Propositional Logic and English

Date: 03/04/2003 at 17:39:23
From: Jason
Subject: Propositional logic

I wish to ask for some assistance in clearing up a couple of issues 
that arose in a class in propositional logic.

Given the propositions:
	P: The sun shines
	Q: The wind blows
	R: The rain falls
	S: The temperature rises

We were asked to write in words various compound propositions, one of 
which was…

Q Xor R         (Q Exclusive-Or R)

Would it be correct to say,

"Either the wind blows or the rain falls"?

This seems correct to me. The 'either' word emphasizes the fact that 
'the wind blows' OR 'the rain falls' but not both (Xor). Even though 
the 'either' word strengthens the sentence, wouldn't anyone who said 
'The wind blows or the rain falls' still be implying that only one or 
the other of the propositions is true and not both? in which case, the 
question is what, in an English sentence, is the difference between Q 
Xor R and Q Inclusive-Or R?

Another problem arose with the proposition

(P Inclusive-Or S) AND R

in which the answer I came up with was

"The sun shines or the temperature rises and the rain falls"

This also seems fine except that the symbolic compound proposition 
allows the possibility of 'The sun shines' and 'the temperature rises' 
together (Inclusive-Or), which seems to contradict what was mentioned 
in the previous problem! Also, how do you differentiate, in English, 
(P Inclusive-Or S) AND R  from P Inclusive-Or (S AND R) "The 
temperature rises and the rain falls or the sun shines"? These 
compound propositions obviously produce different truth tables but 
both seem to produce the same meaning in English, just in a different 
order.

It is only now that I can see just how ambiguous the English language 
can be!

I hope you can help and perhaps give a few more examples to 
demonstrate.

Thanks a lot!

Date: 03/08/2003 at 14:08:26
From: Doctor Achilles
Subject: Re: Propositional logic

Hi Jason,

Thanks for writing to Dr. Math.

For general help in logic, you can refer to the Dr. Math crash course 
in symbolic logic:

   http://mathforum.org/dr.math/faq/symbolic_logic.html 

You are absolutely right that the English language is ambiguous when 
it comes to logical expressions. As if the examples you gave weren't 
enough, we also have a page in our archives describing some further 
ambiguities from inclusive vs. exclusive ORs.

   Exclusive or Inclusive Disjunction?
   http://mathforum.org/library/drmath/view/55692.html 

It's a wonder that we ever communicate anything at all!  The page 
cited above argues that for many English sentences it is almost 
impossible to take the sentence by itself and decide what the 
propositional equivalent is, that the entire context is needed.

But enough of that digression, let's look at the questions you have.

>Q Xor R         (Q Exclusive-Or R)
>Would it be correct to say.
>"Either the wind blows or the rain falls"?

Although it is possible to have an "either...or" sentence in English 
that actually represents an inclusive-or, the default interpretation 
of "either...or" is exclusive-or. So I would say that "Either the wind 
blows or the rain falls" is the best English translation of your 
sentence. You can, of course, get more precise with "Either the wind 
blows or the rain falls, but both do not happen simultaneously," but I 
think that this sentence (while more precise) is awkward English and 
not something one would ever expect to hear outside of a logic class, 
so in my opinion your translation is the best.

>(P Inclusive-Or S) AND R
>In which the answer I came up with was
>"The sun shines or the temperature rises and the rain falls"

This problem is a bit trickier. First, the issue of inclusive- vs. 
exclusive-or: the omission of the word "either" means that the default 
interpretation of this sentence is inclusive-or, but it is somewhat 
ambiguous. You can get more concrete by using the word "and/or" which 
is used with some frequency in written English, but very rarely in 
spoken English, e.g. "The sun shines and/or the temperature rises and 
the rain falls." Once again, although this is more precise, is it also 
very inelegant English, and so I think that you're probably just as 
well off sticking with the simple "or" that you have.

The more difficult problem is the ambiguity between

  (P INC-OR S) AND R

and

  P INC-OR (S AND R)

In the first sentence, the main connective is "AND"; it joins the 
subsentence

  (P INC-OR S)

to the subsentence

  R

In the second sentence, the main connective is "INC-OR"; it joins the 
subsentence

  P

to the subsentence

  (S AND R)

For a review of what "main connectives" are, see

   Main Connectives in a Proof
   http://mathforum.org/library/drmath/view/55623.html 

I'm not sure I can give a definitive set of rules to deal with this 
problem. However, in my experience with English it seems to me that in 
most cases the first connective you use is interpreted as the main 
connective. In cases where it is impossible to do this, you can 
emphasize the main connective with a comma in written English or a 
pause in spoken English.
The sentence

>"The sun shines or the temperature rises and the rain falls"

is ambiguous, but if forced to decide, I would say probably means

  P INC-OR (S AND R)

because the "or" comes first.

In this case, I think you have to change the order of the sentence.  
There are two logically equivalent sentences which are easier to 
understand (in my opinion). Instead of:

  (P INC-OR S) AND R

try translating the logically equivalent:

  (P AND R) INC-OR (S AND R)

or just:

  R AND (P INC-OR S)

The first would translate to something like: "The sun shines and the 
rain falls, or the temperature rises and the rain falls." This 
sentence is pretty clear, and the comma (or pause in spoken English) 
will emphasize that the inc-or is the main connective.  However, I 
don't like this sentence very much because it is awkward, and the 
double use of "the rain falls" is inelegant.

My preferred translation is to start with

  R AND (P INC-OR S)

and turn it into: "The rain falls, and the sun shines or the 
temperature rises." This may be a little more ambiguous than the 
option above, but it is much more elegant.  Also, the comma (or pause) 
before the and will put enough emphasis on it to make it clear that 
the and is the main connective.

A sentence in logic can get much more complicated. For example, you 
can have something like:

  (P AND Q) INC-OR (S AND R)

This you could translate into "The sun shines and the wind blows, or 
the temperature rises and the rain falls."

Worse still is:

  [(P INC-OR Q) AND (S INC-OR R)] INC-OR T

[Where T means "there is a tornado"]

Once you get into sentences which contain more than two layers of 
connectives like this, you pretty much have to break it up into two 
English sentences, e.g. "The sun shines or the wind blows, and the 
temperature rises or the rain falls. Or, there is a tornado."

The worst case for translation, in my opinion, is if you mix 
inclusive- and exclusive-or's in the same sentence. For example:

  (P AND Q) INC-OR (S X-OR R)

One possible translation for this is "The sun shines and the wind 
blows, or either the temperature rises or the rain falls."  I think 
this is inelegant and difficult to understand because of the two or's 
in the same clause. The logically equivalent "Either the temperature 
rises or the rain falls, or the sun shines and the wind blows" is a 
little better, but it is still difficult to make it clear that the 
second or is inclusive.

Finally,

  (P X-OR Q) INC-OR (S X-OR R)

Is really hard. "Either the temperature rises or the rain falls, or 
either the sun shines or the wind blows" sounds just awful and can't 
really be helped without breaking the translation into multiple 
sentences. "Either the temperature rises or the rain falls. Or, either 
the sun shines or the wind blows" is an improvement, which works for 
written English. For spoken English, however, I'm stumped on this one. 
I guess the best thing to do is just have a long pause before and 
after the "Or" at the start of the second sentence.

You can get even trickier, and probably find sentences that can't be 
translated into elegant English and just plain require the use of 
terms like "exclusive-or" in their "English" translation.

Hope this helps. If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/12/2003 at 15:10:12
From: Jason
Subject: Thank you (Propositional logic)

Thanks for the help. The pages you recommended to look at within your 
archives were very useful.
Jason