if I have “God exists” as an axiom, I can prove that God exists.
Or if someone asked you to prove that 5=5, and you told them: “If you accept that A=A is true for any number you substitute for A, then it necessarily follows that 5=5” he may reply: “And why should I accept that A=A?”, and so it becomes clear that you can't convince him that 5=5. The same thing happens with the Law of Contradiction sceptic, you also can't convince them that the Law is true.
Quoting TonesInDeepFreeze
That's just an assertion, the LNC sceptic will demand a proof for it. They want to learn why it's true.
Quoting TonesInDeepFreeze
Could you please show me a proof of the Law of Contradiction that didn't have it as an axiom, and didn't assume that it is true,without proof, in any way?
Quoting Banno
Of course I do agree that we should make language fit what we know about reality and not the other way around. It may happen that, for example, we were wrong in thinking some object had a certain characteristic that we labeled X, because we made a wrong inference based on misleading or insufficient evidence, and later learned that that inference was wrong, and that the characteristics of that object were such that it didn’t fit in the definition of X, and in that case we would no longer say that it has that characteristic X.
But anyway, how does that lead to the claim that it is not possible to prove the existence of anything a priori, that is: that no analytic proposition can also be existential in its content? I suppose you may say that such a proposition would try to make reality fit into language rather than the other way around, but supposing God did exist, that would not be the case, we would in that case make language fit into reality.
Yes, I mentioned it as a possibility. It was ungenerous of me to wonder whether you were trolling when Occam's razor would better suggest that you merely made a typo.
But again it boils down to:
Quoting TonesInDeepFreeze
So I wonder why you still haven't recognize it, as I had already mentioned it twice.
Quoting Amalac
A theory is a set of sentences closed under deduction.
(Some authors define a theory to be any set of sentences. And Enderton says a theory is a set of sentences closed under entailment, which is not syntactical. But they are equivalent with the completeness theorem, so, in that context, no harm done by saying 'closed under deduction' rather than 'closed under entailment'.)
A set of formulas S is an axiomatization of a theory T if and only if all members of T are provable from S. For a given axiomatization S of T, an axiom P is a member of S.
Quoting Amalac
That's a quite broad notion of axiom. It's not what 'axiom' usually means in a context such as formal logic, or as far as I know, even very much, if at all, in an informal discussion about reasoning.
It's not the case that sets of axioms are always independent, but independence of axioms is something we ordinarily desire.
Quoting Amalac
Of course, I understand that in the everyday sense. But there is also a technical (though granted, pedantic) sense in which even axioms are provable. Also, for decidable theories, it is a decidable axiomatization where the set of axioms and the set of theorems are the same set.
Quoting Amalac
That is incorrect. To accept a proof does not require accepting the truth of the premises. To accept a proof may be merely to audit that each step adheres to the rules of the proof system (the logic system). Thus, if the proof system is sound, then the assumption of the premises (no matter whether they are true of false) entails the conclusion.
Quoting Amalac
We can prove LNC is valid (even stronger than true). So we have the consequent of your hypothetical, so we don't need the antecedent. And we prove LNC is valid by proving that it is true in every model. That does not require proof in the logic system itself. However, its validity is entailed by proof in any logic system that that proves only validities (the soundness theorem).
Quoting Amalac
First, sentences of the form 'x exists' are not clear. One way to make them clear is with an existence predicate, but that is usually an advanced topic in predicate modal logic.
So I'll use this instead: G = "there exists a being that is omnipresent, omniscient, omnipotent, and omnibenevolent".
Then, yes, taking G as an axiom proves G as a theorem, but it doesn't prove that G is true. It only proves G is true if all the axioms used in the proof are true. So, since G is an axiom used in the proof, if it is false, then, though we have proved G, we have not proved that G is true.
Quoting Amalac
No, it's not just an assertion. It's a theorem about propositional logic. And it is reducible, in a sense, to a theorem about Boolean algebra. And its proof is reducible to finitary operations, which are reducible to auditing the execution of an algorithm. So (heuristically speaking) we may say that at the root of the question is ability to audit the execution of an algorithm. Of course, it's hard to imagine such an ability in a person who was so delusional that they claimed to witness '0' and '1' written in the same space when only one of them was written in that space. But that is not the same as going all the way back up the chain of reductions I just described to say that LNC must be an axiom.
That might be tedious for me to type out, and if you are not familiar with proof calculi for propositional logic, then it wouldn't be of much use to you.
I recommend 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar. Within about a chapter you could assign yourself the easy exercise of deriving LNC in the natural deduction system there.
Quoting TonesInDeepFreeze
Oh, well if that's what you meant then obviously, if one of the premises is the Law of Contradiction or assumes the Law of Contradiction, the sceptic can just grant the validity of the proof while denying both that premise and the conclusion. He would say: “yes, obviously if the Law of Contradiction is true, then the Law of Contradiction is true, but I'm questioning the claim that the Law of Contradiction is true, not the implication”.
Quoting TonesInDeepFreeze
Right, so the same thing that you say about G, one could say about the LNC: we have proved the LNC, but we have not proved that the LNC is true. The sceptic wants a proof that the LNC is true, that's the proof I was refering to.
However, it seems you say that unlike for G's truth, there actually is a proof that the LNC is true (not merely a proof of the LNC), is that right? It seems it's the one you are refering to here:
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Thanks for the recommendation, I'll read when I have the time (if I can find the book, that is).
Yes, it's the proof of a theorem about propositional logic. And we prove not just that LNC is true in a particular model but moreover that it is true in all models (i.e. that it is a validity).
I realized that a natural deduction proof of LNC is also trivial. This one has different notation, but it is essentially the same as in Kalish, Montague, and Mar and other common systems:
1. P & ~P assumption {1}
2. P simplification {1}
3. ~P simplification {1}
4. ~(P & ~P) RAA [1, 2, 3]
The ease of getting LNC from RAA illustrates the closeness between RAA and LNC. So we are tempted to think LNC and RAA are conceptually "the same". But from LNC to RAA is only one direction. The other direction is from LNC to RAA. So we ask how easy is to get to RAA from LNC. That depends on the particular Hilbert style system we use. A Hilbert style system is a combination of axioms and rules, and RAA in natural deduction systems, unlike LNC, is a rule, not a theorem nor an axiom. So we would be deriving a rule from a combination of axioms and rules.
So there are two tasks: (1) Derive LNC from a Hilbert style system that does not have LNC as an axiom, and (2) Devise a system, without RAA, in which LNC is an axiom, and derive RAA.
Task (1):
A common Hilbert style system (call it 'H1') is given by:
Capital letters are meta-variables ranging over formulas.
Axioms:
A1. P -> (Q -> P)
A2. (P -> (Q -> R)) -> ((P -> Q) -> (P -> R))
A3. (~P -> ~Q) -> (P -> Q)
Rules:
R1. Any axiom can be put on a new line.
R2. If P is on a line and also (P -> Q) is on a line, then Q may be put on a new line.
Definition:
D1. P & Q stands for ~(P -> ~Q).
[end description of H1]
From D1, LNC is:
~(P -> ~P)
And that is what we need to derive. To reduce tedium, the natural suggestion is to first derive RAA as a rule, then apply RAA as previously in this post. Deriving RAA as a rule is a lot of steps, but it is fairly straightforward to do and I think it's fairly common in textbooks. So, I'm not going to type it here.
But since LNC is couched with & and ~, an even clearer approach would be to take & and ~ as the primitives. Then put the axioms in those terms for system H2:
D2. P -> Q stands for ~(P & ~Q)
So A1 - A3 become:
A4. ~(P & ~~(Q & ~P))
A5. ~(P & ~~(Q & ~R)) -> ~(~(P & ~Q) & ~~(P & ~R))
A6. ~(~(~P & ~~Q) & ~~(P & ~Q))
Take an instance of A4:
~(P & ~~(P & ~P))
Eliminate double negation:
~(P & (P & ~P))
Apply associativity:
~((P & P) & ~P)
Apply idempotency:
~(P & ~P)
So, lo and behold, there's LNC.
So we might be tempted to say that H2 itself is a Hilbert style system with LNC as a "subschema" of one of the axiom schemas; that LNC was there all the time, hidden but implicit. Ah, but not so fast there, pardner. First we have to derive rules for double negation, associativity, and idempotency.
A more elegant argument is simply to point to the completeness theorem for H2:
H2 is entailment complete.
LNC is a validity.
Therefore, LNC is a theorem of H2.
But to be convinced of the conclusion, we need to witness the proof that H2 is entailment complete and witness the proof that LNC is a validity.
Task (2)
Offhand, I don't know of a Hilbert style system that has LNC as an axiom. We could add LNC to the axioms of H2, but that would result in system with a non-independent axiom set. That's logically permissible, but it is inelegant and it reduces the challenge, which might not want to do. So the interesting question is to find an independent and entailment complete set of axioms that includes LNC.
Lemon gives this brief natural derivation:
(1) P & ~P Assumption
(2) ~(P & ~P) 1, 1, RAA
There's the predicate version to consider, too:
?(x) ~( F(x) & ~F(x) )
And since modal logic presumes the theorems of predicate and propositional logic as necessary,
? ~(P & ~P)
? (x) ~( F(x) & ~F(x) )
We might go the other way, and take LNC as a given in order to derive RAA.
Right, that was my Task (2). But I mentioned considerations about in my last paragraph.
The natural way to do it is first to derive the deduction theorem:
Where G is a set of formulas and p and q are formulas, then:
If G u {p} |- q, G |- p -> q
Then make that a derived rule. Then deriving RAA as a rule is only a few steps away.
It also occurred to me that someone who thinks LNC is always primary could argue:
Even given the explanations about deducibility, LNC entails (entailment is semantic) a subset of the instances of Axiom A4 and Axiom A4 entails LNC, so they are logically equivalent. That's true, but it doesn't advance any argument, because trivially all the validities are logically equivalent anyway.
OK.
This line of thinking came about as a result of @Bartricks's claim that LNC is true but contingent. Now he doesn't have a consistent leg to stand on, but he might be understood as denying
? ~(P & ~P),
or
? (x) ~( F(x) & ~F(x) )
or both. (edit: ah - the second is just an instance of the first... )
It should be apparent that it is inconsistent to deny these and yet assert ~(P & ~P).
In his case there seems to be a failure to understand that ? P ? ~?~P. I've never seen a denial of this elsewhere.
This is how "...exists" must be used in the ontological argument.
So one version fo my OP would be to ask if (x) ?(E!(x)) - if there is some being that exists in all possible worlds.
But it seems clear from the discussion that E!(x) must mean that x is in the domain; and since the domain is decided more of less by fiat, in effect prior to any deductions.
Hence I don't see that E!(a) can be the result of a deduction.
Quoting Banno
I don't know what you mean by that. There is mention of systems and contradiction, which is syntactic, and mention of worlds, which is semantic. It's not clear to me what relationship between them you are asking about. Also, 'possible world' is relative to a given world and the accessibility relation, so what do yo mean by 'in every possible world' without reference a single world as the "base" from which other worlds are accessible to it? Or maybe you just mean just "in all worlds", so that we could delete the word 'possible'?
But more fundamentally, S4 and S5 are not predicate modal logics. So there is no semantics about individuals.
I'm interested in your question, but I wonder whether you might reformulate it.
Would you please link to it?
Since that post, I am reading to find more about the existence predicate (that I would call just 'E' and not 'E!' as others have), but I haven't yet caught up to seeing exactly how it works.
I'll rewrite that in text only ('A' for universal quantifier, 'N' for 'necessarily', 'X' for the existence predicate.
Ay N Xy
But what do you mean by "if"? Are you asking whether it's a theorem of some given system? (That system wouldn't be S4 or S5, since those are merely propositional logic systems, or do you mean a quantified version of S4 or S5?) Or are you asking whether there are models in which the formula is true?
Why not this trivial model?
w has domain = {0}
W = {w}
R = {
f = {'c' 0}
The thread was https://thephilosophyforum.com/discussion/11355/necessity-and-god/p1. This thread was an attempt to formalise to some degree the intuition in the OP of that thread.
That intuition is that for any individual, one can give consideration to what the world would be like without that individual.
But if there is a necessary being, then it would not be possible to give consideration to a wordl without that being.
Sorting out this incompatibility has helped keep me from doing some paving in the front garden for a few days now.
The discussion with Bart is in that thread, form about here: https://thephilosophyforum.com/discussion/comment/564170
Too hard trying to find where he made the claim about LNC. That's okay, I guess it doesn't bear on this discussion.
It's here:
Quoting Bartricks(My bolding)
But it'd be maltreating a deceased equine to continue. Once LNC is rejected, reasoned discussion follows. Lesson is, don't pay any nevermind to Bart.
https://thephilosophyforum.com/discussion/comment/565970
There's a particular subject in history of ideas that has bearing on this. It has to do with the advent of nominalism and voluntarism in medieval philosophy. The idea is that the scholastics, such as Aquinas, believed that the principles of logic were inviolable. This is not actually to claim, as I understand it, that God was bound by logic, but was in some sense rational, even if in other senses could be supra-rational. However the voluntarists, lead by the Franciscans, regarded this as an artificial limitation on divine omnipotence. This is explored in some depth in Michael Allen Gillespie's excellent 2009 book The Theological Origins of Modernity.
[quote=Christopher Blosser; https://www.goodreads.com/review/show/707174301] Gillespie turns the conventional reading of the Enlightenment (as reason overcoming religion) on its head by explaining how the humanism of Petrarch, the free-will debate between Luther and Erasmus, the scientific forays of Francis Bacon, the epistemological debate between Descarte and Hobbes, were all motivated by an underlying wrestling with the questions posed by nominalism, which according to Gillespie dismantled the rational God / universe of medieval scholasticism and introduced (by way of the Franciscans) a fideistic God-of-pure-will, born of a concern that anything less than such would jeopardize His divine omnipotence.[/quote]
Apparently.
So what remains is for an account to be given of how to talk about a god without such limitation on divine omnipotence.
Can you do that?
The problem is that (p & ~p)?q; the account becomes anything.
Probably not without a 5,000 word essay which nobody would read anyway.
It's just something which I think is a background factor behind what you've been discussing.
One potential problem in my way of thinking about logic is that it ought not be possible to give an account of god in the way proposed. If logic is just grammar, then god not being able to perform contradictions is outlawed simply because if god did do something that was apparently a contradiction, then the response would be to re-formulate the apparent contradiction so that it no longer was a contradiction.
Now exactly what that might look like remains hidden.
One way for me to knut through it would be to give consideration to claimed paradoxes - the Trinity, for example, and to ask what it would really be like for an individual to have three essences, or whatever the explanation is.
I believe that what you are looking for is found in the theorems reflexiveness, which according to https://plato.stanford.edu/entries/logic-modal-origins/#LewiSyst is given as the first axiom in S1.
Its been a long while since I did formal modal logic, and I now just glanced at it, so maybe this is an answer in full unsuited for answering your question. If so, I beg your pardon.
Also, I think your question really is about the metaphysics of modal realism. Maybe it may be worth your while to take a stroll down that street.