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Consider the following argument:

(A) Either sales or expenses will go up.

(B) If sales go up, then the boss will be happy.

(C) If expenses go up, then the boss will be unhappy.

(D) Therefore, sales and expenses will not both go up.

If someone is to apply strict mathematical logic, there would be two possibilities:

  1. If it is possible that the boss can be happy and unhappy at the same time, the argument will be invalid.
  2. If the boss cannot be happy and unhappy at the same time, the argument will be valid.

However, putting mathematics aside, the argument above doesn't make much sense to me anyway I look at it. My reasoning is that, sales/expenses and the boss' mood are independent variables. In other words, sales and expenses can go up or down regardless of the boss' mood.

So my question to you is, if we are to use common sense, is there anything wrong with the argument above? If there is, what exactly?

  • When you say 'is there anything wrong', do you mean is it invalid, or something else? – Eliran Jul 7 at 15:23
  • @Eliran What I am asking is, does this argument make sense? If I get this correctly, the logic of the argument is: "The reason why sales and expenses cannot both go up, is because boss cannot be happy and unhappy at the same time". If you are to consider this reasoning from mathematical perspective, apparently it is valid. But again, if we are to use common sense, argument is just not reasonable (as I explained above, sales/expenses are completely unrelated to boss' mood, they can decrease or increase regardless of how boss will feel about it). – Nelver Jul 7 at 15:57
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    You're right about validity. What you're asking is whether the premises are reasonable. But that's not a matter of logic and that's not something we can tell without further information. I agree however that the premises are too simplistic. But I assume this is just an exercise and not an attempt to convince anyone of anything about the boss or the sales. – Eliran Jul 7 at 16:09
  • So the point there is, the soundness of the premises do not affect the validity of the argument? In other words, even if the premises do not make much sense, if the main condition is satisfied (If all premises are true, the conclusion must be true), the argument is valid? – Nelver Jul 7 at 16:27
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Your argument is obviously valid.

It can be formalised as follows:

(S → H) ∧ (E → ¬H) ⊢ ¬(S ∧ E)

It is also clear that the premises, taken literally, are false. It is not true that each time expenses go up the boss is unhappy or that each time the sales go up the boss is happy. In particular, both won't be true if expenses and sales go up at the same time.

The argument is valid, the premises false, and common sense is safe.

As worded, your premises are unsuitable for deductive logic. We normally understand from context, indeed using common sense, that "If sales go up the boss will be happy" is not equivalent to "sales go up implies the boss will be happy" since it may be true that sales may go up and that the boss remains unhappy.

So, again, the argument is perfectly valid, but we tend to make the usual reading of the premises so we are likely to experience of cognitive dissonance due to the conflict with a deductive logic reading.

In effect, your argument is not one based on form only. We find it difficult to read the premises according to form only. We interpret them in the usual sense and we feel that some fallacy is being committed.

Maybe we can rephrase to make explicit what we think your premises really mean according to common sense:

(A) Either sales or expenses will go up.

(B) If sales go up, then the boss will have a reason to be happy.

(C) If expenses go up, then the boss will have a reason to be unhappy.

(D) Therefore, sales and expenses will not both go up.

This argument is obviously not valid, essentially because the two consequent clauses in (B) and (C) this time are not contradictory.

Common sense is safe.

  • Thank you, you've answered my main question. If it is possible, please help me to clear up one more: assume we are to assign different variables for "boss being happy" (H, for example) and "boss being unhappy" (Q, for example) Would the argument be valid? Because under these circumstances, it would be possible for all premises to hold, and for conclusion to fail. – Nelver Jul 9 at 9:42
  • @Nelver Then the argument would formalise as (S → H) ∧ (E → Q) ⊢ ¬(S ∧ E) and therefore has the same logical form as the rephrasing of your argument I provided in my answer, and it is therefore also obviously not valid. – Speakpigeon Jul 9 at 9:59
  • The reason why I ask is because you are saying that the argument is "obviously" valid. But again, it seems to me that validity of the argument depends on how you assign variables: if you treat "unhappy" just as a negation of happy (meaning that we deal only with one variable, H), then argument is valid. On the other hand, if you treat "happy" and "unhappy" as separate variables, then argument will be invalid. So, at least to me, the validity of the argument is not that obvious. So is there some reason why you say that the validity of the argument is obvious here? – Nelver Jul 9 at 10:15
  • @Nelver Your argument only becomes obvious once you clear up the ambiguity as to whether the consequent clause of (B) and that of (C) are or not the negation of each other. However, the ambiguity of your argument is obvious and takes precedence over the question of validity since there is no use assessing the validity of an argument which is ambiguous. Thus, validity is not initially obvious precisely because of the ambiguity of the argument. – Speakpigeon Jul 9 at 13:53

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