P implies Q, how to read in english

Question

how to read P implies Q in classical logic?

example :

Distributivity:  

Ka(X->Y) -> (KaX -> KaY)

This is modal logic which uses classical logic rules.
KaX : a knows the that X is true.

I m curious about how to read implication in english? if then else?

Edit : in Modal Logic, Ka becomes Box, well it s boxed shape sign, that symbolizes necessiation rule, Rule N, that means, box P , if you have P in a world Delta then all the acessible worlds should also have P.

THere is also Diamond P, which means possibility, that there exists one world which has P accessible from the world that Diamond P has.

Answer 1

P implies Q. You have the English in front of you.

Answer 2

"P implies Q" is equivalent to "if P, then Q".

Answer 3

Not P Or Q. This version you want?

Answer 4

P implies Q is true if P and Q are true, or if P is false.

It is false if P is true and Q is false.

*edit: Basically, what Svisstack said.

Answer 5

Translating your example:

Distributivity

Ka[Z] : ‘A knows Z is true’

Ka[(X->Y)] -> (Ka[X] -> Ka[Y])

If A knows that from X follows Y, then from A knowing that X is true it follows that A knows that Y is true.

Answer 6

To me, P => Q is best read as P is false, or Q is true

Answer 7

Perhaps it helps you to understand that if you imagine a small example from the real world:

Fire implies Heat

That means if you have fire, there must be heat. If there is no fire, there can be heat, due to other effects (e.g. sun is shining :) ), but there could as well be no heat.

If you have fire but no heat, somethings wrong. The implication is false then.

Answer 8

Sometimes these distributivity laws, and other axioms of modal logic, are easier to grasp if you use the comodalities, which are the De Morgan dualities of given modalities. The comodality of necessity is then necessity. For a to coknow P means that a does not know not P: intuitively it means that a's knowledge does not contradict P, so a could learn P without coming to know a contradiction. Say Ca P if a coknows P.

Then using classical logic, distributivity is equivalent to:

Ka(X or Y) -> (KaX or CaY)

This form is often easier to handle that the form with implication in formal manipulations.

Answer 9

Are you looking for a definition of the P -> Q or advice on how to actually say this express it in words when writing or speaking? If it's the former, there are already some good suggestions.

However, if the latter, I would suggest simply saying "P implies Q" as you've already used in your post. It's succinct, and unless your talking to someone with limited or no familiarity with mathematical logic, it's meaning is clear.