# What is modal logic?

1. Aug 31, 2004

### Math Is Hard

Staff Emeritus
This philosophy course is listed as a possible elective for my major, but I don't have a clue about this description.

First course in two-term sequence (also see course 176). Topics include various normal modal systems, derivability within the systems, Kripke-style semantics and generalizations, Lemmon/Scott completeness, incompleteness in tense and modal logic, quantificational extensions.

Can anyone tell me briefly what a modal system is? Thanks!

2. Aug 31, 2004

### Tom Mattson

Staff Emeritus
I'm off to bed due to an early class but for now you can chew on this, which is from one of my favorite websites.

Modal Logic

3. Sep 1, 2004

### Math Is Hard

Staff Emeritus
what a great site!

Hi Tom,
My goodness! I don't think I have ever seen anything like that. Modal logic looks very complex, but interesting. I am just now finishing my very first course in logic. I have seen the arrow symbol, as in P -> Q, but that's about the only notation that looked familiar.
It says that "an understanding of modal logic is particularly valuable in the formal analysis of philosophical argument". What sort of questions do you tackle with modal logic?
Thanks for that link. I have been looking though that entire site and there's an abundance of good information!! I have a final exam tomorrow and I think that site will be very helpful in preparing for it.

4. Sep 1, 2004

### Tom Mattson

Staff Emeritus
Yes, the Stanford Encyclopedia is really great. When I found it, I tracked down all the articles dealing with logic, printed them out, and put them in binders. It's like getting a few free textbooks.

The other logical symbols are defined at the beginning. Basically, it is regular propositional/quantificational logic, with two new operators:

It is necessary that (denoted by a box)
It is possible that (denoted by a lozenge)

The operators obey rules that look formally similar to the universal and existential quantifiers. If I knew how to TeX, I'd post those rules here. Later I'll dig through that document and find them.

Questions of necessity and of possibility. :tongue:

In addition, some accounts of modal logic include all of the operators listed at the beginning of the document. You probably noticed when taking your course in logic that translating statements from English to logic sometimes resulted in a loss of meaning. For instance, you would translate the statement "Radiation causes cancer" as "If you are irradiated, then you will get cancer", or "r-->c". But the conditional has nothing to do with causality, so that shade of meaning is lost when translating into propositional logic. Modal logic is the result of an effort to allow for greater depth of expression in formal statements.

More later, must get back to work...

5. Sep 2, 2004

### Math Is Hard

Staff Emeritus
There definitely seems to be a finer level of granularity in these techniques. What *I think* I am seeing is a distillation in the meanings of the statements and a refinement of the connections through these symbolic representations.

This brings me to a question that's been nagging at me Tom. I hope you can respond when you have time to answer. No rush - it's just a curiosity I have.
I realize that math and logic are two totally separate entities, but this is something I wonder about:

Math can be said to have the property of being logical. Can logic be said to have the property of being mathematical?

6. Sep 3, 2004

### Tom Mattson

Staff Emeritus
Yes, look at any theorem in your math books, you can see that the structure of the statements is such that they can be written using the operators "not", "and", "or", "if...then", and "iff". In fact, I take my Calc II classes through conversion, inversion, and contraposition of conditional statements before we do infinite series, because Calc II students consistently screw up the logic of the n-th term test for divergence.

There's at least one other guy out there who agrees with me, because he wrote an online calc book that has a whole chapter on formal logic. I'll dig up the link and post it here, if you want.

There's not a single, well-defined answer to the question "What is mathematics?" In truth, mathematics is whatever mathematicians decide it is! But certainly formal logic has all the hallmarks: abstract objects, unary and binary operations, axioms, and theorems. I'd be surprised if there was not universal agreement that logic is a branch of mathematics.

7. Sep 3, 2004

### Math Is Hard

Staff Emeritus
Ah, Tom. What a great pity it is that I will never be able to take one of your math classes! I am so awed by the care and thought that you put into preparing your students.
Thanks for your response to my question. I definitely need to think about this some more - especially the comparisons that can be made between logical and mathematical operators.

Last edited: Sep 3, 2004
8. Sep 3, 2004

### Philocrat

Something very fundamnetal is missing in modal logic:

The LOGIC OF POTENTIALITY!

Think of this:

"A horse is Potentially a unicorn"

The proximity of this is tighter than is of the Logic of Possibility. It's better than a mere possibility. Although this may be enhanced by preloading it with a domain or a boundary, but I think this is irrelevant. Perhaps this is better called 'TRANSITIONAL LOGIC' to widen its scope to also cover facts captured in transit from possibilty to necessity.

Last edited: Sep 3, 2004
9. Sep 3, 2004

### Math Is Hard

Staff Emeritus
That is a VERY interesting thought, Philo. But I want to clarify a little bit so that I might understand better. When you say a horse is potentially a unicorn, do you mean

a) The particular horse in question is already a unicorn and we just haven't noticed because we haven't inspected the horse closely enough to see the horn budding from it's head?

b) The (ordinary) horse has the capacity to become a unicorn through manipulation of its DNA.

c) None of the above.

Can we say formatically and very generally speaking, that all unicorns are horses but not all horses are unicorns? Or does an animal cease being a horse when it is classified as a unicorn?

10. Sep 4, 2004

### Math Is Hard

Staff Emeritus
I had a similar thought along those lines. How does the horse potentially becoming a unicorn logic compare to or contrast with this statement:

A caterpillar is potentially a butterfly.

Two things can be derived from this:

It is possible that a caterpillar will become a butterfly. (unless someone steps on it and squashes it, for instance).
It is necessary that a butterfly was once a caterpillar.

The problem that's bugging me is that the process of caterpillars becoming butterflies has been observed. Horses becoming unicorns has not been observed.

It seems that if we use the logic of a horse potentially becoming a unicorn, then we could also just as well say that a horse is potentially a cup of Earl grey tea.

But maybe my thinking is too limited here?

11. Sep 4, 2004

### chronon

I would hope so.

Aristotle made a list of 19 true syllogisms (out of a possible 256). These would have been studied by a great number of people over the next two millennia. It was only with the work of Boole in the mid nineteenth century, writing them in symbolic form, that one of them was found to be wrong (or at least ambiguous). Presumably other forms of logic can be dealt with in a similar way.

One thing that I constantly find myself thinking when reading philosophy is that it would make more sense if it were reduced to some sort of symbolic notation and treated mathematically.

12. Sep 4, 2004

### plover

If the above has any bearing on your tea-making practices, 'limited' is not the first word that would spring to mind...

13. Sep 4, 2004

### plover

My impression has always been that logic is considered (at least by logicians) to be more fundamental than mathematics. I assume the argument would be that you have to know what underlying logical axioms you are using before you can define anything so particular, so mundane, and, as it were, ahem, not to put too fine a point on it, so concrete, as a mere mathematical object.

I imagine that, in the end, there is a fair amount of room for defining the scope of what is meant by 'logic' and by 'mathematics' and that this ambiguity leaves room for either (or neither) to be defined as fundamental. But I would, in fact, be surprised if there were universal agreement that logic was a branch of mathematics.

14. Sep 4, 2004

### chronon

Clearly logic has to be described in some language. If you accept mathematics as a language, which can be used to describe logic, then you might think of mathematics as being more fundamental.

15. Sep 4, 2004

### plover

This is where the scope for definition appears. Dpending on how you think about it, the reverse of your statement makes just as much sense:
Clearly mathematics has to be described in some language. If you accept logic as a language, which can be used to describe mathematics, then you might think of logic as being more fundamental.
I don't have a strong preference. Both seem workable. I suspect it's a matter of context which is more useful.

16. Sep 4, 2004

### Tom Mattson

Staff Emeritus
I found the book in my files. It is called Calculus for Students of Mathematics by Michael Dougherty. Unfortunately, he took it off the web (but I did save a copy). Symbolic logic is covered in the very first chapter.

I wasn't commenting on fundamentality. What I meant was that if one mathematician says to another, "What's your specialty?" and the other says, "Logic", the first mathematician is not going to balk and say, "You're no mathematician, you're a philosopher!" or some such.

This is interesting. What you describe sounds a lot like the dialectical logic of Hegel/Marx/Engels. Is that close to what you mean?

17. Sep 4, 2004

### Tom Mattson

Staff Emeritus
I don't think that's really a problem, because we're talking about a deductive system here. Deductive systems aren't concerned with determining the truth values of statements, but rather with the validity of inferences. So if it is impossible for a horse to become a unicorn, then the statement, "A horse is potentially a unicorn" is simply a false statement.

18. Sep 4, 2004

### Tom Mattson

Staff Emeritus
Me too! Especially when the amateur philosophers on this Forum try to talk about physics! That's one of the things I really like about the Stanford Encyclopedia--the arguments are so clear and precise.

19. Sep 4, 2004

### Math Is Hard

Staff Emeritus
You're right. I was getting distracted by the content. Garbage in, garbage out.

20. Sep 4, 2004

### Math Is Hard

Staff Emeritus
remind me to tell you my really horrid "Koala tea of Mercy" pun some time!