Originally Posted by

**fast**
What I see is analysis at work. They’ve taken a concept (what you call validity) and insightfully separated the structural integrity (or what allows for sensible entailment and implication) from the relevance of truth. They’ve taken a combined, intertwined, symbiotic thing and broken it down into two major component parts.

No, it is much less glorified and much more simple. Sort of pathetically simple. The story is that they made basically one unwarranted assumption and one unwarranted hypothesis. That was clearly doomed from the start. The unwarranted hypothesis is well known and seems very reasonable. The unwarranted assumption, well, it's an assumption, they still as of this day don't even know they've made it. They haven't a clue.

The difference between mathematics and empirical sciences is that in mathematics, once you've settled on the axioms, you're unlikely to have any surprise. It's just hard work trying to find out the logical consequences of your axioms. None of your conclusion is likely to contradict your choice of axioms, unless you had been very careless in selecting them. Counterexample, Cantor's theory of sets, which included several contradictions. It must have been a traumatic experience, and you still see today mathematicians insisting intuition isn't reliable. So, when they find a contradiction, what do they do? They just invent a machinerie to encapsulate the contradiction to make it ineffective. It's imagination at work and our imagination is a curse. Mostly, imagination is laziness. It allows you to circumvent the problem without starting all over again. Laziness.

When scientists realise their pet theory is junk, like for example the jewel in the Crown, Newton's Theory of Gravitation, they grumble and keep looking at nature waiting for someone bright to figure out what the problem is and what might be a "reasonable" solution. Imagination plays a part but mostly not anything you can imagine will do.

Mostly, there's just one possibility given what you think you know. So, wild imagination is kept in check by the requirement that the theory fit with nature, though this is not so obvious in the case of Quantum Physics and Relativity (and even less in the case of String theory). But, broadly, if anyone thinks they know better, they're welcome to explain themselves. But here, scientists will throw the falsified theory in the junkyard and the redesign the theory from top to bottom. That's the only way.

Mathematics is an ever-growing fat mass of axioms and theorems. You even have theories which are essentially a repeat of each other! By comparison, science is slim. Relatively easy to do it all over again when the need arise. Two very different world. Logic is science. Not maths.

In the case of mathematics, who is going to tell them that what they've imagined is wild junk? There's no check and balances. Anything goes as long as it is "logical", and now therefore anything at all goes because logic is "arbitrary" and you can justify any idiot theory by making up a new kind of meaningless logic. Every mathematician their "logic". They don't even bother to keep track of the mess because it's such a mess. It makes me think of the explosion of Protestant Churches because unchecked by the authority of the Pope.

You can't even make any generality about it, notwithstanding what you just said. But, let's limit ourselves to "classical logic". They haven't analysed anything. Their definition of validity doesn't make sense, which is why some mathematicians invented relevance logic, which is just as pathetic. What you call "*sensible entailment and implication*" doesn't even exist. You couldn't give an example of that. It is easy. Look at truth-table proofs and try to make sense of why the result is what it is. There's no sense to it. The rules are simple, so you understand the calculation. But there's no sense in it. It just doesn't mean anything at all. Do the exercise. Write the truth tables of all sorts of formulas. And you will see. No sensible entailment and implication. None at all. The only reason they've adopted this calculus is that it seemed to give the correct results. They adopted it because they are mathematicians. No sense at all but apparently the correct results. However, it's not true for all formulas. So, really, nothing at all "insightful". It's plain moronic.

Originally Posted by

**fast**
However,

If the house is wet, the house won’t burn down.

The house is wet.

Therefore, the house won’t burn down.

Is that valid? Is that sound?

The first premise is true, for the particular house in question is impervious to fire, wet or not—and a house impervious to fire, wet or not, will not burn down. The second premise is true because it’s raining on the house and the rain is wet. The conclusion is true for the same reason the first premise is true.

Nah, you can't do that. If you consider a particular house, then it's no longer the same argument. In effect, you are smuggling in a new premise, "the house is impervious to fire". You need to analyse the argument as it is, without redacting it.

If you want an argument with all true premises and conclusion, here it is:

p1 True;

p2 True;

C Therefore, true

Now you have an argument which is trivially valid and literally doesn't "mean" anything at all, which is what you just called "insightful" and "sensible". But I'm fine with it, as long as you don't redact it. It's valid, but useless, except for methodological considerations.

Originally Posted by

**fast**
The structure seems to make sense, but before knowing the reasoning for why the first premise is true, there’s so much doubt because there’s the appearance that the house being wet is somehow relevant when it’s not.

You don't seem to be aware that "relevance" is precisely the word used by mathematicians doing mathematical logic, not exactly mainstream but still "classical". Why don't you abide by their notion of "relevant" validity?

Further, I never bought the notion of relevance myself and I now understand why it is... irrelevant. And it is easy to demonstrate why. Very easy.

So, you are beating a dead horse here. All these notions like "relevance" mathematicians have imagined are just that, imagined. They are not properly justified because to justify them they would need the empirical evidence provided by an investigation of logic as the logic of the human mind, something mostly they think would be suicide.

Originally Posted by

**fast**
It goes to show that truth is a requirement for your sense of validity (as it should be with your sense of validity) and are led to think it’s not valid because it appears untrue whereas validity (mathematical style) won’t make for a sound argument unless all premises are true.

Nah. You don't get it. It's not and never was a question of the premises being true or false.

Originally Posted by

**fast**
When we use the very same words to convey different things, a convoluted interpretation is bound to ensue. I suppose many mathematicians might not have become so well-versed on non-glossary terms. If all they know is their specific usage and it’s engrained in them, they probably will deny such seemingly uneducated interpretations. They do have a word for your use of validity: “sound.” So, it’s not that they wouldn’t recognize validity—it just goes by another name. Truth must be separated from structure in order to isolate them. They’ve just adopted the term and chizzled away truth pertainment.

Still, there are arguments that are clearly unsound because of truth alone that you find valid. Right? Curiously right? So, there is still something deeper about your view that I haven’t quite gotten down.

Because I don't explain. Sorry for that but I'm not here to teach logic. I am investigating. Of course some arguments are valid and unsound:

Trump is a Martian;

All Martians are presidents;

Therefore, Trump is president.

We all know this argument is valid and unsound, yet, our reasons are different from the reason given by mathematical logic.Truth table logic says it's valid because the premises are false, which is just plain meaningless and not at all "insightful" and "sensible". We will all say instead it is valid because the conclusion is necessarily true given the premises. Which makes sense. And it's all formally valid and validity has nothing to do with whether the premises are true or false.

Originally Posted by

**fast**
I recall once you saying something seemingly derogatory about statements including the word “if” and how it factored in somehow—something about needing to be tested against the real world? Kind of like my first premise not sounding particularly probable.

Possibly, but that would be a different context, perhaps the online test where you have to say which cards would need to be flipped over to check whether there is a given implication? Yes, and the test is just wrong, made by people who don't understand logic because they didn't even try to understand it.

EB