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An argument is invalid if and only if there is a possibility where its premises are true and its conclusion is false.

Then is the following argument technically valid?

  • a: Alfred has exactly 20 mice
  • b: Alfred has exactly 30 mice
  • c(conclusion): Bob has 5 mice

I'm inclined to think that it's not valid because it seems that we can say that Bob has 10 mice without contradicting the first two premises and then show that the conclusion is false. On the other hand, because the premises have nothing to do with the conclusion, I am not sure if the argument is then technically not invalid.

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    The fact that the conclusion does not follow from the premises itself means the argument is invalid. It is a fallacy known as a "non-sequitur". Sep 16, 2023 at 17:02
  • Where did you find that definition for invalid? Typically "valid" is defined "an argument is valid if and only if it is necessary that if all of the premises are true, then the conclusion is true" or something similar, and anything that isnt that is "invalid". That's not the same as your definition of invalid.
    – JMac
    Sep 16, 2023 at 18:34
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    This question has come up a few times, e.g. here: philosophy.stackexchange.com/questions/84490/… As benrg's answer says, in classical logic an argument with inconsistent premises is always valid. Provided your premises are provided with the background arithmetic needed to prove a contradiction, then the argument is valid. In paraconsistent logics it is not.
    – Bumble
    Sep 16, 2023 at 22:59
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    Please note that "If Alfred has both exactly 20 mice and exactly 30 mice, then Bob has 5 mice" is not equivalent to "Alfred has both exactly 20 mice and exactly 30 mice, thus Bob has 5 mice". The former is true, and the latter is false.
    – Stef
    Sep 17, 2023 at 19:09

5 Answers 5

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For an argument to be valid just means that its conclusion follows from its premises, not that its conclusion is true. Your argument is valid in classical logic by the principle of explosion, but it's a moot point because you'll never encounter a real-world situation where the premises are true, so you'll never be able to use the argument anyway.

Paraconsistent logics don't have the principle of explosion, and your argument would be invalid in those.

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    “you'll never encounter a real-world situation where the premises are true, so you'll never be able to use the argument anyway.” Arguments from contradictory premises can certainly be used in real life, just indirectly — most often, to refute a hypothetical which would imply those premises. E.g. if Eve claims Alfred has 20 mice and Frank claims Alfred has 30 mice, then you can conclude that either Eve or Frank is wrong. Sep 18, 2023 at 9:51
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Here's an argument you might find interesting.

  1. The only even prime is 2
  2. Gollum appears in The Idiot
    Ergo,
  3. God exists or God does not exist

If the conclusion is a tautology (necessarily true) then it's impossible for the premises to be true and the conclusion false. That is to say, the above argument is valid.

EDIT

Also (gracias Bumble) ...

If one of the premises is a necessary falsehood (contradiction?), the argument is valid for the same reason - impossible for the premises to be true and the conclusion false (the synatctic avatar of ex contradictione sequitur quodlibet?)

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    Indeed. Sometimes called the principle of implosion. Though, as with explosion, it holds for classical logic, but not for paraconsistent logics.
    – Bumble
    Sep 17, 2023 at 12:59
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Can an argument be valid even if its conclusion has nothing to do with its premises?

No. The proponent just ends up with three true, but unrelated, sentences.

There are rules for a valid syllogism. There are typically six, seven, or eight of these, depending on what you are reading.

The example in the question has an "undistributed middle." This syllogism fails to distribute the middle term ("mice"), and distribution must occur in a valid syllogism. Thus there is nothing to connect the premises to each other.

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Yes and no.

Most arguments in life outside academia are enthymematic, i.e., incomplete. That is, part of the argument is left implicit, for the audience to guess.

Given this, the explicit argument may have an explicit conclusion unrelated to the explicit premises. However, once you put back together the complete argument, if it is to be logical, then the conclusion will follow rigorously from the premises, which requires that the truth of the premises be relevant to the truth of the conclusion.

So, "Caesar didn't die, so the moon is made of green cheese" is not only unsound but also logically invalid, while "Caesar didn't die, so he is still alive" is equally unsound but perfectly valid once you put back into the argument some common-sense assumptions about life and death.

This does not preclude using the phrase "the moon is made of green cheese" to make perfectly valid arguments. For example: "If he has money on the bank, then the moon is made of green cheese". Presumably, we all understand the implication. But this is another story.

Your argument:

  1. Alfred has exactly 20 mice
  2. Alfred has exactly 30 mice
  3. (conclusion): Bob has 5 mice

This is self-evidently not valid. If we don't equivocate, the premises are inconsistent, and no conclusion can be derived logically from the same inconsistent premises.

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  • Which of the two inferences do you think are invalid: “A holds, so A holds or B holds,” or “A holds or B holds; A doesn’t hold, so B holds”? You can’t accept that the premises are inconsistent and that both of the mentioned reasoning patterns are valid without also accepting that at least some inferences can be made from inconsistent assumptions.
    – PW_246
    Sep 17, 2023 at 19:17
  • @PW_246 This is the implication A ⊢ A ∨ B and the implication (A ∨ B) ∧ ¬A ⊢ B, and both are true implications, so any inference based on that will be valid. 2. "You can’t accept" Yes I can. In fact, nobody ever infers anything from inconsistent premises and everybody accept these two implications as true. Mathematical logic is nonsense. Sep 18, 2023 at 9:48
  • Are you using ‘infer’ differently than I am? I mean that from A one may infer B iff under any circumstance in which A holds, B also holds. This clearly includes the case when A never holds. If by ‘infer’ you mean “to derive new knowledge from previously established facts” or something to that effect, then ok, but at least be clear about it.
    – PW_246
    Sep 18, 2023 at 11:42
  • @PW_246 "Are you using ‘infer’ differently than I am?" You are. "This clearly includes the case when A never holds" See? "but at least be clear about it" Why would I need to specify that I am using words according to what they mean in English?! It is mathematicians who should stop equivocating on the English vocabulary. Sep 18, 2023 at 14:56
  • If you assume 0=1, you may validly infer 10>100, and anything else you want. You don’t even need the concept of negation to prove anything that can actually be written down by a person doing arithmetic, so long as you assume 0=1. When you have an opinion/use a definition that almost no one else uses, the onus is on you to be clear.
    – PW_246
    Sep 18, 2023 at 14:59
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Please note the subtle difference between the following two propositions:

P1. If Alfred has both exactly 20 mice and exactly 30 mice, then Bob has 5 mice.

P2. Alfred has both exactly 20 mice and exactly 30 mice, thus Bob has 5 mice.

Proposition P1 is true, and proposition P2 is false.

Now let us examine your argument:

a) Alfred has exactly 20 mice

b) Alfred has exactly 30 mice

c) Conclusion: Bob has 5 mice

Now the question is whether this argument corresponds to P1, which is true, or to P2, which is false.

Someone who reads your argument literally would argue that your argument corresponds to P2. You stated as a fact that Alfred had both exactly 20 and exactly 30 mice, and this is false. Thus your argument is wrong.

However, someone who has seen syllogisms before could argue that your a) and b) are not stated as facts, but as hypothesis. In this case, they would read your argument as "If a and b, then c", which corresponds to P1 and is correct.

So, whether your argument is correct or wrong is a matter of convention: if its presentation in the classical form of a syllogism implicitly leads the reader to read your a) and b) as hypothesis rather than fact, then the argument is correct; but if the presentation is not enough for the reader to understand this implicitly, then your argument is wrong.

Personally, I have a strong dislike for anything that is intentionally obscured so that it can only be read by the initiated. So I would argue that your argument would only be correct in context, i.e. within a text that first introduced the convention that arguments that are formulated like syllogisms should be understood as "if a and b, then c" rather than be read literally as "a and b, thus c".

Alternatively, you could make your argument both more explicit and entirely self-contained by indicating within the argument whether a and b are hypotheses or facts, instead of relying on a convention about syllogisms that may or may not be shared by the reader:

a) Hypothesis: Alfred has exactly 20 mice

b) Hypothesis: Alfred has exactly 30 mice

c) Conclusion: Bob has 5 mice

CORRECT

or

a) Fact: Alfred has exactly 20 mice

b) Fact: Alfred has exactly 30 mice

c) Conclusion: Bob has 5 mice

WRONG

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  • You are confusing truth with validity. Your use of the word 'thus' in P2 appears to be a kind of turnstile, so P2 is not a proposition but two propositions. It consists of a premise "Alfred has both exactly 20 mice and exactly 30 mice" and a conclusion "Bob has 5 mice". As an argument, this is valid in classical logic, but it is unsound, since not all its premises are true. The question is asking whether the argument is valid. At the end, you use the word 'wrong' to describe an argument, but it is wrong by virtue of being unsound, not invalid.
    – Bumble
    Sep 17, 2023 at 20:05
  • Thank you for your comment. Had it not opened with an accusation, it might have been welcome.
    – Stef
    Sep 17, 2023 at 20:07
  • @Stef "Proposition P1 is true" I gather that you say this because the antecedent "Alfred has both exactly 20 mice and exactly 30 mice" is false. This is the horseshoe theory, but I doubt anyone would ever dare assert this sort of meaningless implication in real life contexts, say in a job interview or campaigning to get re-elected chairman of banking association. We don't do P1 for the same reason that we don't do P2. Sep 20, 2023 at 5:48
  • @Speakpigeon I don't understand your comment. I say "Proposition P1 is true" because proposition P1 is true. No interviewer has ever faulted me for being logical. I don't know what you mean by "do P1" or "don't do P1". P1 is a proposition, not an action. It's a true proposition, regardless of why we "say this", regardless of your political inclinations, and regardless of horses and their shoes.
    – Stef
    Sep 20, 2023 at 8:58
  • @Stef "* I say "Proposition P1 is true" because proposition P1 is true. " Clearly not, because it is not. 2. "*No interviewer has ever faulted me for being logical" Did you ever say something like P1 in a job interview? This is what I mean by "do P1". 3. "P1 is (...) a true proposition, regardless of why we "say this"" But P1 is not true, whatever you say. Sep 20, 2023 at 10:08

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