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I'm trying to translate an argument into sentential logic. It's of the form $$\text{sentence }1:\text{ } p\\\text{sentence }2: \text{ If so, then } q$$ What I want to know is, do I translate this as a single premise, i.e. $p\rightarrow q$, or as two premises, i.e. $1.$ $p$, $2.$ $p\rightarrow q$?

Edit: To clarify, the second sentence makes me wonder if $p$ is declared as true in the first sentence, or if it's really a conditional split into two (English) sentences.

Edit 2: Here's the full argument.

$$1.\text{ Either cats are the best animal or dogs are the best animal or snakes are the best animal.}\\2.\text{ If cats are not the best animal, then it will rain tomorrow.}\\3.\text{ But it will not rain tomorrow}\\4.\text{ The temperature will be warm tomorrow}\\5.\text{ If so, then dogs are not the best animal.}\\6.\text{ It follows that snakes are the best animal}$$

My confusion is with the interplay between $4$ and $5$.

jessica
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  • Yes, the first sentence tells you that $p$ is true. Does the "if so" in the second sentence mean "if sentence 1 is true"? – Adam Francey Jan 21 '16 at 01:10
  • @AdamFrancey I'll make an edit to give the exact context. – jessica Jan 21 '16 at 01:11
  • I take 5. to mean "if 4., then dogs are not the best animal". That is, "if the temperature will be warm tomorrow, then dogs are not the best animal." Since the antecedent is already given, the "if so" part is redundant. – BrianO Jan 21 '16 at 01:24
  • @BrianO I understand that. I want to know if "the temperature will be warm tomorrow" is a premise in itself, or if it's only the conditional you describe. That is, ($4$ and $4\rightarrow 5$) OR ($4\rightarrow 5$) – jessica Jan 21 '16 at 01:25
  • is a premise too. Both 4. and 5. are premises. NOTE: not "4 $\to$ 5" as you have it, because 5 itself is the implication "4 $\to$ dogs are not the best animal".
    • – BrianO Jan 21 '16 at 01:26
  • @BrianO So, say if I wanted to prove the validity of this, I could derive that dogs are not the best animal from $4$, $5$ and modus ponens? – jessica Jan 21 '16 at 01:27
  • Is this by any chance an example of a poorly constructed argument? The conclusion is wrong since lines 2 and 3 imply that cats are the best animal. – Adam Francey Jan 21 '16 at 01:28
  • Yes. As I read it, the "if so" is redundant: in the presence of 4., 5. is equivalent to "dogs are not the best animal". // Note that 3. and 2. as written imply that cats are the best animal. Is 2. correctly transcribed? – BrianO Jan 21 '16 at 01:30
  • @AdamFrancey This is just sentential logic, there's no predicate. So technically snakes could be the best animal AND cats could be too. – jessica Jan 21 '16 at 01:30
  • Yes, we could wind up with "cats are the best animal and snakes are the best animal". But I see no way to infer "snakes are the best animal" -- it does not follow, contra 6. – BrianO Jan 21 '16 at 01:31
  • The first line says "cats OR dogs OR snakes", though. – Adam Francey Jan 21 '16 at 01:32
  • @AdamFrancey Right, those are not exclusive ORs. – jessica Jan 21 '16 at 01:32
  • @BrianO I never said the argument is valid or invalid. I just wanted a translation. – jessica Jan 21 '16 at 01:33
  • @AdamFrancey Right. From 1 4 5, we know that cats are the best or snakes are the best. If 2 said instead 2-alt "if cats are the best animal then...", then we could conclude from 2-alt and 3 that cats are not the best animal, and then we could conclude 6. – BrianO Jan 21 '16 at 01:35
  • @jessica Good, because it isn't a valid argument :) – BrianO Jan 21 '16 at 01:36
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    @BrianO I know :) – jessica Jan 21 '16 at 01:36
  • @jessica I think having "Either" at the beginning of statement 1 means it is exclusive OR. Anyhow, I think we can't get a straightforward translation because statement 5 is ambiguous, since we have to pick an interpretation of "if so". Who knows if we can assume "if so" should be taken to mean "if the temperature is warm" or not? – Adam Francey Jan 21 '16 at 01:40
  • @AdamFrancey That is not how "either" is typically translated, at least in most sources I've seen. – jessica Jan 21 '16 at 01:44
  • Hmm, it looks like "Either" is ambiguous too, where did you find this argument? – Adam Francey Jan 21 '16 at 01:49
  • I don't think "either" is there to indicate exclusive or. It's just for naturalness — it signals that a disjunction is coming up, so you're not surprised when you hear/read "or". Anyone would be more likely to say this than they would the version that begins "Cats are the best animal or...". Granted, nobody is very likely to say... either of them. – BrianO Jan 21 '16 at 02:01
  • "If so" in the presence of 4 preceding statements is ambiguous, true, and might even refer to the conjunction of 1 through 4. It might mean, "granting all the foregoing, then..." It's not very good usage, in this situation. BUT in any case, whether you render "if so" as 4. or as 1. $\land$ 2. $\land$ 3. $\land$ 4., the (in)validity of the argument is unchanged, because in any case it's "just noise". – BrianO Jan 21 '16 at 02:03
  • @BrianO Would you mind taking a look at a new question I posed? http://math.stackexchange.com/questions/1643758/modal-logic-translation-example – jessica Feb 06 '16 at 22:45
  • @jessica I just looked -- I was too late ;), you deleted it. Feel free to ask again though. – BrianO Feb 07 '16 at 00:14
  • @BrianO I undeleted it. I thought I had figured it out and didn't want to waste others' time, but I made a mistake in my solution. – jessica Feb 07 '16 at 16:17
  • OK I'll take a look (a bit later). – BrianO Feb 07 '16 at 16:25
  • @BrianO don't worry about it. I figured it out. – jessica Feb 08 '16 at 19:22