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Consider the following argument.

$$\begin{array}{rl} & P & \text{All roads lead to Rome.} \\ & Q & \underline{\text{E4 is a road .}} \\ & \therefore & \text{E4 leads to Rome.} \end{array}$$

So we're asking if $ P \wedge Q \Rightarrow C.$

I am arguing that the argument is not valid due to it not being a tautology because of the fact that P is false. All roads do not lead to Rome and therefore $P \wedge Q$ is false and also C becomes false. The statement is therefore not always true and thus it can not be a tautology.

I am forked on what exactly it means for an argument to be valid so I am not sure at all about my answer.

Air Mike
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NoName123
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  • You have “All” thus you need predicate logic. – Mauro ALLEGRANZA Sep 08 '20 at 12:14
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    $\forall x (P(x) \to R(x))$ and $P(E)$ imply $R(E)$ – Mauro ALLEGRANZA Sep 08 '20 at 12:16
  • @MauroALLEGRANZA Can you elaborate on what you mean? I am new to this field and am not yet all too familiar with all the terminology. Would you say that the argument is a valid one, if so why (or why not)? – NoName123 Sep 08 '20 at 12:17
  • Do you know predicate logic? – Mauro ALLEGRANZA Sep 08 '20 at 12:30
  • If you are restricted to propositional logic, then you are right: the argument above is not a alid argument of propositional logic – Mauro ALLEGRANZA Sep 08 '20 at 12:31
  • I think this argument is part of what is sometimes called syllogistic logic (or something like that). It includes some simple quantification but is way short of full first order logic. – Ned Sep 08 '20 at 12:43
  • Are you using the word "valid" in the formal sense or just informally? You are showing the argument is not "sound." https://math.stackexchange.com/questions/281208/what-is-the-difference-between-a-sound-argument-and-a-valid-argument – roundsquare Sep 08 '20 at 14:31

1 Answers1

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The Soundness Theorem in Logic states the following.

The derivability of an argument implies its validity.

So, if you prove that such argument is derivable, you are also proving that it is valid. So let’s try to prove that the argument that you gave is derivable.

Our premisses are $P = \text{“All roads lead to Rome”}$ and $Q = \text{“E4 is a road”}.$

Using Universal Instantiation, we know that $\text{“E4 leads to Rome”}.$ (Because E$4$ is a road and all roads lead to Rome $-$ this is straightforward).

What we are doing here is getting the conclusion of the argument from the premisses, using rules of inference. This is a derivation of the argument.

Since our argument has a derivation, then it is derivable and therefore it is valid.

Also note that even if $P$ is false, then the premisses are false. So it doesn’t matter if the conclusion is true or false ($P \to Q$ is true if $P$ is false, regardless of the truth value of $Q$.)

Air Mike
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