Not actually Philosophy, but Logic.

[lawtonfogle]

This is a logic question, and knowing not where it went, I have placed it here.

What does "A only if B" mean?

Does it mean "A if and only if B", or "If B, then A"?

 

[The Nihilist]

I'm not that sure that those are terribly different. Why do you ask?

 

[Silhillian]

It's been ages since I've done logic, but let me see:

A only if B means that for B to occur is neccesary (but not sufficient) for A to occur.
For example "It is 1.00am only if the minute clock hand points to 12" (the minute hand HAS to point to 12 at 1.00 (so the hand pointing to 12 is a neccesary condition) but the minute hand can also point to 12 at 2.00, or 3.00, etc so it's not always 1.00 when the hand is at 12 (so it's not a sufficient condition).

If B, then A (or A if B) means that if B happens, it is sufficient for A to occur - we can use the last example here: "If it's 1.00am, then the minute hand points to 12" (1.00 is sufficient but not neccesary for the hand to point to 12)

Then A if and only if B - here both are neccesary AND sufficient conditions of each other: "A shape is a triangle if and only if there are three sides." There must be three sides for a shape to be called a triangle, and a triangle can only have three sides.

I hope this helps, (I was never any good at explaining things! ) but logic often gave me plenty of headaches!!

 

[Sojourner<><]

This is a logic question, and knowing not where it went, I have placed it here.

What does "A only if B" mean?

Does it mean "A if and only if B", or "If B, then A"?

The statements 'if A then B' and 'if B then perhaps A' could be deduced from 'A only if B', but they are not equivalent.

Just out of curiosity, what about logic interests you?

 

[EverlastingMan]

I think, been proved wrong quite frequently though, that "A only if B" means "If B, then A" is the only statement that would be true consistently. Whereas "A if and only if B" may or may not be the case.

 

[Sojourner<><]

I think, been proved wrong quite frequently though, that "A only if B" means "If B, then A" is the only statement that would be true consistently. Whereas "A if and only if B" may or may not be the case.

You're right to say "if B, then A". The problem is a little more interesting than I first thought....

If solving for A, the operator 'only' can be ommitted since 'A if B' and 'A only if B' would be equivalent.

If solving for B, the operator 'only' establishes B's dependence on A since if A is true we know B must be true because A is true only when B is true.

I think the 'only' operator would then seem to establish a codependence between A and B. "A only if B" is equal to "A if B and B if A".

 

[EverlastingMan]

Aye. I believe that was what I was saying.

 

[Sojourner<><]

You're right to say "if B, then A". The problem is a little more interesting than I first thought....

If solving for A, the operator 'only' can be ommitted since 'A if B' and 'A only if B' would be equivalent.

If solving for B, the operator 'only' establishes B's dependence on A since if A is true we know B must be true because A is true only when B is true.

I think the 'only' operator would then seem to establish a codependence between A and B. "A only if B" is equal to "A if B and B if A".

If "not A if not B and not B if not A" can be worked out of it then I think that can be even further simplified by "A = B". "not A if not B" is pretty obvious but I'm not sure about "not B if not A".

 

[Sojourner<><]

If "not A if not B and not B if not A" can be worked out of it then I think that can be even further simplified by "A = B". "not A if not B" is pretty obvious but I'm not sure about "not B if not A".

Wait a sec. "A if B" is true so "not B if not A" must also be true. So then "A only if B" should be equal to "A = B". Is that a little weird or what?

 

[sparklecat]

This is a logic question, and knowing not where it went, I have placed it here.

What does "A only if B" mean?

Does it mean "A if and only if B", or "If B, then A"?

A -> B is, I believe, all you can derive from the statement.


Let's substitute an actual example to illustrate this. Let A = 'It is a cat.' Let B = 'It is a mammal.' Thus we get 'It is a cat only if it is a mammal.'

A -> B would therefore be 'If it is a cat, then it is a mammal.'
B -> A would be 'If it is a mammal, then it is a cat.'
A <-> B would be both of those, and not work.

 

[EverlastingMan]

A -> B is, I believe, all you can derive from the statement.


Let's substitute an actual example to illustrate this. Let A = 'It is a cat.' Let B = 'It is a mammal.' Thus we get 'It is a cat only if it is a mammal.'

A -> B would therefore be 'If it is a cat, then it is a mammal.'
B -> A would be 'If it is a mammal, then it is a cat.'
A <-> B would be both of those, and not work.

A->B, assuming you mean the "B if and only if.." one, is the only statement that is always true, but you could deduce "If B, then A" in certain circumstances. For instance:
A= The hour hand is at a 180* angle from 12-o'clock.
B= It is 6-o'clock.
Thus: The hour hand is at a 180* angle from 12-o'clock if and only if it is 6-o'clock. (A if and only if B.)
And: If it is 6-o'clock, then the hour hand is at a 180* angle from 12-o'clock. (If B, then B.)
Though, looking at your example and others we can see that the latter statement is not always so. Anyhow, I think the only statement that can actually be deduced is the "if and only if" one. The other cannot really be reasoned out by mere logic.

 

[sparklecat]

A->B, assuming you mean the "B if and only if.." one, is the only statement that is always true, but you could deduce "If B, then A" in certain circumstances. For instance:
A= The hour hand is at a 180* angle from 12-o'clock.
B= It is 6-o'clock.
Thus: The hour hand is at a 180* angle from 12-o'clock if and only if it is 6-o'clock. (A if and only if B.)
And: If it is 6-o'clock, then the hour hand is at a 180* angle from 12-o'clock. (If B, then B.)
Though, looking at your example and others we can see that the latter statement is not always so. Anyhow, I think the only statement that can actually be deduced is the "if and only if" one. The other cannot really be reasoned out by mere logic.

No, I meant 'If A, then B.' A if and only if B would include both 'If A, then B' and 'If B, then A,' and since the latter is not always the case and thus cannot be logically derived, if and only if doesn't work.

 

[Sojourner<><]

No, I meant 'If A, then B.' A if and only if B would include both 'If A, then B' and 'If B, then A,' and since the latter is not always the case and thus cannot be logically derived, if and only if doesn't work.

I think that the statement "A only if B" is case specific and can't be expected to apply universally to all similar situations. No matter which way you apply it to cats and mammals it won't make sense.

I think that it's more applicable to less complex examples -
Let A be "I will go to the store"
and let B be "You will go to the store"

A only if B would mean if I go to the store, you go and vice versa.

 

[sparklecat]

I think that the statement "A only if B" is case specific and can't be expected to apply universally to all similar situations. No matter which way you apply it to cats and mammals it won't make sense.

I think that it's more applicable to less complex examples -
Let A be "I will go to the store"
and let B be "You will go to the store"

A only if B would mean if I go to the store, you go and vice versa.

But that's the point of logic. If an argument is valid, it's valid universally.

Take your example: A only if B would come out to I will go to the store only if you go to the store, yes? We'll take this as given.

Now, we're looking at three statements to see if they can be logically derived from this statement. If A, then B is one, if B, then A is two, and A if and only if B is three.

So, we've established that the only way I'm going to the store is if you go. That means that if I am at the store, you must necessarily be there as well. A -> B. However, it does not mean that you cannot go on your own. That would be B -> A; if you are at the store, then I am necessarily there as well. But that's not the case. As A <-> B includes both A -> B, and B -> A, and the latter is not a necessary conclusion of A only if B, A <-> B is not a correct representation of the statement.

 

[Sojourner<><]

But that's the point of logic. If an argument is valid, it's valid universally.

Take your example: A only if B would come out to I will go to the store only if you go to the store, yes? We'll take this as given.

Now, we're looking at three statements to see if they can be logically derived from this statement. If A, then B is one, if B, then A is two, and A if and only if B is three.

So, we've established that the only way I'm going to the store is if you go. That means that if I am at the store, you must necessarily be there as well. A -> B. However, it does not mean that you cannot go on your own. That would be B -> A; if you are at the store, then I am necessarily there as well. But that's not the case. As A <-> B includes both A -> B, and B -> A, and the latter is not a necessary conclusion of A only if B, A <-> B is not a correct representation of the statement.

Are you sure? I would think that if I go to the store alone after I say "I will go only if you go", it would mean I didn't stick to my word and it was a false statement.

 

[sparklecat]

Are you sure? I would think that if I go to the store alone after I say "I will go only if you go", it would mean I didn't stick to my word and it was a false statement.

Sorry, I switched around which of us was A and which of us was B, and made it more confusing. My last post was written from the perspective of myself as A and you as B: I'll go to the store only if you go, but you can go on your own.

 

[EverlastingMan]

No, I meant 'If A, then B.' A if and only if B would include both 'If A, then B' and 'If B, then A,' and since the latter is not always the case and thus cannot be logically derived, if and only if doesn't work.

Right; I accidentally reversed it; agree entirely with you; we're saying the same thing here.

 

[sparklecat]

Right; I accidentally reversed it; agree entirely with you; we're saying the same thing here.

We all seem to be switching things around and confusing the issue tonight

 

[StrugglingSceptic]

"A only if B" is written symbolically as "A=>B". It is exactly equivalent to "If A, then B", and tends to be used so that symbolic conditionals can be read without having to transform the syntax in any way:

(P & Q) => R

reads correspondingly as

P and Q, only if R

whereas a bit of mental reordering is needed for the rendering:

If P and Q, then R.

The statement "A if B" is written symbolically as "B=>A", and is just a transformation of "If B, A". Naturally, then, the biconditonal "A<=>B" which is equivalent to the conjunction of "A => B" (A only if B) and "B => A" (A if B) can be read as "A if and only if B".

There is some confusion, however, in natural language. You almost never hear "if and only if" in daily usage, because we instead rely on context to express biconditionals. The parents who tell their children "we will go to the sea tomorrow only if it is sunny", formally mean they will go to the sea tomorrow if and only if it is sunny. Indeed, the children would be much surprised if the next day it was sunny, but they were told "just because it is sunny, does not mean we get to go to the sea: that wasn't a biconditional I gave you yesterday, y'know!"

Even in mathematics, context is relied upon when it comes to definitions. For instance, the following is taken from one of my textbooks:

"T is a topology on X if the following three conditions hold: ..."

In fact, the "if" here expresses the biconditional, since the conditions failing to hold guarantees that T is not a topology on X. It is standard in mathematics that in definitions, the biconditonal is assumed unless otherwise stated, likely for the sake of conciseness.

It would seem then that the OP's confusion lies with this natural language understanding of "only if", and the reliance of that understanding on context. But in formal logic, context is thrown out the window, and all connectives are given unique interpretations. So

"A if B", "A only if B" and "A if and only if B"

are mutually non-equivalent, being English versions of the statements "B => A", "A => B" and "A <=> B" respectively.

 

[Sojourner<><]

"A only if B" is written symbolically as "A=>B". It is exactly equivalent to "If A, then B", and tends to be used so that symbolic conditionals can be read without having to transform the syntax in any way:

(P & Q) => R

reads correspondingly as

P and Q, only if R

whereas a bit of mental reordering is needed for the rendering:

If P and Q, then R.

The statement "A if B" is written symbolically as "B=>A", and is just a transformation of "If B, A". Naturally, then, the biconditonal "A<=>B" which is equivalent to the conjunction of "A => B" (A only if B) and "B => A" (A if B) can be read as "A if and only if B".

There is some confusion, however, in natural language. You almost never hear "if and only if" in daily usage, because we instead rely on context to express biconditionals. The parents who tell their children "we will go to the sea tomorrow only if it is sunny", formally mean they will go to the sea tomorrow if and only if it is sunny. Indeed, the children would be much surprised if the next day it was sunny, but they were told "just because it is sunny, does not mean we get to go to the sea: that wasn't a biconditional I gave you yesterday, y'know!"

Even in mathematics, context is relied upon when it comes to definitions. For instance, the following is taken from one of my textbooks:

"T is a topology on X if the following three conditions hold: ..."

In fact, the "if" here expresses the biconditional, since the conditions failing to hold guarantees that T is not a topology on X. It is standard in mathematics that in definitions, the biconditonal is assumed unless otherwise stated, likely for the sake of conciseness.

It would seem then that the OP's confusion lies with this natural language understanding of "only if", and the reliance of that understanding on context. But in formal logic, context is thrown out the window, and all connectives are given unique interpretations. So

"A if B", "A only if B" and "A if and only if B"

are mutually non-equivalent, being English versions of the statements "B => A", "A => B" and "A <=> B" respectively.

That being so what is the formal syntax for "A if B and B if A"?