# Logical translation with possibly one or two premises

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 $5$.

• 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 1:10
• @AdamFrancey I'll make an edit to give the exact context. – jessica Jan 21 '16 at 1:11
• 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 1:24
• @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 1:25
• @BrianO I know :) – jessica Jan 21 '16 at 1:36

## 1 Answer

Simple Statements (propositions):

• $C$ : cats are the best animals
• $D$ : dogs are the best animals
• $S$ : snakes are the best animals
• $R$ : it will rain tomorrow
• $W$ : it will be warm tomorrow

Note: Line 1 is ambiguous but, with the assumption that "best" is unique, it can be interpreted as using the exclusive or connective.

$\boxed{\begin{array}{l|l|l} 1. & C\veebar D\veebar S &\text{ Either cats are the best animal or dogs are the best animal or snakes are the best animal.} \\ 2. & \neg C \to R & \text{If cats are not the best animal, then it will rain tomorrow.} \\ 3. & \neg R & \text{But it will not rain tomorrow} \\ 4. & W & \text{The temperature will be warm tomorrow} \\ 5. & W\to \neg D & \text{If so, then dogs are not the best animal.} \\ \hline 6. & \therefore S & \text{It follows that snakes are the best animal} \end{array}}$

PS: The argument is invalid, as $\neg C\to R, \neg R \vdash C$ by modus tollens (2,3), and we would require neither cats nor dogs to be the best to conclude that snakes are.

$C, \neg D, C\veebar D\veebar S \nvdash S$