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So I am currently attacking a question from the first chapter of my logic book. I know that the question is true, but I am having a hard time actually proving it. The question is as follows.

If a argument is valid, then it cannot be made invalid by adding premises.

If someone could please give me some guidance on how I might approach this proof. This is as far as I could get. If V is an argument then there must not exist a premises P that could be added to the a valid argument V such that when P is added to V V' becomes invalid.

All help would be much appreciated. I am of course not looking for a full solution but perhaps some insight or guidance.

Thank You

EDIT: Also, I suppose the reason I'm having trouble is because I cannot show why such a P cannot exist. I don't know what kind of contradiction it may offer if it did either.

Valentino
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1 Answers1

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Consider these two claims:

(1) There is no possible situation in which $A$ is true and $C$ false.

(2) There is no possible situation in which $A$ is true and $B$ is true and $C$ false.

Question: do you see that if (1) is true then (2) is true too?

Question: what does that tell us about the relation between the validity of the argument $A \therefore C$ and the validity of $A, B \therefore C$?

Question: how do we generalize the answer to the last question to answer your original question about adding more than one premiss to a valid argument?

Community
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Peter Smith
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  • Okay, so I see that if (1) is true then (2) is true as well. This would tell us that if A$\therefore$C is valid then A,B$\therefore$ C is valid as well. The third question is what I suppose I am having the most trouble with. Would this be possibly a proof by induction? However, I don't know if that is a viable or efficient route. Thank you for the Response by the way Peter. – Valentino Feb 10 '14 at 19:25
  • @Valentino - It is not necessary to invoke a "proof by induction". The reasoning proposed in Peter's answer is simply this : "there is no possible situation in which ...". When you "add" a new premise, you "restrict" the set of "possible situation" to be considered (and not enlarge it). But if there are none ("there is no possible situation ..."), you cannot restrict this set further, becuase it is already "empty". This is why you can add as many premises you want, and the relation of logical consequence between $A$ and $B$ will still "be there". – Mauro ALLEGRANZA Feb 10 '14 at 19:40
  • Wow that was awesome Mauro. I see what you are saying. Thank you very much the both of you for helping me with that. I wish I could rate you guys up, but I can't yet. – Valentino Feb 10 '14 at 20:02