Hi Tim,

I add "If S is consistent then G (is not provable)" to S as an axiom. It then becomes clear why we can't prove the consistency of S within S because it would lead to a proof of G which is impossible by the First Incompleteness Theorem.

Suppose per hypothesis that, tomorrow, "p & ~p" becomes true. One response is to reject the hypothesis. That is, to say that such a scenario is impossible and thus cannot obtain. Another response is to change the rules of logic to accommodate the scenario. — Andrew M

Both impossible if p & ~p becomes true because then we couldn't talk/think straight/meaningfully.

More and more it seems that the main difference between induction and deduction is this: If we imagine an induction to be false tomorrow then we can at least comprehend what it would mean for tomorrow while with deduction we could only say that if it happens then tomorrow would 'black out' for us. But that's not enough for me to exclude such a possibility. It would be like if you would exclude the possibility to get insane tomorrow just because you couldn't say anything about that tomorrow.

Can you give an example of ostensive talk that doesn't assume non-trivialism? — Andrew M

I can't, you are right.

My line of thinking goes like this: Imagine tomorrow the proposition "p & ~p" becomes somehow true! At that point our logic would collapse, we couldn't even talk or think about it (trivialism), we'd be literally insane. But we can today - under the assumption that our logic still holds - verify that this very scenario could happen tomorrow and that it would destroy our logic. Isn't that a proof by example that logic is not necessary?

I do believe we were created but it should be noted if you roll a million sided die one million times trying to roll a 566 you will almost definitely roll a 566. This was pointed out in the book "a brief history of time" by Stephen Hawkings. — christian2017

But we assume there's only one world, therefore just one roll of your million-sided die which we assume as random. And if you roll there a 566, what would be the rational conclusion of a mathematician using probability theory & statistics?

In turn, accepting inconsistency seems to take one out of the bounds of meaningful language. — Andrew M

Not really. It just means trivialism, i.e. everything becomes true, but I can still talk and be understood on an ostensive level. That sounds enough meaningful to me.

Back to Hume. How does he prove that logic is not a matter of fact but something higher? IMO he can't, so his "fork" is pretty much made up from speculation and tradition.

@EEE: I don't care about the creator at all. My argument only is that because of statistical reasons it is very probable that we were created non-random, not more, not less, no further conclusions. And I ask if this argument is sound or not.

No, we would just look for ways to model the world that avoided inconsistency. — Andrew M

But your answer implies that it could happen and so we'd need to adjust our logic and math, but that means the problem of induction also applies to logic and math so why did Hume not agree? For him only our experiences are inductive, not logic and math, but we see that it's wrong. We can easily imagine a scenario where our world of tomorrow has different logical/math rules than the world of today. Was David that blind?

Again: Why is it impossible that we wake up tomorrow in a world where one tiny particle has the property to be and not to be (which would make the whole world inconsistent)? That would make MP invalid and kill all our logic, right?

Let me re-emphasize my thought-experiment: Suppose the world changes overnight so that it becomes impossible to model an implication (per se and of course especially for our human minds). It's hard to see why and how, but just bare with me. Wouldn't that mean that MP becomes impossible as well, in contrast to a day before where it was not only possible, but necessary? Doesn't that prove the induction problem for logic as well?

Again, I don't understand why the induction problem is never seen as problem for logic and math as well, there must be something I do miss since far wiser people than me dealt with this before.

As long as P does not change, then Q will keep necessarily following. — alcontali

Why? P, P -> Q | Q is just right because it follow from some rules. But these rules can change overnight, can they? So MP could be true today but false tomorrow. Imagine - overnight - our world becomes weird in the way that it becomes impossible to construct any implication P -> Q (~P v Q). I know it's hard to imagine, but I can just write it down and say: so be it from henceforward. Obviously in such a world you could not conclude anymore Q from P & P -> Q because it wouldn't be a wff at all. But somehow Hume and basically all philosophers after him disregard such a possibility.

Free will is impossible with or without determinism and it's not circular reasoning ... it's a basic argument. Namely:

(1) Ultimately, to control your actions you have to control your fundamental nature.

(2) But you can't control your fundamental nature.

(3) So, ultimately, you can't control your actions.

This is true with or without determinism. — luckswallowsall

But from your (3) it also follows that you can't control your very argument, so how can you believe in it? That's exactly my problem.

Just that we not forget it: the question is if one can hold determinism rationally because it seems in case one holds determinism, i.e. thinks determinism is true, one has no reason to trust ones thinking since it's determined itself. It seems we need to hold free will as a position, it seems our epistemological apparatus presupposes some freedom to work properly from our perspective. But if that is the case then determinism is not a rational position at all, as well as its counterpart (since it's presupposed in us), so the whole free/unfree will problem is undecideable.

So would you say that you're choosing to believe the principle of noncontradiction, for example, where you could just as easily choose to believe the opposite? — Terrapin Station

No, I choose to believe in non-contradiction, because it compares better to the alternative (contradictions). Without an alternative you can't compare, without comparison you have very little information, because you just have some p, which can be T oder F, but it's wide open if you just got p and no alternative like ~p.

The whole notion of "free reasoning" seems rather odd. That doesn't seem to mesh with the logical notions of validity, soundness, implication, etc. We don't choose what follows logically. — Terrapin Station

Actually we do, because we choose our rules of logic out of many possible alternatives, because they work better than others and because they persuade better than others. If we had just our rules and couldn't think of alternatives, then how to trust them since we couldn't compare them to alternatives? It seems the notion of freedom is built into our logic and proof systems thru bivalance: you never have just p, you always have p v ~p available. But determinism undermines this notion and leads to one-dimensionality we can't make sense of and can't trust therefore.

If I understand these determinists rightly they basically say this: XYZ is true and from there it follows determinism/unfree will; yes, that also means this very proof/logic is determined but we just trust that the proof/logic is determined to be correct, giving us the truth, because ?!? we just trust, period. So basically all deteminism is religious and irrational eventually. That doesn't help libertarism though since that position would be just circular since we just saw that freedom is a-priori within our logical apparatus. So it's kind of like with God: nobody can prove anything rationally, so atheists and theists just toss around their irrational ideas that are more or lesss popular/believed. Is it that?

Here's a better version of my proof of nihil ex nihilo:

1. We postulate the empty set as an universe (= representation of nothingness).
2. We assume some t in the universe which contradicts 1.
3. So it follows that no t can be in the universe.

If you want to add time it's pretty much the same:

1. We postulate the empty set + time as an universe. (= representation of nothingness within time)
2. We assume some t in the universe which contradicts 1.
3. So it follows that no t can be in the universe.

@terrapin station:

Ok, let's assume only ∅ + time. Still it's (logically) impossible for a thing to exist from these premises.

@fooloso4: We just can reason within our intellectual scope, i.e. logic and meaningful concepts. My proof wants to show that within our scope it's impossible that something can come from nothing. Of course the world could be so different from our reason, our logic could be far off of how the world works, our concepts could be so flawed in comparison to what the world really is all about that we'd be like dogs in a library. We will never conquer that limit but it shouldn't touch us for we can't reason about it anyway, it's just like a big wall we can't look behind. There could be anything, there could be nothing.

No. If we assume only ∅ then nothing else matters, not even time. No thing can come out of this assumption, no matter how one twists it.

Aside from the problem of reifying abstractions and positing some questionable definitions there, you don't actually present any sort of argument as to why something can't "come from nothing." — Terrapin Station

1. Let's postulate only ∅ (Nothingness).
2. Let's assume some t, but that's contradicting 1., so it's impossible.
3. Conclusion: If only ∅ then nothing can exists no matter in which way or modus, nihil ex nihilo.

p.s. It's obvious for me that only ∅ can represent nothingness, one could prove that with raa since any other set would have elements and certainly unable to grasp our intuition of nothing.

p.p.s. Also be aware that Nothingness is relative here since ∅ is not really nothing, but that's our only chance to reason about it; we cannot grasp or refer to absolute nothing since that reference alone would make it something.

p.p.p.s. This proof uses set theory as a model. We could proof the same with just predicate logic by using "There exists nothing" as our assumption, we could even use propositional logic to use "~p1 & ~p2 & ..." to model it roughly.

1. Let me present you a more simple version of my argument "nothing can come from nothing" which renders a creatio ex nihilo impossible, an argument without sets:

Let's define nothingness as the conjunctions of negations of any possibly or actually existing things: ~p1 & ~p2 & ~p3 & .... From that definition is follows trivially that no object can exist out of nothingness.

2. I still think the set S (representing the Being) of all existing things is neither the set of everything nor does it lead to it. It's different, because it only assumes things that are already existing and therefore non-contradictory, while the set of everything doesn't. Because S exists, ~S exists (~S = empty set = nothingness) and from there it follows trivially as well that if we assume ~S we cannot assume anything out of this empty set.

Yes, this model cannot grasp total nothingness (the same with 1.), but that's how far we can imagine nothingness anyway, there's no consistent way to imagine some more total nothingness because we always need something to define nothingness. Total nothingness is actually meaningless like triangle with four angles, it just looks like it means something due to its letters, but contentwise it's the same as "%$%/&%$/$/$" - meaningless.

Definitely appreciate your input.

1) By definition my set contains only things that exist (non-contradictory). That excludes Russell's set. No Russell's set, no problem. In other words this set is defined like: containing everything that does not lead to a contradiction somehow. It should be clear that such a set is clean of problems by definition alone, don't u think?

2) I think nothing has to be the empty set very naturally since otherwise only sets with members could be available that obviously couldn't serve as nothingness.

3. You are right that the empty set is itself a thing and just basically postulated. But as I wrote in my note: that's how we have to interpret nothingness, there's no better way. We simply cannot postualte a further-going nothingness since it would lead to contradictions/falseness. What we mean when refering to "nothing" is the empty set (or e.g. in logic the conjunction ~p1 & ~p2 & ...), that's "our" nothing, beyond that is just a brainf*ck that doesn't mean anything, just like when we talk about the universal set that SEEMS alright but isn't (as Russell showed).

Russell's paradox is no problem since I don't talk about the set of everything, but the set of everything that exists. That's a huge difference. Obviously Russell's paradoxical set doesn't exist since it's contradictory.

I also wonder if my proof could just start to define Nothingness as just the empty set. Why bother?

How do you prove G to be true in T? You can't for then you'd prove G and contradict to its content. So G is true or false in T but we cannot know it (within T). We also have no position beyond T for T does already contain logic & arithmetic. Therefore I think we cannot conclude that G is true, but its truth is unprovable and that's not a proof of G's truth.

First of all I disagree with Srap that you can't have a predicate "is true/false" in PL. I think you can have, but sometimes it leads to inconsistencies or violation of formal rules and only in that last case it is forbidden. That's why "This statement is false" is forbidden in PL, because you couldn't assign a truth value to it which is necessary to apply PL.

I disagree with Meta that one can construt the liar paradox in propositional calculus. It's too weak.

I think I found the solution (at least for myself). "All statements are false" can never be equivalent to "All statements are false AND this statement is false" because the last sentence would not be a wff in predicate logic. From "Alle statements are false" one cannot deduce "All statements are false AND this statement is false" because of the same formal reason. Therefore I cannot prove "All statements are false" to be illogical. I could only argue that when you say "All statements are false" you really mean "Alle statements are false AND this statement too" and that's indeed illogical. And of course that's what a truth skeptic would argue. But this is not logic anymore, but how you formalize everyday-language into PL.

@srap: Why can't I have "is a statement" and "is false" in predicate logic?

@Meta: Your formulation in propositional logic is too simple, you can't express there what "All statements are false" want to say. In propostional logic, as a matter of fact, the liar paradox is just plain false since it says there: p <-> ~p.

If your point is only that this sentence can't be represented within classical logic, then duh. — Srap Tasmaner

That's my point. The problem is this: To prove that (S) "All sentences are false" is not a wff, I have to assume that S = S' with (S') "All sentences are false and this sentence is false". Then I can prove that S' is not a wff and since S = S' then S can't bei either, but how can a not well formed formula like S' be equal/equivalent to anything at all? It's not working. This is the reason why I wanted to formalize "All sentences are false" right away as a conjunction "All sentences are false and this sentence is false" and wondered if it would be possible and if not why.

So you're thinking that since you have, in essence, "[the Liar] & P" as your conjunction, we'll be unable to construct a truth table because the first conjunct is not truth-apt. True. — Srap Tasmaner

Yes, that's my point.

Now, a conjunction of the form "[liar] & P" is neither a conjunction nor a proposition at all, because a conjunction is formally defined as "p & q" and p,q need to be propositions with truth values. That's just predetermined in the language of predicate and propositional logic. So "[liar] & p" can't be true and can't be false, it's just syntactically not a wff formula if I am right. I see your point and you are quite right from a practical point of view, but formally it's not possible I think.

@Michael: It may not be true, but it wouldn't be false either. It would hang in the middle between the two values.

They're not. Whether you want to say "all statements are false" is false or "all statements are false" is neither true nor false, it is still the case that "all statements are false" isn't true and "at least one statement is true" is true. — Michael

Yeah, but that statement wouldn't be false either! And it's a huge difference if a statement is false or not false (no matter if it's true or not besides that).

In summary, my problem is why nobody interprets the statement "All sentences are false" as "All sentences are false and this very sentence is false", because in this version the whole thing wouldn't be true or false. I just don't see an error in infering one from the other.

I assume a set of all statements as the domain. Then indeed ∀x(Sx∧Fx) means: everything is a statement and everything is false. But that's exactly what "all statements are false" says, isn't it? Why can't I formulate it that way? My guess is there is no real reason, it's just an agreement that we formalize universal statements as implications (and use universal sets as domains).

But the consequences are huge, because if my case is formally possible then ∀x(Sx∧Fx) fails to be true or false in the one and only case of applying it to itself. It'd be false because it says so, it'd be true because it's false and says so. Wouldn't that be enough to make ∀x(Sx∧Fx) a not-wff-formula?

Because ∀x(Sx→Fx) and ∀x(Sx∧Fx) don't say the same thing. — Srap Tasmaner

Ok, but why is my formulation of "All statements are false" as ∀x(Sx∧Fx) impossible and only ∀x(Sx→Fx) the correct formulation? Is there any reason? Because, again, both formulations lead to different results for the proposition. In case of ∀x(Sx→Fx) it's false, in case of ∀x(Sx∧Fx) it's not well formed.

Here is what I found out so far:

The logicians formulate "All (S)tatements are (F)alse" as follows: All x: (Sx -> Fx). If you do that you can indeed prove that this is just plain false since if it's true it's a contradiction and so by RAA it's its negation that is consistent.

But why do I have to formulate the statement like above? Why can't I just formulate: All x: (Sx & Fx). This statement is not false, it is not well formed since it entails the liars paradox.

Both versions of the upper statement say roughly the same - that every x in the set of statements is false - but their form is different and so are their results. So who is right? Or why am I wrong?

So not having a truth value is the third option. If we have the conjunction p ∧ q and if p is false and q doesn't have a truth value then the conjunction as a whole is false. — Michael

Not in classical logic. Not having a truth value is not a third option there. I think 2. of my proof is dubious, maybe nagase will check that out.

@Michael: We are dealing just in the scope of classical (two-valued) logic and there a statement like "This statement is false" is just without a truth-value as far as I know.

As I mentioned in a private message, I'm not entirely sure this step holds. What is the reference of "this" above? If it is "this statement is false" (taking it to have small scope), then your proposed rule would take us from "All statements are false" to ""This statement is false" is false". But the latter one is false, not truth-valueless. So the whole conjunction is false. — Nagase

You are right with the reference, but if the part "...this statement is false" is false then it'd be true (and vice versa false if it's true), isn't it? So therefoe it can't have a truth value and so does the conjunction.

Just for reference the proof again:

1. We assume (A) "All statements are false" has a truth value.

2. Since we can always go from "All x are y" (x may stand for 1,2,3 here) to for instance "All x are y and 2 are y" without changing the truth value, it must hold that A is logically equivalent to (A') "All statements are false and this statement is false", so A <-> A'.

3. Now we prove that A' can't have a truth value because of its part "this statement is false". This is self referential and can't have a truth value (if it's true it would be false and vice versa), so it follows for the whole conjunction of A'.

4. That leads to a contradiction, because it's impossible that A <-> A' (2.) and A' doesn't have a truth value (3.). That means that the assumption, A has a truth value, leads to a contradiction and is false, A has no truth value.

University logic says you did not prove anything. Your statements are meaningless. — Meta

Why?

1. We assume (A) "All statements are false" has a truth value.

2. Since we can always go from "All x are y" (x may stand for 1,2,3 here) to for instance "All x are y and 2 are y" without changing the truth value, it must hold that A is logically equivalent to (A') "All statements are false and this statement is false", so A <-> A'.

3. Now we prove that A' can't have a truth value because of its part "this statement is false". This is self referential and can't have a truth value (if it's true it would be false and vice versa), so it follows for the whole conjunction if A'.

4. That leads to a contradiction, because it's impossible that A <-> A' (2.) and A' doesn't have a truth value (3.). That means that the assumption, A has a truth value, leads to a contradiction and is false, A has no truth value.

That looks like a legit proofsketch to me.

So you're saying S can't be false because S', the equivalent statement, can't be false because of the "S is false" part. — TheMadFool

Yes, but maybe this proof makes everything clearer:.

If "All statements are false" would have a truth value and therefore would be a statement, it'd follow by universal instantiation + introducing conjunction: "All statements are false and this statement is false". But that deduction has no truth value since it's a conjunction containing "this statement is false" which has no truth value which poisons the whole conjunction eventually. Therefore "All statements are false" cannot be a statement with a truth value as well.

Again: This is HUGE, basically all philosophers agree that truth skepticism (All statements are false) is refuteable, but that seems just false, because it's not even a statement. Is nobody here with real university logic knowledge?

But I say that "All statements are false" has no truth value and therefore can't be false.

And my proof is simple: We know that (S') "All statements, but this one, are false and this statement is false" has no truth value because of the paradoxical sencond half sentence. But wouldn't we all agree that S' means the same like "All statements are false"? But then "All statements are false" must have the same fate: no truth value.

And that would be huge, e.g. truth skepticism "All statements are false" would be non-refutable instead of plain false.

But if (S) "All statements are false" is illogical - like I want to prove - then it can't be false (and therefore it's negation be true) like the majority wants us to believe. And (S) "All statements are false" seems to have the exact same meaning like (S') "All statements, but this statement, are false and this statement is false", just that the last one explicitly shows what's implicit in the first one. If S' is an illogical statement - which it is - then since S' has the same content than S, S must also be illogical.

I hope it makes my point clearer.

~p -> p is not equivalent to ~p & p, that's my point.

Now if creatio ex nihilo is modeled as: ~p & ~p -> q then it's logically possible.

Is there any book, article or link that describes how to formulate a creation out of nothing in a formal way? I mean I wonder that I can't find anything that proves or disproves the old proverb that nothing can come from nothing in a logical way.