What is Modal Logic? - Explained for Major Elective

[Math Is Hard]

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!

 

[quantumdude]

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

 

[Math Is Hard]

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.

 

[quantumdude]

Hi Tom,
My goodness! I don't think I have ever seen anything like that.

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.

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.

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.

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?

Questions of necessity and of possibility.

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...

 

[Math Is Hard]

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?

 

[quantumdude]

Math can be said to have the property of being logical.

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.

Can logic be said to have the property of being mathematical?

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.

 

[Math Is Hard]

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.

 

[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 possibility to necessity.

 

[Math Is Hard]

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?

 

[Math Is Hard]

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?

 

[chronon]

Can logic be said to have the property of being mathematical?

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.

 

[plover]

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?

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

 

[plover]

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.

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.

 

[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.

 

[plover]

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.

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.

 

[quantumdude]

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.

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.

But I would, in fact, be surprised if there were universal agreement that logic was a branch of mathematics.

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.

Something very fundamnetal is missing in modal logic:


The LOGIC OF POTENTIALITY!

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

 

[quantumdude]

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?

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.

 

[quantumdude]

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.

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.

 

[Math Is Hard]

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.

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

 

[Math Is Hard]

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

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

 

[Philocrat]

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.

Well, (a) is typically your trademark...it just cracked me up; I just couldn't stop laughing. (b) is the close better. Transitional logic picks every fact in transit from possibility to necessity. Can you imagine trying to sell (b) to your fellow scientists and philosophers before the Gene (DNA) Gold Rush...they would think that you're mad.

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?

Good question! I suspect that Transitional Logic is Bidirectional while in transit, and I think only in the 'final perfect state' may the pathway back in history be permanently closed. If you can show that the state of being a unicorn is the final or absolute perfect state, then the pathway in the spatiotemporal history from unicorn back to horse is closed under this condition. Otherwise I cannot see any reason why the genetic engineer who turned a horse into a unicorn could not reverse the process, unicorn back to horse. I cannot see this being construed otherwise.

One more thing:

The Logic of Necessity does not necessarily imply the 'Final Perfect State' (FPS). I always equate FPS with a state where physical things overcome physical destruction along with what is left of their causal and relational properties. The logic of necessity could range over to PFS. There is a dispute about this in philosophy, but I just prefer to steer clear of it.

 

[Philocrat]

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.

Well, Aristotle had already shown that Actuality never undermines potentiality, that things retain their potentialities while in actuality. That, for example, when a horse is running it still retains its potentiality to do so. Wheter this goes down well with everyone is a matter of taste. In terms of a caterpillar potentially becoming a butterfly, transitional logic gracefully accommodates this. For the facts of being a butterfly is wholly contained in the facts of being a caterpillar otherwise there would not be a transition from one to the other.

Two things can be derived from this:

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

But you do know that intelligent beings have the natural capacities to model physical things ahead of their physical realisation or production. We do this all the time. A unicorn is a mental model of reality. It is a mental modification of real horse in the outside world. We create plans of houses and bridges in our heads before we built them. And the fact that we humans carry a mental image of a unicorn in our heards for thousands of years seems to me to be irrelevant.

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.

Transitional Logic does not allow things to wander this far...especially things wanting to be facts. However, this may not necessarily be completely ruled out in the logic of possibility. I have no stamina for such controversy.

 

[Philocrat]

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

Somewhat close...but the Logic of Potentiality is not just about mere interactions of opposites in your very dialectic sense; rather, it takes a more concrete stand.

 

[Math Is Hard]

Thanks, Philo. I think I am seeing this a little more clearly now. The "mental model" is still a little confusing to me though, so I'll have to look into that more.

When you said earlier that the logic of potentiality was missing from modal logic, does that mean that modal logic does not contain the symbols to represent something like:
A is potentially B
?

 

[Philocrat]

Thanks, Philo. I think I am seeing this a little more clearly now. The "mental model" is still a little confusing to me though, so I'll have to look into that more.

When you said earlier that the logic of potentiality was missing from modal logic, does that mean that modal logic does not contain the symbols to represent something like:
A is potentially B
?

Perhaps, it does. I have speculated around a lot of symbols, but I still can't find one. Can you think of one, if any? Unless you are suggesting that the Logic of Potentiality is already contained in those of possibility and necessary. Well, I am thinking otherwise.

 

[Math Is Hard]

Perhaps, it does. I have speculated around a lot of symbols, but I still can't find one. Can you think of one, if any? Unless you are suggesting that the Logic of Potentiality is already contained in those of possibility and necessary. Well, I am thinking otherwise.

I honestly couldn't even begin to conjecture on that. I am just taking my first baby steps in logic. I am really interested to hear what others would say on this, though!

Another thing I was wondering about: is there any symbolic representation for the probability of something? For instance, could you express that
A is likely to be B
or
A is not likely to be B
?
Or is that completely out of bounds? I am thinking that it is too ambiguous, but I don't know for sure.

 

[Philocrat]

I honestly couldn't even begin to conjecture on that. I am just taking my first baby steps in logic. I am really interested to hear what others would say on this, though!

Another thing I was wondering about: is there any symbolic representation for the probability of something? For instance, could you express that
A is likely to be B
or
A is not likely to be B
?
Or is that completely out of bounds? I am thinking that it is too ambiguous, but I don't know for sure.

You can. Some logicians use venn diagrams to try to explain it, and some use symbols similar to yours...and there are those who just default to hard statistical symbolisation. I will find some links that explain this well later.

But I say this;

although it is good to give to the King what is King's, however, symbolisation in all forms of logic is better understood when you think about it properly and play around with your own imaginary symbols: they just help you understand things much more clearly. But when it comes to exams, stick to your tutor's symbolisation.

 

[Philocrat]

But then again if you ask me "what is the epistemological status of probality and its symbolisation?", my immediate answer would be; "it shares the same epistemological status as the logic of possibility." I lock them up in the same cage of knowledge.