The problem is quite simple. S is not defined. To be defrined, a sentence must be set equal to a string of symbols, such that all the symbols are themselves defined.
S is defined in terms of S. Before we can know what, exactly, S is defined as, we need to know what S is. So there’s no way to ever know what S is defined as. We are so used to filling the holes in other people’s logic that we don’t even notice that there is a hole.
Quinne provided an laternative formulation:
Oy. Many posts (including mine) have offered symbological restatement of the proposition. Symbologizing is trivially easy.
Simplifying it as far as I see possible, all that this is is saying A => B where A is (A => B). The statement then becomes (A => B) => B, which becomes ((A => B) => B) => B, and so on ad infinitum. It would be silly to apply logical rules to a statement before it has been fully stated, which in this case never happens.
Or, perhaps to phrase it in natural language:
“God exists if the existence of God is implied by the statement “God exists if the existence of God is implied by the statement “God exists if the existence of God is implied by the statement… ”
At least that’s how I think it would go.
I guess, having seen all these posts, it’s obvious where the “proof” goes wrong. Starting a logic proof with “let S = S => B” is like starting a math proof with “let n = n + 1”. No such number n exists, so any conclusion could be drawn by hypothesizing the existence of such a number. Similarly, no sentence S exits such that S = S => B, so the bizarre consequences of assuming such a sentence exists should not be surprising. Thanks everyone for helping me with this.
its turtles, all the way down…
SENTENCE S
IF S is true, i am the emporer of the world
S is true
logical isnt it?
Or at least, it is if you don’t pay attention to what we’re saying.
S=S=>B may be a symbolisation of the statement, but it is not one which is allowed in first order (or indeed any standard form). Why? Because equality of strings is not a valid logical predicate.
You must be able to reduce the string to symbols involving ONLY:
[ul]
[li]Atomic variables[/li][li]Negations[/li][li]Implications[/li][/ul]
(And, Or and iff are allowed as well, but these can be constructed from negation and implication).
Further, they must be combined in a valid way (explained in a previous post) and there must only be finitely many of them.
You can certainly define a string as S = (A=>(Not B Or C) ) And B for example. (Yes, it’s not a terribly interesting string. It’s equivalent to Not A, but that’s not the point). A, B and C are all also logical strings. The point is though, that by referring to strings as letters and using equality you’re making a simplification. It’s perfectly ok to do so - Writing out the entire string would be confusing and counterproductive. However, it must be in principle possible to reduce your sentence to a first order statement. (i.e. Atomic variables, negations and implications), otherwise it’s not a valid sentence.
I explained before why this isn’t possible. If S is the first order sentence such that S=S=>B then you have 2 more terms on the right hand side of the equality than on the left hand side, so it’s not a finite sentence. If you were allowing infinite sentences then you could have S=(S=>S=>S=>…)=>B, but that isn’t a valid sentence in first order logic.
You’re preaching to the choir, Kitarak. If we’re talking first order logic, then S can be considered an open wiff. If we’re talking truth-functional propositional logic, where bound variables may range over predicates, then S is a proposition. Unfortunately, it is also an equivocation. As such, it reduces to a first order identity.
There have already been several excellent responses. My only contribution left is to recommend the work of Jon Barwise and John Etchemendy. They explore the meaning of self referential sentences such as the Liar Paradox. My pet topic, non-wellfounded sets, even (ha ha) makes an appearance. In particular:
The Liar: An Essay on Truth and Circularity. Jon Barwise and John Etchemendy. Oxford University Press, 1995.
Isn’t (A=>(Not B Or C) ) And B equivalent to ( B And (A => C) ) ?
Yes, provided that you are speaking of classical logic, where ‘=>’ is the symbol for material implication.
Tyrell: Oopsie. I didn’t look particularily closely at the string I was using for an example, as I just plucked it out of thin air. You’re right. Doesn’t do wonders for my credibility, does it? 
Libertarian: Well, seeing as we disagreed it seemed I wasn’t preaching to the choir. To be honest my grasp of mathematical logic is still in a developing stage - My course doesn’t teach it until woefully late, so I’m learning it on my own. I’m reasonably competent at it, just over a somewhat limited area. Still, I’ll try and see if I understand what you’re saying. ::struggles for a bit::
Ok, I’m not sure what you mean by a wiff, and can’t find the term in any of my textbooks, so I assume it’s an abbreviation of some sort. Well founded formula or something of that nature?
The argument however was entirely statement calculus. In principle any logical statement you make should be reducable to a (admittedly rather long) string of statement calculus, and thus it suffices to consider the problem in that context?
The problem with allowing equality of formulae in a logical statement would seem to be that equivalent statements aren’t neccesarily equal, which looks like it should cause difficulties.
Tyrell: Oopsie. I didn’t look particularily closely at the string I was using for an example, as I just plucked it out of thin air. You’re right. Doesn’t do wonders for my credibility, does it?
Libertarian: Well, seeing as we disagreed it seemed I wasn’t preaching to the choir. To be honest my grasp of mathematical logic is still in a developing stage - My course doesn’t teach it until woefully late, so I’m learning it on my own. I’m reasonably competent at it, just over a somewhat limited area. Still, I’ll try and see if I understand what you’re saying. ::struggles for a bit::
Ok, I’m not sure what you mean by a wiff, and can’t find the term in any of my textbooks, so I assume it’s an abbreviation of some sort. Well founded formula or something of that nature?
The argument however was entirely statement calculus. In principle any logical statement you make should be reducable to a (admittedly rather long) string of statement calculus, and thus it suffices to consider the problem in that context?
The problem with allowing equality of formulae in a logical statement would seem to be that equivalent statements aren’t neccesarily equal, which looks like it should cause difficulties.
A wiff is a Well Formed Formula. Do not be the least bit apologetic of your excellent command of logic. I’m certainly not criticizing you in that regard. I commend you for studying on your own. You and I are saying the same thing in different ways: statement calculus - identity. What happens when S = (S => C) is that S = S because if Not C, then Not S.
I hope to see you in my new ontological argument thread. I would be very interested in your take on Tisthammer’s analysis.
Thanks. I try. I’m more of an analyst than a logician, but all maths is interesting (unfortunately. Too much to learn… :eek: ).
Ok. I’m possibly being pedantic here, but would you mind defining equality for me? I’m not entirely clear on how you’re using it in this context. It looks like you’re using it to mean equivalence, while I would take it to mean that they are the same string. i.e. You look at the first character, it’s the same. Move one to the right, they’re the same, repeat until you get to the end of the string (which much be at the same point for each string), whileas I by S is equivalent to T I would mean (S=>T) And (T=>S). Certainly equal statements are equivalent, but equivalent statements aren’t neccesarily equal. e.g. Not Not A is equivalent to A, but not equal to it.
By the way, I probably came across as somewhat patronising earlier, and if so I apologise. Bad habit of mine…
Not to be pedantic or anything, but I’m curious about your use of the word “maths”. I guess you’re British, so you’re treating it as a plural word, but then shouldn’t it be “all maths are interesting”?
No. Just because the word ends in an S doesn’t mean it’s a plural. We don’t pluralise math, you just drop the s from maths. 
You may look at it as a tautology symbol.
Not at all. You came across as eager.
Ah, yeah. As I thought, that’s where the confusion is coming in. I don’t think that tautologically equivalent sentences should neccesarily be considered as equal sentences. They mean the same thing, but meaning and form are two seperate properties. For example, in english rather than symbolic logic, would you consider “This door is open.” and “This is an open door.” to be the same sentence, even though they mean the same thing?