Therefore, there is a god

[Speakpigeon]

I've been trying for quite some time to assess how much formal logic, as opposed to just our intuitive logical sense, is necessary to maths, computing sciences and physics. I drew a blank. Maybe if you have information on that...

The law is an example of arbitrary rules and it works fine. Chess has arbitrary rules and it works fine. So, you'll have to be more specific about what your argument is here.
EB

Chess is not in the business of predicting outcomes. Neither is the law (which isn't entirely arbitrary, but regardless). Now, if we came up with completely arbitrary rules, one would expect that the universe would not oblige and behave in accordance to models developed through extensive usage of arbitrary rules. It would be hugely surprising.

Physics, on the other hand, is meant to be able to (at least) predict phenomena, on the basis of some amount of information. It does that through a mathematical model. And it is extremely good at it. The question of how much formal logic is required is not relevant in this context, because mathematics also uses our intuitive logical sense (what else do you think formal logic is based on?), and math does not use the interpretation of 'if' in the ordinary sense, in which your first premise would be intuitive - but the argument invalid -, and because math does use the sense of 'if' in which the argument would be valid (and you have no good reason to even suspect that the first premise is true) is pretty common.

"mathematics also uses our intuitive logical sense (what else do you think formal logic is based on?), and math does not use the interpretation of 'if' in the ordinary sense, in which your first premise would be intuitive - but the argument invalid -, and because math does use the sense of 'if' in which the argument would be valid (and you have no good reason to even suspect that the first premise is true) is pretty common."

I'm a bit lost as to what you're trying to say. As I see it, there's just one logic and that's the logic that's done by our brain and essentially all our neurons individually, and that comes out as logical intuitions. Any reasoning that we do, right or wrong, relies on it. You can take any reasoning as a crude formalisation of logic. Aristotle went a step further but no one has improved on that since Aristotle. The mathematisation of logic done in the 19th and 20th centuries is obviously a progress but on form only, not on substance. So, as I see it, mathematicians can indeed only use their own intuitive and essentially Aristotelian sense of logic to do any maths but I fail to see where would be the difference with the kind of reasoning done routinely by ordinary people. People ordinarily do all sorts of reasoning, including with premises they think are false (counterfactuals) or that they just assume as true for the sake of the argument. I expect mathematicians to do exactly the same thing.

Of course, any such reasoning can be expressed through language, and so in effect formalised, but without going into the kind of formalisation using truth tables. So, my guess is that mathematicians don't need and and don't use truth table logic. You seem to agree at least with that last bit. This leaves open the question of the practical use of truth table logic, if any. I haven't found any example of that. Truth table logic seems to be just an object of study without any application whatsoever. Maybe I'm wrong on this but I haven't any example that I would be.
EB

[Speakpigeon]

I've been trying for quite some time to assess how much formal logic, as opposed to just our intuitive logical sense, is necessary to maths, computing sciences and physics. I drew a blank. Maybe if you have information on that...

The law is an example of arbitrary rules and it works fine. Chess has arbitrary rules and it works fine. So, you'll have to be more specific about what your argument is here.
EB

Whitehead and Russell, Frege, Peano, Cantor, and others. There is a large and technical corpus of works on the subject of the logical foundations of math. You will need to do a little homework to really have some understanding about logic and it's connection to mathematics. An online forum is unlikely to be able to supply you with a royal road to understanding the subject I am afraid.

Thanks for the advice and that's indeed what I'm doing. I have a good scientific library nearby where I live. It's fun to see how little these people understood of logic.

Please note that I make the distinction between logic and methods of logic. I take logic to be what our brain does and as such there's just one logic. I'd love to see any conclusive counterexample to that. I also believe Cro Magnon also had the same logic as us. Methods of logic on the other hands are multiple and at best efforts to find a way to calculate logical formulas. However, unfortunately, all existing methods seems either irrelevant or just plain wrong.

Forums have their limitations but they are definitely useful. I can't argue but Russell himself you know and all these people have been necessarily biased in their views. I think talking to ordinary people like here is necessary. That would be true whatever the subject.
EB