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