He says the "if" at the beginning because the logical structure of the axiom is one big material conditional (indicated by the arrow in the middle) where the antecedent is □(P->Q) and the consequent is □P->□Q.
@ACARook ...That is, if you think of the antecedent as I wrote above as A and the consequent as B the structure of the axiom is just A->B (“if A then B”) but A happens to be “it’s necessary that if P then Q”...so if you plug that in for ‘A’ in the English translation of the axiom you get “if it’s necessary that if P then Q...” for the antecedent, that’s why he started out by saying “if”, it’s not a part of what □ "says"
Professor Campbell, First and foremost, thank you for making this video series. You're doing the TH-cam community a great service by offering your time and expertise in such an accessible way, and I commend you for that. That aside, I can't help but notice a potential issue with your phrasing of these modal propositions. They may be minor issues, but nevertheless I'll share them with you since they've come to mind many times while watching this video. My first issue is a general issue and it concerns your expression of modal propositions. You translate □P as "If it is necessary that P", and I find this somewhat cumbersome. While it may be easy for you to read it this way, to many people □P may be more easily read as "Necessarily P". I feel the benefits of reading it this way is twofold: it is true to the notion being expressed, and it has the added convenience of being much shorter and less difficult to follow. My second issue is with your offhand remarks concerning P. More that once you let P be some object (eg. at 7:35-7:50 you refer to P being some object that exists). I feel this may be confusing to viewers as they'll later learn P need not be an object. For example, we could let P be the antecedent to some conditional statement (i.e. Let P be "If 4 is greater than 2"). Given we can express necessary propositions that lack objects, I feel it's important to avoid use of "exist" here [especially since ∃ may confuse students down the road already]. That aside, thanks for sharing and keep it up! Adam
An existential qualifier (the notation of an object's existence) is different from saying that it is or is not the case. If P=it is raining, and we make the statement, P, it is to say that it is the case that it is raining. If we say ~P, it is to say that P is not the case or that it is not the case that it is raining. This is different from saying that the negation of an object means that it doesn't exist. Maybe it's different in modal logic. Some clarification would be appreciated.
Good job. But you shouldn't say f.ex. "it is necessary that P exists" for "necessarily P". The reason is that P cannot be an object, but is instead a proposition (or a formula, sentence, whatever). The existence of an object is not at all the same as the truth of a proposition, otherwise any type of description would be an existential hypothesis.
He says the "if" at the beginning because the logical structure of the axiom is one big material conditional (indicated by the arrow in the middle) where the antecedent is □(P->Q) and the consequent is □P->□Q.
@ACARook ...That is, if you think of the antecedent as I wrote above as A and the consequent as B the structure of the axiom is just A->B (“if A then B”) but A happens to be “it’s necessary that if P then Q”...so if you plug that in for ‘A’ in the English translation of the axiom you get “if it’s necessary that if P then Q...” for the antecedent, that’s why he started out by saying “if”, it’s not a part of what □ "says"
Professor Campbell,
First and foremost, thank you for making this video series. You're doing the TH-cam community a great service by offering your time and expertise in such an accessible way, and I commend you for that.
That aside, I can't help but notice a potential issue with your phrasing of these modal propositions. They may be minor issues, but nevertheless I'll share them with you since they've come to mind many times while watching this video.
My first issue is a general issue and it concerns your expression of modal propositions. You translate □P as "If it is necessary that P", and I find this somewhat cumbersome. While it may be easy for you to read it this way, to many people □P may be more easily read as "Necessarily P". I feel the benefits of reading it this way is twofold: it is true to the notion being expressed, and it has the added convenience of being much shorter and less difficult to follow.
My second issue is with your offhand remarks concerning P. More that once you let P be some object (eg. at 7:35-7:50 you refer to P being some object that exists). I feel this may be confusing to viewers as they'll later learn P need not be an object. For example, we could let P be the antecedent to some conditional statement (i.e. Let P be "If 4 is greater than 2"). Given we can express necessary propositions that lack objects, I feel it's important to avoid use of "exist" here [especially since ∃ may confuse students down the road already].
That aside, thanks for sharing and keep it up!
Adam
An existential qualifier (the notation of an object's existence) is different from saying that it is or is not the case. If P=it is raining, and we make the statement, P, it is to say that it is the case that it is raining. If we say ~P, it is to say that P is not the case or that it is not the case that it is raining. This is different from saying that the negation of an object means that it doesn't exist. Maybe it's different in modal logic. Some clarification would be appreciated.
Jason, thanks for posting these...very helpful.
Great job. Enjoying your videos.
@ACARook oh yeah!! duh! i totally missed that (though not sure if doc explicitly mention that), thx
aren't those basically the Q.N. rules?
But does the necessity operator in N(p->q) Distribute to the implication? :P that is the question.
fantastic series, a huge help!
Good job. But you shouldn't say f.ex. "it is necessary that P exists" for "necessarily P". The reason is that P cannot be an object, but is instead a proposition (or a formula, sentence, whatever). The existence of an object is not at all the same as the truth of a proposition, otherwise any type of description would be an existential hypothesis.
You are a logic god
I think that's the Bayesians and their calculus.
Well, if Modal Collapse obtains then that's not true.
negative plus negative is positive thats it