Atheism and the existence of peanut butter

[R00tKiT]

In a more modern take, the technique of forcing in set theory (which was used to show the independence of the axiom of choice) relies on a version of multivalued (even infinitely valued) logic.

Next, the whole point of paraconsistent logics is exactly that the 'explosion' phenomenon of classical logic (where one false statement implies everything), is avoided. i can give you some references if you are interested in looking at what this leads to, including some very interesting aspects of mathematics including how to deal with Russell's paradox.

From the bit of research I did on the topic of multi-valued logic, it seems it doesn't introduce any fundamentally new concept, it's just a different way of looking at the semantics of some logical propositions, which can be perfectly described using classical logic. You might have come across Suszko's thesis, it literally says than certain categories of multi-valued logic can be provided with bivalent semantics, that is, classical logic.

Interestingly, the idea of positing three or more truth values was motivated by contingent propositions such as; P: "It will rain tomorrow". Some logicians suggest assigning a new truth value to P: as-yet-undetermined. It's clear that this is a redundant way to interpret P, because P contains the label "tomorrow" by construction, and we know what tomorrow is, it's the next 24 hours. If, at some point of time within this interval, there is rainfall, then P is true, and vice versa. We just don't need a third truth value for that.

If you have a reference on some substantial result in mathematics that can't be proven using classical logic, then I am definitely interested. So far, I am convinced that anything beyond classical logic is based on some redundant (maybe clearer) description of logical statements.

Look up intuitionist mathematics some time. A great deal of math can be done without excluded middle. And, yes, like math without AC (which, by the way is NOT an axiom of logic, but of set theory), such mathematics tend to be stilted and lack beauty. But don't forget the paradoxes produced by AC (like the Banach-Tarski paradox).

That said, some of the recent developments in set theory are based on looking at alternatives to AC and seeing how such alternatives apply to the more traditional math.

There are many more alternatives to the 'laws of thought' than you seem to think exist.

I agree. But the simple fact is, we will mostly be content with classical logic. Physicists don't look for alternatives to these laws of thought when they formulate their theories. This of course doesn't mean alternative axioms are useless, but I frankly don't see why we should go to such lengths when discussing short arguments with simples premises. 

In general, these laws of thoughts are the foundation of rational discourse. That is, one cannot have a meaningful argumentation if they start second-guessing the very building blocks of how one evaluates logical propositions. It's just an exercise in futility, or, as they say, mental masturbation.

Since classical logic is enough to properly formulate all modern theories of physics, including GR, QM and the Standard Model. It's safe to say classical logic is enough to describe reality, and so can be used when formulating premises in a posteriori arguments about metaphysical entities.

Well, as above, it is far from clear that the universe 'began to exist' in the sense you have give: was there a *point* when the universe did not exist and was it in fact in a causal chain?

if so, where did that causal chain begin?

But, more specifically, causality is something that happens *within* the universe and *within* time. And no, not every event is caused. It is clear that many events at the quantum level cannot be said to be caused in any classical sense.

Again, if you accept the causality principle, then a universe that began to exist has to be in a causal chain. The causality principle rules out an uncaused universe that began to exist.

You say, "causality is something that happens with the universe", and I will again ask you why? Why do you think that things outside of this spacetime can somehow spontaneously arise from nothing -in the philosophers' sense, not Krauss's, really nothing- ? Even Hume didn't go this far AFAIK.

Where is the contradiction? And no, thi sis NOT a difference between 'potential' and 'actual' infinities. I am talking about an *actual* infinite past. My example has an *actual* infinite past. Look at the set of *all* integers, both positive and negative. That is a countably infinite set and every element has infinitely many precursors. And yet, you still have 0 and 10 and -100.

I mean.. seriously ? An infinite set of numbers is the archetype a potential infinity, it's just a collection of stuff, that's it. In the real world of actual infinities, it's not just a collection of stuff, it's a causal chain of stuff, where each element in this chain is a necessary condition for the existence of the next element. In a set of numbers, 1 didn't cause 2, nor did 9 cause 10. 

And regarding an infinite past, saying that any wait between two events is finite doesn't solve the problem, we're not discussing two events, we're discussing the entire duration of time that needs to elapse for us to get to the present moment, an infinite past doesn't have a lower bound, so a universe with an infinite past actually went through an infinite wait -impossible.