Logic Lecture: Introduction to Modal Logic - 5

Logic Playlist: http://www.youtube.com/view_play_list?p=C88812FFE0F526B0

Category:

Education

Tags:

• mathematical logic
• necessity
• possibility
• system
• iteration
• s4
• s5
• axiom
• axioms

[francopalombo]

it great, thenks.!!! but, when A is posible, and then it is posible that A is posible.. the redundance is significative. Im Iost?

[francopalombo]

@francopalombo about the chocolate: if you say taht it is posible that you may give me a chocolate ( or a civil right, to makeit more human science..).. then it is not necesari truth that you will. It is posible that it is posible. It is NOT necesari posible.

get what a im asking?

[drjasonjcampbell]

@francopalombo All that's being stated is that if X is possible, then it must be the case that it is possible (necessarily possible) but using the rules of simplification we can reduce the string of operators to the last operator in the string. If something can happen, it must be the case that it can happen

[drjasonjcampbell]

@francopalombo What the process of simplification says is that there is no need to have repeating operators. possibly possible can be simplified to possible. necessary necessary can be simplified to necessary. That's all that's being stated.

[alawrence89]

Thank you for the series. It's not as obvious or a matter of common sense for me. The distribution axiom doesn't make good sense to me. I understand that someone can distribute the necessity but I'm not convinced that the result is true.

The other thing that I found confusing was the concept of a claim being necessisarily necessary. That just makes no sense, really. Something being necessarily possible does, though. It doesn't just happen to be the case that it's possible. It must be.

[drjasonjcampbell]

@alawrence89 You're right. Something being necessarily necessary is redundant so the simplification rule says there's no need to have repetition in the operators. possibly possible can be simplified to possible. necessary necessary can be simplified to necessary. That's all that's being stated. Peace

[Rekenbij]

@RonnnieBuckley i don't believe something can possibly be necesarry :) because possibly necesarry does not mean that it is also possible that it is not necesarry, but it does mean that it is not necesarry that it is necesarry, which is impossible. (therefore PN=P does not exist)

[Rekenbij]

Thanks man :) i really appreciate this ghetto philosophy. (btw. in proving synchronic contingency, the redundancies are not redundant. (but i know you know that there are many uses for the redundancies).

[RonnnieBuckley]

Question:

I was thrown off when you said to regard the last operator, as I'm sure this isn't always the case. Regarding the last operator of the string, what if something is possibly necessary? Wouldn't that make it possible, rather than necessary?

NN = N

NP = P

PP = P

PN = P

Also, thanks for the lesson!

[oxxxxymoron]

Thanks doc, very informative and really gets to the heart of the subject. I hope others will follow in your example!

[francopalombo]

ok..



[theobamathellama]

@francopalombo with the chocolate it think you are confusing the terms of necessity and possibility with truth. if it is possible that he will give you chocolate then it is necessarily possible that he will give you the chocolate. This means in order for it to be possible for him to give you chocolate then it is necessary for the possibility of him giving you chocolate to be in existence.

[Empathica1]

I don't quite understand S4 and S5. is it that you can replace A with box A, thus m becomes box box A arrow box A, and 4 is the converse?

[Ubernoob85]

Thanks for this series! I have but one question; where do you suggest one goes from here for a deeper explanation. Do you have any recommended books or (better) free web-based materials?

Thanks in advance! :)

[LaureanoLuna]

@alawrence89 I think it was Oskar Becker, a german mathematician, who proposed these laws:

If p is possible, then it is necessarily possible

If p is necessary, then it is necessarily necessary

Modal concepts are a matter of necessity: unlike the size of the Earth or the color of snow, they are not subject to variation from one possible world to another because they are logical in nature and every logical issue logical is a matter of necessity. This is the philosophy behind S5.

[alawrence89]

@LaureanoLuna hmm. I will have to think about this one for some time. Can you "dumb" it down for me some? haha. Thank you.

[LaureanoLuna]

@alawrence89 The distribution axiom (also called axiom K) is intuitive: all it says is that if a conditional is necessary and its antecedent also is necessary, then the consequent is necessary.

As Jason says, in this system all strings of modal operators can be reduced to the rightmost one. But this does not mean that iterated operators are semantically redundant. Philosophically, it is important that necessity and possibility are a matter of necessity: any modal claim is necessary if true.

[colourmegone]

Thanks very much, I enjoyed the series a lot. I started reading the Wikipedia article on ML and began to get lost, this is a much simpler presentation. I'll go back to Wiki and see if I can slog through it.

[ACARook]

I could see somebody saying that something that was true in all possible worlds (2+2=4, say) was "necessarily necessary" whereas something contingent to the structure of our universe (like the gravitational constant) was merely "necessary" (i.e. we have no reason to think that there couldn't be another universe in which the gravitational constant was some other number but in our universe [by some necessity of it's arrangement] it's, let's say, 'contingently' necessary.

[amjiva]

Another great series. Thanks.