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Too much modal logic is bad for your health

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1/21/2015 6:14:52 AM
Posted: 2 years ago
Let's talk about some other logic.

Modal logic is all anyone ever talks about on DDO, and it get's boring after a while (in every possible world). I'm not sure how many people take the time to acquaint themselves with logic, but if you plan to do so, don't go with modal logic. Start with Categorical logic, then move to propositional logic, then finish with the modern system of predicate logic.

So, I'm raising the contention that all philosophers are stupid. 'Tis a simple statement that is represented differently in categorical logic and predicate calculus. CL would show the statement as Sp, but predicate logic is far better as you can make universal and existential claims, and then apply laws of inference by using nifty techniques such as universal/existential instantiation/generalisations to draw valid conclusions that categorical logic, and modal logic, cannot draw. (lots of other stuffs, but intro first)

(Note, 'A' means universal quantifier and 'E' means existential quantifier, since I can't use the actual symbols).
(Ax)(Px->Sx) Is how the contention is represented. If x is P, then x must be S too.
The existential quantifier is used this way: (Ex)(Px /\ Sx), which reads: 'There exists an 'x' where x is a philosopher and is stupid.
Note that Fx is bound, you can't even pull simple sh1t like modus ponens when all you have is bound variables. That's why instantiation and generalisation saves the day.
let's suppose your contention is:
1. (Ex)Ux->(Ax)(Sx->Lx)
In words: If there exists at least one unicorn, then all scientists are liars. Now lets apply drugs, and show that Bob is a liar, because he is a unicorn scientist.
2. Ub /\ Sb
Definition, Bob is a unicorn and a scientist.
How do we prove that Bob is a liar? You might think it follows that since Sx->Lx, Sb->Lb, but that's illegal logic right there. Can't use the bound stuff, since it is enclosed in the brackets. There is an easy answer, simplify and then generalise. We are told that Bob is a unicorn, so let's make that premise 3, because we can generalise this statement:
3. Ub
Existential generalisation is where we denote the constant with a variable, and state that there is at least one of it. Since Bob is a unicorn, we can generalise and say that there is a unicorn that exists:
4. (Ex)(Ux)
Above reads: a unicorn exists. Since we can establish that, modus ponens can be used, since the following operator isn't bound. (Ex)Ux->(Ax)(Sx->Lx)
5. (Ax)(Sx->Lx)
It follows, as you can see. Now we want to work with this statement, and show that Bob is a liar. To do so, we need to do a universal instantiation to make Bob a scientist:
6. Sb -> Lb
From line 2, we have Ub and Sb are both true. So line 7 can be:
7. Sb
C: Lb
And that's why Bob is a liar, courtesy of predicate logic. Of course, this has been massively simplified, you're never going to have to analyse such simple statements if you go on to do some further study. Not a lot of what makes predicate logic what it is has been discussed. So I invite you to do something about that, and sacrifice the DDO worship of modal logic.

Yup. ^.^
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