# Are there only three types of statements?

I don't have a very strong background in formal logic, so I'm no expert in this field. According to some websites, there are three "types" of statements:

1. Simple statements (e.g "it is raining" or "the dog is red")
2. Compound statements (e.g "it is raing and the dog is red")
3. If-then statements (e.g "if it is raining then the dog is red")

Is there a formal definition of a type of statement? Are they all just special cases of something? Are these really all of them? Sorry if my question sounds a bit vague, but I'm just wondering if theres a statement that doesn't fit into these three types.

• You may be looking for an analysis of logical constants (eg true, false), modifiers (eg not) and connectives (eg and, or, implies) from which more complex statements are built up. The question of a minimal set is not, I think, widely addressed, because, provided the notions are (provably) mutually consistent, redundancy is not a problem. To say "there are three types of statements" is a category mistake, because there are clearly much more complex statements which can be built up fro elementary building blocks. Commented Sep 9, 2023 at 21:53
• Also there are only three kinds of fruits! (1) Single-seed fruits (peach, avocado, cherry, etc.) (2) Multiple-seed fruits (apple, raspberry, etc.) (3) Citrus fruits (lemon etc.).
– MJD
Commented Sep 9, 2023 at 22:01
• I'm not sure if this helps, but on Page 35, the textbook Book of Proof says, "A statement is a sentence or a mathematical expression that is either definitely true or definitely false. You can think of statements as pieces of information that are either correct or incorrect. Thus statements are pieces of information that we might apply logic to in order to produce other pieces of information (which are also statements)." I don't think the textbook mentions simple or compound statements like what you learn in elementary school. Commented Sep 9, 2023 at 22:17
• @Accelerator That's not an optimal definitition of a statement. Commented Sep 10, 2023 at 1:50
• @ryang Thank you for taking the time to share that with me. But are you saying the author of that textbook is wrong, at least when it comes to formal logic? I read the link but I wasn't sure what precise definition of a statement you had in mind. Commented Sep 10, 2023 at 4:33

## 2 Answers

Here is a typical treatment:

We allow some set of atomic sentences, say $$p,q,r....$$. These exactly correspond to your simple statements. Now take some logical operators (which take as input sentences, and return sentences) such as $$\land, \neg, \implies$$. We now close the set of atomic sentences under the logical operators, obtaining sentences such as $$p \land q, \neg (p \land q), [\neg (p \land q)] \implies q,...$$.

The last logical operator applied can be uniquely recovered due to parantheses. In the sentences above, they are $$\land, \neg, \implies$$ respectively. This is presumably the "type" of your sentence, where compound sentences are obtained by taking two sentences and applying $$\land$$, and if-then sentences are obtained by taking two sentences and applying $$\implies$$. You can then recover all "types" of a sentence by looking at which logical operators your language allows.

• You probably also wanted to mention that another "type" is obtained by putting a negation sign in front of a sentence.
– Alex
Commented Sep 9, 2023 at 22:28

That webpage you linked to is utterly, utterly dreadful.

Daniel Velleman's How to Prove It can be strongly recommended as a terrific book for beginning maths students on basic logical ideas you ought to know.

Or there are many very reliable straight logic texts out there. Versions of the relatively brief forallX text are freely downloadable: I particularly like this one https://www.homepages.ucl.ac.uk/~uctytbu/forallxcam.pdf.

Or my more expansive book (was CUP, now freely downloadable) can be found at https://www.logicmatters/net/ifl.

• this was like a week ago but this hurt my feelings im not a complete beginner in proof writing im just dumb😭 Commented Sep 20, 2023 at 23:43
• @iwjueph94rgytbhr I know you commented on this a week ago, but please don't call yourself dumb. You didn't do anything wrong. People in general, including myself, do idiotic things on occasion but that's how we learn. I was corrected in the comments of your post and I took it as a positive thing. Commented Sep 28, 2023 at 5:17