Thursday, February 10, 2011

Conjunction and natural language

One of the least controversial rules of logic is conjunction-introduction: from the premises A and B, one infers the conjunction A&B. But now consider the following argument in ordinary English:

  1. (Premise) The following claims are all false: 1=2, 2=3, 3=4.
  2. (Premise) 5=5.
  3. The following claims are all false: 1=2, 2=3, 3=4 and 5=5. (By conjunction-introduction)
Premises 1 and 2 are true, but the conclusion is false.

Of course what we should say is that (3) is not in fact a conjunction of (1) and (2). This shows an interesting thing: it's not that easy to define a conjunction syntactically in natural language. The obvious rule of taking sentences "A" and "B" and forming the sentence "A and B" fails (even if we bracket the fact that in written English we often need to adjust the capitalization of the first word of the second sentence, and adjust punctuation), as (3) shows. In logic/mathematics-influenced written English we can conjoin "A" with "B" by forming the sentence "(A) and (B)". But that's not grammatical ordinary English.

We can, of course, do some paraphrase. Thus, we can say one of the following:

  1. 5=5 and the following claims are all false: 1=2, 2=3, 3=4.
  2. It is false that 1=2, it is false that 2=3, it is false that 3=4, but 5=5.
But while (4) and (5) express propositions that are obviously logically equivalent to the conjunction of the propositions expressed by (1) and (2), it is not clear that they express the conjunction of the proposition expressed by (1) with the proposition expressed by (2). Moreover, it is not clear that we can in purely syntactic terms specify how to give such a paraphrase in every case.

This is another reason to think that logic within natural language is at least somewhat tricky. The neat distinction in artifical languages between syntactic and semantic properties is much harder to draw. The notion of a conjunction of two sentences may well have no syntactic characterization.

Moreover, there may be sentences that in ordinary English have no conjunction or that have no disjunction. This is because the order of operations in English is foggy. In spoken English, we can do something with tone of voice and emphasis, but it is clear that this cannot be made to work always. In particular, if A,B,C,D,E,F,G,H are ordinary English sentences, I doubt that there is an ordinary English equivalent to "((A or (B and C)) and D) or (E and ((F and G) or H))". Thus, at some point we will have a failure of forming conjunctions or a failure of forming disjunctions.

This is relevant to this post.


Heath White said...

There are similar oddities when we use 'and' and 'or' in modal contexts. "How do I get to the store?" "You can go on First Street, or you can go on Elm Avenue." What we really mean is that both of those methods will get you to the store. If anyone uttered a disjunction like that, where only one disjunct was true, we'd think he was pretty obnoxious.

Alexander R Pruss said...

Nice example. I am told some students find disjunction-introduction baffling. (Here's a post of mine on that, but your example is structurally different.)

Demonstratives provide another slew of cases:
"This sentence is short."
"This sentence is short and Fermat's Last Theorem eluded mathematicians for a long time before it was proved."
(A correct formation of the conjunction would require adjustment of "This" to "That", or the addition of a finger pointing to the first sentence.)

Other cases come from combining sentences from different contexts.
"He is a great logician." (about Goedel)
"He got 30% on my logic final" (about a student)
So: "He is a great logician and he got 30% on my logic final. I made the final too hard!"

The lesson in all of this continues to be that conjunction and disjunction are semantic concepts. The cool thing about formal languages then is there is a syntactic concept that happens to be co-extensive with the semantic concept.

Alexander R Pruss said...

Another fund of examples: One may be able to make an assertion by nodding one's head or by a rhetorical question. But ordinary language makes it hard to truth-functionally combine such sentences.

There is another point to all this, besides wearing away at the syntax/semantics distinction. There are Liar sentences which grammatically are conjunctions whose conjuncts are true. See my post for tomorrow.

Matt said...

I think your examples are interesting, but I'm not sure how much they say about whether conjunction is syntactic or semantic. They clearly show that we can't always form a conjunction of two sentences by simply writing "and" in between them and correcting for capitalization. But can't this be explained syntactically? The reason your example failed was that "5=5" was being conjoined to something subsentential, rather than the sentence itself. We would see this syntactically if we diagrammed the sentence. We get the right conjunction if we start the sentence with "5=5." I'm not sure why this kind of syntactic explanation of the phenomenon is inadequate. Maybe the thought is that we need semantics to explain why putting the "5=5" first rather than second yields the right conjunction? That's an interesting thought, but it's not immediately clear that there is no syntactic explanation.

Alexander R Pruss said...

I doubt that one can diagram sentences purely syntactically. There are standard problems, such as the "Time flies like an arrow"/"Fruit flies like butter" problem.

Another problem is that there are a lot of ways of signaling conjunction in English, even leaving aside my cases:
"Both of the following claims hold: A, B."
"A but B."
"A while B."
"Both A and B."
"A (and B)."
"A and B." (Emphasis should generates a new sentence type in the correct syntactic theory.)
"A and B."
"A and B."
Is the list finite or infinite? I don't know.

Jarrett Cooper said...

I'd definitely be upset if someone uttered to me the example Heath White used. If so he better be emphatic that he's using an exclusive disjunction instead of a nonexclusive disjunction. I'd still be upset that they just didn't tell me the correct option.

But, yes, there are some things that just can't be captured (fully, adequately, etc.) by symbolic logic. However, I don't see a problem with this. Just as long as we are precise enough whenever having philosophical discourse (or any discourse for that matter), then we should be able to capture enough of the argument for it to be broken down syntactically.

Alexander R Pruss said...

"Just as long as we are precise enough whenever having philosophical discourse (or any discourse for that matter), then we should be able to capture enough of the argument for it to be broken down syntactically."

That sounds very good.

Another example.

1. There is a bank on Austin Ave.
2. The Brazos has two banks.

Try to conjoin 2 with 1:
"The Brazos has two banks and there is a bank on Austin Ave." :-)

Jarrett Cooper said...

I got the thought from reading Elementary Symbolic Logic. It's a great autodidactic book.

The authors note that there are ambiguities that can occur with symbolic logic that in conversation can be overcome by emphasis on certain words and body gestures. In written contexts we can overcome such ambiguities by commas, capitalization, and/or the underlining of words.

Furthermore, idiomatic expressions and homonyms, what you used for your last example, are hard (impossible?) to capture syntactically.

Another example:

1. Some trees can produce bark
2. Some dogs can produce a bark.

Conjoined: Some trees and dogs can produce barks.

Alexander R Pruss said...

Right, but (a) the various devices can only go so far (once we get too much nesting, they don't work), and (b) it is far from clear that we can make them go algorithmically.

Jarrett Cooper said...

I definitely agree (you're preaching to the choir). That's why we need to be precise as possible, especially with philosophical arguments, so it may be possible for us to capture them algorithmically. (This is the best a logician can hope for.)