## God exists, Whatever thinks exists, Fiction: Free Logic

• 15.1k
Free logic: logical systems that do not insist that individual variables denote items in the domain of discourse.

So in our usual logic, "We detect no motion of the earth relative to the ether" is false, because the ether does not exist. In a free logic, it is true, because the singular term "ether" is allowed to refer to things that do not exist or are not part of the domain of discourse.

This fell off the back of a truck about Being.

It has it's problems, of course, but is presented here as a curiosity. Has anyone come across it before?

It introduces

E!t=df∃x(x=t)

___________
Edit: A rearrangement of the title to better reflect the content. This time to include fiction.

For God and Descartes, see here.
• 15.1k
So for any singular term t, E!t is true if t is an element in the domain, and otherwise false.

And further, since E!t=df ∃x(x=t), E!t is defined using identity: "=".

So there seems to be some support here for this sort of reasoning:

existence is not treated as a predicate in logic. That is, there is no simple way to parse. "Xtrix exists".
— Banno

∃ Xtrix = there exists Xtrix
∄ Xtrix = there does not exist Xtrix

(@Xtrix)

We get

E!(Xtrix)= ∃(x)(x=Xtrix).

This by way of undermining my own point that "∃ Xtrix" is ill-formed.
• 41
What do you know first color od the car or its engine ?
• 6.5k
Meinong's Jungle.
• 15.1k
Indeed!

Meinong's logic - it seems that free logic is a part of the jungle - the jungle is broken into two domains, inner, the things that exists, and outer, the things that don't - but free logic allows us to say things about stuff that doesn't exist. The puzzle is, at what cost?
• 729
Yes, although the technical innovation of free logic is not the introduction of an existence predicate (you can define such a predicate in any first-order logic easily, and so the dictum that 'existence is not a predicate' doesn't mean much). The innovation is a split domain, one over which the quantifiers quantify, and one over which they don't though you can still refer to the individuals in the 'outer' / 'non-existent' domain using individual constants.

Like any logical formalism, there's no question of it being right or wrong, just apt to certain purposes.
• 15.1k
Cheers. The split domain is a consequence of introducing E!, no? Since it leads directly to existent and non-existent individuals. The history section of the SEP article has E! preceding the dual domain.

SO we have :
Classical logic
• Existence is not a first order predicate
• All singular term refer to members of the domain
• The domain is not empty

Free logic
• Existence is a first order predicate, and hence
• Singular terms can refer to things which are not members of the domain
• The domain is not empty

Inclusive logic
• Existence is a first order predicate, and hence
• Singular terms can refer to things which are not members of the domain
• The domain may be empty

IS that right?
• 6.5k
Like any logical formalism, there's no question of it being right or wrong, just apt to certain purposes.
:up:
• 729
As far as I know, you don't need the dual domain to have E! – you can define it in a regular old first-order predicate logic, which is pretty much what you did above.

So we say, as a postulate governing its interpretation, that E!x iff ∃y[y=x].

Where the free logic comes in is that you can interpret ∃ as quantifying over only the domain of 'really existing things.' So if you say something like 'the aether exists,' and the aether is referred to by 'a,' you can put the aether in the non-existent domain, so that:

E!a comes out false, because, we interpret it as:

∃x[x=a]

And this will come out false. Why? Because the quantifier ranges only over individuals in the domain of real existents, which the aether is not in (it's in the domain of non-existent things). So we find no individual x in the domain that is identical to a. So the translation of 'the aether exists' is false.

But we can still say, e.g. 'John doesn't believe in the aether,' if we interpret that as ~B(j,a), and have that come out as true – so long as we allow individual constants like 'a' to refer to things both in the domain of existents and the domain of non-existents.

The result is that we can refer to non-existent things, and say true things about them, while we cannot existentially quantify over them, meaning that it will be true to say they don't exist.

This is all from memory – anyone can correct me if they're looking at the material.
• 729
Free logic
Existence is a first order predicate, and hence
Singular terms can refer to things which are not members of the domain
The domain is not empty

My understanding is that free logic has a split domain, one of existents and one of non-existents (although you can interpret the domains however you want – there are just two of them). The difference between them is that individual constants can refer to members of either domain, while quantifiers range only over the individuals in one domain (the domain of 'existents').
• 3.6k
This by way of undermining my own point that "∃ Xtrix" is ill-formed.

Oh wow! Banno is giving some thought to his posts. Miracles happen... So it turns out that existence is (or can be, amongst other things) a logical predicate after all...
• 13.7k
I wish I had read enough and also endowed with a good memory to give some "real" world examples but, for better or worse, I'll have to do this on the fly. Bear with me, please.

1. Real world:

Someone called me.

Px = x is a person. Cx = x called me.

$(\exists x)(Px \wedge Cx)$

2. Fictional world: From the nonexistent book Leprechauns and their peculiar habits

Some Leprechauns eat poo.

Lx = x is a leprechaun. Ex = x eats poo.

$(\exists x)(Lx \wedge Ex)$

Something's not quite right.

Hence, I suppose, Free Logic.
• 15.1k
Ah, so defining E!a as ∃y[y=a] in a standard logic, E!a would just be true in the case that a is a member of the domain?

At this stage there are no individuals that are not part of the domain. Introducing things that are not part of the domain leads to what has been called two domains - the things that are E! and the things that are not.

Cool.
• 15.1k
So it turns out that existence is (or can be, amongst other things) a logical predicate after all...

Yeah, but there is a problem for the most common sort of philosophical arguments that try to treat existence as a predicate. Free Logic does not permit the expression of existence conditions. That is, that something exists cannot be the conclusion of an argument in free logic.

SO arguments that have such things as the existence of god, or the existence of the self, or the existence of reality as their conclusion receive no solace from free logic.

And we are left in much the same position as I have pointed out in the thread on Being: things that exist because they are the presumed individuals in the domain, and things that exist because they are the subject of a predicate.
• 15.1k
You've made leprechauns part of the domain by presupposing the predicate "...is a leprechaun".

That is, fictional species are part of the conversation, so you can talk about them in your scheme.

In free logic leprechauns would not be members of the domain of things that exist - E!. But you could still make inferences about them.

However as noted above, you could not infer their existence, even in free logic.

In classical logic, to make the inference you would have to presume the predicate "... is a leprechaun". How you understand that predicate remains moot; and one can play on that ambiguity.

If one supposes that all ∃(x)(Lx) says is that something is a leprechaun, one need not conclude that one might meet a leprechaun walking down the street. That there are leprechauns says nothing more in this context than that we can predicate being a leprechaun to something - fictional or otherwise.

Some folk see this as problematic. Seems to me to be just an ambiguity in the use of "is". That Shamus is a leprechaun does not imply that you might meet him in the pub.
• 15.1k
, this is where I'm up to.
• 2.5k
It's been an impressive and amusing discussion. Sometimes like a colonoscopy of the mind but I think I have retained my girlish enthusiasm. I just wish I understood it... :groan:
• 15.1k
I just wish I understood it...

Me, too.
• 3.3k
In classical logic, to make the inference you would have to presume the predicate "... is a leprechaun". How you understand that predicate remains moot; and one can play on that ambiguity.

This is the ambiguity ↪bongo fury apparently traded on in the Being thread.

If one supposes that all ∃(x)(Lx) says is that something is a leprechaun, one need not conclude that one might meet a leprechaun walking down the street. That there are leprechauns says nothing more in this context than that we can predicate being a leprechaun to something - fictional or otherwise.

Some folk see this as problematic. Seems to me to be just an ambiguity in the use of "is". That Shamus is a leprechaun does not imply that you might meet him in the pub.

What?

You are claiming there’s an ambiguity to avoid existential quantification meaning exactly what it says and what everyone agrees it means. @bongo fury wasn’t trading on any ambiguity; sentences found in fiction are literally false, and that’s fine. (Lawrence Block wrote a book about fiction writing called “Telling Lies for Fun and Profit”.)

If you take away existential import there’s no way to put it back just when you want. “There is something under the bed” will just no longer mean there is something under the bed.

Also, what is “presuming a predicate”?
• 3.6k
And we are left in much the same position as I have pointed out in the thread on Being: things that exist because they are the presumed individuals in the domain, and things that exist because they are the subject of a predicate.

To simplify even further: things are always presumed to exist. Existence cannot be proven.

When people say things like "X exist", they usually mean "I cannot think through this issue without making the assumption that X exists." Or simply: "I would rather make the assumption that X exists." They are positing the existence of God, the mind, the Higgs boson or true love. And oftentimes they know perfectly well it is simply an assumption, rather than some deep ontological commitment.
• 13.7k
You've made leprechauns part of the domain by presupposing the predicate "...is a leprechaun".

That is, fictional species are part of the conversation, so you can talk about them in your scheme.

In free logic leprechauns would not be members of the domain of things that exist - E!. But you could still make inferences about them.

However as noted above, you could not infer their existence, even in free logic.

In classical logic, to make the inference you would have to presume the predicate "... is a leprechaun". How you understand that predicate remains moot; and one can play on that ambiguity.

Free logic, as far as I can tell, seems to have an issue with the existential quantifier because, taking the fictional statement "some unicorn ate my lunch", translating that would require a committment we can't (afford to) make viz. existence. The translation of the above example statement is $(\exists x)(Ux \wedge Ax)$ where Ux = x is a unicorn and Ax = x ate my lunch.

Free logic would, in my humble opinion, open up the world of fiction - Tolkein's works, Doyle's works, etc. - to logical analysis.

You say that's not all free logic can do. It seems that you're under the impression that with free logic we can begin an analysis of a given domain of objects and then, almost miraculously, switch the topic to something outside that domain. I'm not sure I understood you correctly but if this is what you mean, what's the point, really? It appears to be something but, in fact, it is, I suspect, nothing. I'm out!
• 15.1k
The ambiguity referred to is the domain of discourse, which may be whatever one desires.

...sentences found in fiction are literally false

Well, it's false that Holmes lived at 222 Baker Street.
• 15.1k
things are always presumed to exist.

Yep.

Interesting, then, that introducing E! does not provide solace for those who wold do otherwise.
• 15.1k
Free logic would, in my humble opinion, open up the world of fiction - Tolkein's works, Doyle's works, etc. - to logical analysis.

You seem to think this would be problematic. Why shouldn't fiction be logical?

Holmes lived at 221b Baker Street. Why shouldn't we consider this to be true, within the context of the writings of Doyle and their derivatives? Is there an argument against this?

It seems that you're under the impression that with free logic we can begin an analysis of a given domain of objects and then, almost miraculously, switch the topic to something outside that domain.

Free logic defines two domains. There's no magic here, just an attempt to make explicit the grammar around saying that something exists.

What is amusing about this is that in attempting to build a sensible account of E!(x) a string of logical difficulties arrises, allowing us to look with greater care at the problems of treating existence as a property.

To be sure, it's the problems of free logic that are fun.
• 15.1k
A few notes, for my purposes as much as for you.

If we confine ourselves to the stories of Tolkien, then we can make inferences about fictional characters without making use of free logic. So the example from the SEP article is that within the context of LOTR, Gollum hates the sun, and we can validly infer that something hates the sun*.

The two issues noted in the article are that there is no compunction on fiction to be logically consistent; and that the properties of fictional individuals are incomplete.

The first of these is apparent from watching Marvel or DC movies. There's much to be said for the usefulness of applying logic to these scripts in order to set out their inconsistencies.

The second is of use to authors, presumably, since it allows the characters they create to develop in unexpected ways. It presents problems for logicians if they erroneously treat fictional characters as complete individuals.

Free logic might be employed if one is attempting to talk across both fictional and non-fictional individuals. The example given in the SEP article is
Gollum is more famous than Gödel.
Treated in ordinary logic this statement would be false, since Gollum does not refer to anything. But we can make use of Free logic, since here Gollum is fictional, but not Gödel, hence E!(Gödel) but ~E!(Gollum). In a positive free logic Gollum is more famous than Gödel.

Keep in mind that positive free logic is one which an individual is identical with itself, even if that individual doe snot exist: one in which (Gollum = Gollum) is true; a negative free logic is one in which (Gollum = Gollum) is false. Hence in a negative free logic one can infer from (Gollum=Gollum) that Gollum exists.

*I wonder if @Srap Tasmaner thinks this inference is invalid...? How seriously should we take the claim that all fictional statements are false?
• 12k
I don't understand the title.

If it's Fiction Logic within Free Logic, then aren't we committing ourselves to Epistemic Logic at this point to "conclude" anything about the domain of discourse, which is the mind of the author and/or literature or script?
• 15.1k
My apologies, I'm not sure what to make of this.

So Epistemic Logic is the various logics that include predicates for belief and knowledge - yes? Where's that fit here?

But it's not true, nor even helpful, to think of fictional worlds as only existing in the mind of the author. Were this true, both Holmes and Frodo would have ceased to exist along with the minds of their respective authors.

Isn't it simply the case that the domain of discourse for LOTR includes the individuals Frodo, Gandalf and so on, and the predicates "...is a hobbit" and "...is a wizard", and that it is true within that domain that Frodo is a hobbit and false within that domain that Gandalf is a hobbit? And isn't that settled extensionally?
• 3.3k
For the logical analysis of literature, I would start here.
• 11.3k
Holmes lived at 221b Baker Street. Why shouldn't we consider this to be true, within the context of the writings of Doyle and their derivatives? Is there an argument against this?

More aptly referred to as being stipulated or written, rather than being true. no? We have good reason to believe it is true that it is written that Holmes lived at 221b Baker Street.

If there is a real 221b Baker Street, then it is false that Holmes ever lived there. If that address is also fictional then I would say it is merely stipulated that the fictional character lives at the fictional address.
• 15.1k
Very droll.

1. That a post shall accomplish something and arrive somewhere. But your post accomplishes nothing and arrives in air.

:wink:
• 15.1k
More aptly referred to as being stipulated or written, rather than being true. no? We have good reason to believe it is true that it is written that Holmes lived at 221b Baker Street.

"It is written, that's why" - Brian's mum, Life of Brian.

Well, the stipulation is that we are talking about LOTR or whatever; and the question is as much about what we might infer as what is true and what is false.

So I think it not unreasonable, given that Frodo walked into Mordor, that we might infer that something walked into Mordor. That is, that the logic of existential generalisation might be expected to hold in a fictive world.

The alternative suggestion is that it is not true that Frodo walked into Mordor, and hence that we cannot make such an inference.
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