Actually there's only one.
I use the word 'logic' in the sense that logicians use it. People who are mathematicians, computer scientists, and -- gasp and horror -- even philosophers. I myself was a logician once and spent several years in a university department doing research on a specific corner of symbolic logic. In particular, I spent most of my time trying to find ways to use different logics to solve actual real world problems. Problems like: "is it possible that this distributed computer system goes into a deadlock?" Doing that I became very aware of the limitations of classical logic.
You, on the other hand, use the word 'logic' in some sense that no actual logician who is currently alive uses it. And you apparently know so little about logic that you don't even realize it and actually believe that it is a single unified monolith. Right now you are acting like the guy in the other thread who goes on about "hard problem of consciousness" without having any idea what brain research has found out during the last decades.
[What in the blue (or yellow) hell are you talking about?
What I mean is that with classical logic it is not possible to say that something is usually true but that there may be occasional cases where it isn't because something exceptional happens. The commonsense reasoning that each and every person does can handle these exceptions without a problem. Classical logic can't.
Let me give a real example and use what happened to me yesterday morning. I went to fridge and noticed that there was no milk. My thought process after that can be approximated with:
P1. I need milk
P2. There is milk in shops
P3. It's 7:30 am on a weekday
P4. The nearby shop opens at 6 am on weekdays
Conclusion 1. The shop is open now
Conclusion 2. I can go to the shop and buy milk
Simple commonsense reasoning. I do reasoning like this every day.
You do reasoning like this every day. How could we represent this with classical logic? Well, we need to throw in a bunch of predicates but I can't be arsed to do it for the whole example. Instead, let's focus at P4 in more detail.
Let's use 's' to denote that particular shop. Then the straightforward way to model P4 is:
P4.1: forall D, T . (current_day(D) and weekday(D) and current_time(T) and T > 6 implies open(c, D, T))
Intuitively, this says that if the current day is D and current time is T and D is a weekday and T is later than 6, then the shop 'c' is open.
The problem here is that the 'forall' quantifier requires that the formula holds for each and every possible value of the variable. But in the real life there are always exceptions. Even though the shop's posted opening hours say that it opens at 6 on each and every weekday, it doesn't actually do so. Two months ago there was some accident there and they were closed for a day for repairs.
The proposition P4.1 is not a suitable way to formalize P4 because using it lets me to conclude that the shop is open also on days that it is actually closed.
We could also try to model it with the other quantifier:
P4.2: exists D, T . (current_day(D) and weekday(D) and current_time(T) and T > 6 implies open(c, D, T))
This has the problem that the 'exists' is true when there is at least one day D when the shop is open at time T > 6. I can't use this rule to infer that the shop is open during any specific day and time. It is completely useless from the knowledge representation point of view.
Those are the only two possible ways to introduce variables in classical logic. Either something has to be always true, or it is enough that it is true only once. One doesn't happen in real life and the other is useless for reasoning.
To catch the reasoning in the argument, we need to be able to represent the concept 'usually'. A way to say that "this statement is true unless some exceptional situation happens" and then a way to introduce the exception when it actually happens. You can't do these with classical quantifiers. You can't do these by adding auxiliary predicates. If you are really interested in finding out why not, you can read any primer on
nonmonotonic logics.
Researchers spent a lot of time between 1960s and 90s trying to build AI system on top of classical logic. On the whole the attempts failed. They could get some toy problems working but practically nothing that would be useful in real life. The main reason for that is that classical logic is a damned fragile system where your models break if you look at them in a wrong way. And a big reason for that you have only 'for all' and 'exists' for quantifiers.
Those are not forms of logic. You simply have no idea what we're discussing. It's odd because we've explained and detailed what we're discussing. I can only surmise that you didn't read anything in this thread.
I freely admit that I haven't read each and every post in this thread. I'm just objecting to your misuse of the word 'logic' that you have repeated time after time.
Stop and listen to yourself for a while. You are saying that Linear Temporal Logic is not a form of logic. Doesn't that sound any strange to you? Not only that, it's a logic that wasn't invented by one of those filthy stinking useless philosophers, but by a computer scientist who then went to win the most prestigious award in computer science for that very invention. The ACM thought that the logic was worth of the Turing Award. But no, the omniscient Belz knows better than some propeller heads. I guess that the board of ACM are actually philosophers in disguise.
You certainly are doing a good impression of a "why are you speaking about brains when we are speaking about minds" guy.
But anyway, I'm not going to spend further hours of my life writing posts your you to ignore in 10 seconds.