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.

Ideology

I'm voting to close this question as off-topic because it isn't directly about programming. (It would have been a good fit on either Philosophy or Mathematics, though in fairness those sites didn't exist back when it was asked.) – Mandatory

Answer 1

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

Knothole

which is logically equivalent to ~P or Q – Pigpen

"if P, then Q" is correct in principle, however be sure not to confuse it with the if condition then statement code construct. While this construct executes statement only if condition evaluates to true, the implication operator takes two boolean expressions. The result is even then true, if P is false, independent of the value of Q. The colloquial term "if P, then Q" doesnt really reflect that behavior. ~P or Q is much better in this regard. – Acrospire

Another alternative: In plain english "P implies Q" means "if P holds then Q must hold, too. If not, Q can be whatever" – Acrospire

@Philip Daubmeier's is by far the best plain-English translation. In particular explicitly states out the often-misunderstood part "If not, Q can be whatever" – Resound

Answer 2

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.

Acrospire

+1: That's a very good explanation for something that is often explained very badly – Pigpen

Answer 3

Not P Or Q. This version you want?

Tomfool

+1 since with nested implications is probably easiest to translate. – Moquette

Answer 4

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.

Dan

Answer 5

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

Elevon

:) can you convert the above formula to that ? – Ideology

converting the formula to English would sound redundant and confusing. Otherwise implication would have a better adjective. Boolean logic is learning how to abstract your thoughts out of english :D – Elevon

then why do AI researchers or logicians use first order logic to describe things in bidirectionally? what are the purposes of prolog ? – Ideology

Because it is difficult to express effectively in English ;) Prolog allows you to codify that logic in a program. Translating a "standard" program into English is way easier than in prolog just for this reason. – Elevon

Exactly, because you need to convert p->q to english, becuase p implies q doesnt mean anything except that p implies q. – Ideology

Answer 6

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

Tanberg

Answer 7

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.

Hereditable

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.

Snowfall

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.

Amaya

Whoops. I see that Byron Whitlock said essentially the same thing. – Amaya