🚀 go-pugleaf

In article <rkaivola.718902434@mits> rkaivola@mits.mdata.fi (Risto Kaivola) writes:
>
>  The following is no doubt a little trivial for the participants of this
>group, but I hope someone is willing to help me, nevertheless.
>  Exactly how should one formalize the English notion of 'only if'
>in the language of the propositional calculus?
>  That is, given the sentence
>"Tom will visit us only if we invite him.",
>is the correct formalization
>
>A = "Tom will visit us."
>
>B = "We invite him."
>
>A  -->  B
>
>
>Or, should this perhaps be B --> A (not in my opinion), or A <--> B
>(this last alternative is more likely to be correct than B --> A, in my
>opinion)?
>
>This isn't homework.
>
>--
>Risto Kaivola
>(Internet address:   rkaivola@mits.mdata.fi)

I'd go for:

(B --> A) & (-B --> -A)


Antoni Diller

In article <Bw0r39.G2I@cs.bham.ac.uk>, ard@cs.bham.ac.uk (Antoni Diller) writes:
|> In article <rkaivola.718902434@mits> rkaivola@mits.mdata.fi (Risto Kaivola) writes:

	[ stuff about translating conditionals deleted ]

Here are the usual translations found in logic texts for the material
(and similar) conditional and biconditional:

     --------------------------------------------

     P -> Q

     if P then Q
     if P, Q
     Q if P
     Provided that P, Q
     Q provided that P
     On condition that P, Q
     Q on condition that P
     P only if Q
     P only on condition that Q

     ---------------------------------------------

     P <-> Q

     P if and only if Q
     P just in case Q
     P when and only when Q
     P exactly on condition that Q

     -------------------------------------------

There are probably other common translations as well, but I
cannot recall at the moment.

For some reason 'only' tends to confuse people (I suspect it
is because they don't really know how it works in English).
Sometimes it is easier to get the correct sense of an 'only'
expression by considering a quantified version, as in

	Only a dog is a labrador retriever.

	(x is a labrador retriever) -> (x is a dog)

	Bob is a labrador retriever only if he is a dog.

On the other hand, in my experience, many people find this
even *more* confusing.  In reading 'P->Q' as 'P only if Q'
it helps to emphasize the 'only':  P *only* if Q.  (If you
say the 'only' loudly enough you start to get the idea.)

-- 
Gary H. Merrill  [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC  27513 / (919) 677-8000
sasghm@theseus.unx.sas.com ... !mcnc!sas!sasghm

rkaivola@mits.mdata.fi (Risto Kaivola) writes:

>A = "Tom will visit us."

>B = "We invite him."

You want "A only if B"

Think: if not B?       then not A
so !B -> !A

Think: if A?			  if not B then not A, so we must assert B is true.
A -> B

Both of these are equivalent.

In article <rkaivola.718902434@mits> rkaivola@mits.mdata.fi (Risto Kaivola) writes:
>  Exactly how should one formalize the English notion of 'only if'
>in the language of the propositional calculus?
>  That is, given the sentence
>"Tom will visit us only if we invite him.",
>is the correct formalization
>
>A = "Tom will visit us."
>B = "We invite him."
>
>A  -->  B
>
>Or, should this perhaps be B --> A (not in my opinion), or A <--> B
>(this last alternative is more likely to be correct than B --> A, in my
>opinion)?

Right the first time.

B --> A would be "Tom will visit us if we invite him"; the "only" makes the
implication go the other way.

Note that mathematicians say "if and only if" (usually written "iff") for
A <--> B.  It can be argued that some actual instances of "A only if B" in
speech mean A <--> B, but I think a better interpretation in such cases is
A --> B, with the B --> A part being taken for granted.

The problem with A --> B is that we think of --> as representing *causal*
implication, which would mean "Because Tom will visit us, we will invite
him."  But the propositional implication does not have this meaning.

A-->B = if A then B
A<-->B = A if and only if B
A<--B = A only if B = if B then A  This is read: A is true only if B is
true since a true statement cannot imply a falsehood.

In article <burt.718950333@aupair.cs.athabascau.ca>, burt@aupair.cs.athabascau.ca (Burt Voorhees) writes:

|> A<--B = A only if B = if B then A  This is read: A is true only if B is
|> true since a true statement cannot imply a falsehood.

I don't know what to make of this, except ...

	1.	A <-- B

	   certainly is non-standard notation, so I guess you can
	   impose on it whatever semantics you like.

	2.  If the claim is that

		A only if B

	    and

		if B then A

are equivalent (in the truth-functional, or any other, sense), then
a serious mistake is being made.
-- 
Gary H. Merrill  [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC  27513 / (919) 677-8000
sasghm@theseus.unx.sas.com ... !mcnc!sas!sasghm

>A-->B = if A then B
>A<-->B = A if and only if B
>A<--B = A only if B = if B then A  This is read: A is true only if B is
>true since a true statement cannot imply a falsehood.

  Appologies, this is clearly wrong. :(  That's what I get for posting
late at night and thinking about things like necessary and sufficient.
Guess that puts me one up on Dogwood (Much Ado About Nothing) as he only
wanted to be written down an ass.  :):):)
  Let's try again:  How about A only if B = (B-->A)&(-B-->-a)
bv

>>A-->B = if A then B
>>A<-->B = A if and only if B
>>A<--B = A only if B = if B then A  This is read: A is true only if B is
>>true since a true statement cannot imply a falsehood.

>  Appologies, this is clearly wrong. :(  That's what I get for posting
>late at night and thinking about things like necessary and sufficient.
>Guess that puts me one up on Dogwood (Much Ado About Nothing) as he only
>wanted to be written down an ass.  :):):)
>  Let's try again:  How about A only if B = (B-->A)&(-B-->-a)

  Nope, this is just if and only if

>bv

  Okay, let's go for the hat trick here.
1.  Tim will come to dinner if we invite him
2.  Tim will come to dinner if and only if we invite him
3.  Tim will come to dinner only if we invite him.

A = Tim will come to dinner; B = We invite him
so we have
1.  B-->A that is, tim will come if invited
2.  B<-->A  tim will come if invited, and if tim comes he was invited
3'.  B<--A  if tim comes he was invited
  Claim is that 3' = 3
  Tim will come only if we invite him means that if tim comes he was
invited.  Or, contrapositive, if he is not invited, he will not come:
i.e., -B-->-A  But this is equivalent to --a-->--B; i.e., A-->B
  The Phoenix rises
confusion of logical and causal thinking.

bv

In article <rkaivola.718902434@mits> rkaivola@mits.mdata.fi (Risto Kaivola) writes:

>  Exactly how should one formalize the English notion of 'only if'
>in the language of the propositional calculus?
>  That is, given the sentence
>"Tom will visit us only if we invite him.",
>is the correct formalization
>A = "Tom will visit us."
>B = "We invite him."
>A  -->  B


Exactly.  The only assignment of truth-values which makes 'A only if B' false
is when A is true and B is false.  Same truth value assignments as A->B.

--Dan

In article <rkaivola.718902434@mits> Risto Kaivola,
rkaivola@mits.mdata.fi writes:
>"Tom will visit us only if we invite him.",
>is the correct formalization
>
>A = "Tom will visit us."
>
>B = "We invite him."
>
>A  -->  B

How about:
~A --> ~B
or
"If we don't invite Tom, he will not visit us."  == "Tom will visit us
only if we invite him."

In article <1992Oct15.224329.20578@agt.uucp>, Brian F G Bidulock <bbidulo1%agt@cs.ualberta.ca> writes:
|> In article <rkaivola.718902434@mits> Risto Kaivola,
|> rkaivola@mits.mdata.fi writes:

|> How about:
|> ~A --> ~B
|> or
|> "If we don't invite Tom, he will not visit us."  == "Tom will visit us
|> only if we invite him."

In logic courses (where I tended to spend a *lot* of time on issues
of translation to and from English) I emphasized as a fundamental
principle of translation that one should attempt to have the structure
of the formal representation mirror (as closely as possible) the
structure of the English.  In particular, one should not end up with
a formal representation containing negations if there were no explicit
negations in the original English.  Cleaving to this principle avoids
a number of nasty confusions and problems.  Among other things, it
tends to remove *inference* from the translation process and keep
it where it belongs (post-translation, in the formalism).  And this
is particularly important as the complexity of the sentence to
be translated increases.

Accordingly, a good translation of

	If we don't invite Tom, he will not visit us.

is

	~A -> ~B

since the two negations are explicit in the English.  But this is
*not* a good translation of

	Tom will visit us only if we invite him.

since there are no explicit negations in the English.  These are,
of course (truth-functionally) equivalent, but it is *not* the
case that any logical equivalent to a good translation is itself
a good translation.
-- 
Gary H. Merrill  [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC  27513 / (919) 677-8000
sasghm@theseus.unx.sas.com ... !mcnc!sas!sasghm

In article <rkaivola.718902434@mits> rkaivola@mits.mdata.fi (Risto Kaivola) writes:

     Exactly how should one formalize the English notion of 'only if'
   in the language of the propositional calculus?
     That is, given the sentence
   "Tom will visit us only if we invite him.",
   is the correct formalization

   A = "Tom will visit us."

   B = "We invite him."

   A  -->  B

Yes.

Think about it this way: When is "Tom will visit us only if we invite
him" false?  It's certainly false if Tom visits without being invited,
i.e., if A is true and B is false.  If the sentence is true, any of the
other cases -- Tom is invited and visits, Tom is invited but doesn't
visit, and Tom is not invited and doesn't visit -- is possible.  Well,
those are precisely the truth conditions for A -> B.  So if you're going
to treat "Tom will visit us only if we invite him" as a truth-functional
compound of "Tom will visit us" and "We invite him", it's simply a
conditional.  (Of course some, including myself, would argue that
attempting to treat this as a truth-functional compound is not good
enough for some purposes.  For example,

    Tom will visit us only if we invite him.

might be thought of as indicating a general disinclination of Tom to
visit uninvited, making this something like a universal generalization
of a conditional rather than a conditional.  But this is beside the
point, which is that "only if" corresponds to -> in the same way that
"if" corresponds to <-.)

                                                        -- rar

In article <1992Oct15.224329.20578@agt.uucp> Brian F G Bidulock <bbidulo1%agt@cs.ualberta.ca> writes:
>In article <rkaivola.718902434@mits> Risto Kaivola,
>rkaivola@mits.mdata.fi writes:
>>"Tom will visit us only if we invite him.",
>>is the correct formalization
>>
>>A = "Tom will visit us."
>>
>>B = "We invite him."
>>
>>A  -->  B
>
>How about:
>~A --> ~B
>or
>"If we don't invite Tom, he will not visit us."  == "Tom will visit us
>only if we invite him."

Half right.  What you symbolized by ~A -> ~B would be translated as,
"If Tom does not visit us, we did not invite him."  This would be false
only if ( I just love this construction!) Tom didn't visit us, and we did
invite him, and true otherwise.  This means that the case in which Tom
visits us even though we did NOT invite him is true for ~A -> ~B, but is
the sole false assignment of truth values for A -> B.  In other words, 
they aren't logically equivalent propositions.

Your English translation of ~A -> ~B is incorrect as a translation, 
but it IS logically equivalent to the original statement.  It's actually 
of the form ~B -> ~A (which is very likely what you meant to write in the
first place), and has exactly the same truth conditions as A -> B.

--Dan