Maverick Philosopher

Footnotes to Plato from the foothills of the Superstition Mountains

Does the Validity of an Argument Depend on the Order of its Premises?

Suppose you have a valid argument. Can you render the argument invalid by changing the display order of the premises?

I should think never. The Dark Ostrich, however, offers the following putative counterexample. He says he got it from Sainsbury; I should like to see a reference.  And if there is a literature on this, I should like to see a bibliography.

(A) Some Greek is called ‘Mark’, Mark is an evangelist, therefore some Greek is an evangelist. (VALID)

(B) Mark is an evangelist, some Greek is called ‘Mark’, therefore some Greek is an evangelist. (NOT VALID)

(A) is valid and (B) is not. But this is not evidence that premise order affects validity. For while the sentences are the same, the premises of the two arguments are not the same.  Made explicit, (A) becomes

(A*) Some Greek is called 'Mark', this same individual called 'Mark is an evangelist, therefore some Greek is an evangelist.

Clearly, (A*) and (B) have different premises. So it is not the different order of the premises in (B) that causes it to be invalid.

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Bill Vallicella

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18 responses to “Does the Validity of an Argument Depend on the Order of its Premises?”

  1. CAMcIntosh Avatar
    CAMcIntosh

    I had a professor as an undergraduate who, to teach Anselm’s ontological argument, distributed its dozen or so premises individually and asked the class to order them so as to make the argument valid. I objected that the order of a deductive argument’s premises has no relevance to its validity. The professor said I was wrong!

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  2. CAMcIntosh Avatar
    CAMcIntosh

    Anyone who’s learned propositional logic by doing proofs can just see that order is irrelevant. I remember doing multiple 80-something step proofs under a strict time limit. The best strategy was to derive, as quickly as possible, everything you can from what you’re given at the start. Sometimes you see a conjunction first, sometimes a disjunction, sometimes an implication, etc. You might number 10 derivations 1-10, then refer back to each one by number as you complete the proof. But because you numbered them in the random order you saw them, it’s clear order doesn’t matter!

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  3. BV Avatar
    BV

    Chad,
    The professor might have distributed lines in the proof where a line is either a premise or a sub-conclusion. If that is what he did, then order would matter.
    But I agree that the order of premises is irrelevant to validity.
    Still and all, the evangelist example above is very interesting. I take it you accept my diagnosis.

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  4. CAMcIntosh Avatar
    CAMcIntosh

    Yes, I agree your diagnosis.
    But I’m not sure order matters even if premises are sub-conclusions of a larger argument. Suppose we have a 20-premise argument for conclusion C, and 2 of those premises are sub-conclusions of other premises. I don’t see why their order should matter. All 20 premises are equally prior to C in logical space.

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  5. BV Avatar
    BV

    We need an example.
    1. Something is green. Conclusion. From 4
    2. Either Hillary is honest or grass is green. Premise
    3. Hillary is not honest. Premise
    4. Grass is green. From 2, 3, by Disjunctive Syllogism. Subconclusion-premise
    So I think you are right, Chad. What I said wasn’t right. A subconclusion functions as a premise, and display order of premises does not matter logically. So your undergrad prof was wrong.
    There is nothing spatial about logical space, and nothing temporal about logical consequence.
    It is interesting that our thoughts, which must be temporally sequential, are able to grasp atemporal logical relations such as entailment, consistency, and inconsistency.

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  6. The light ostrich Avatar
    The light ostrich

    I borrow the term from Sainsbury (‘Fregean Sense’, in Departing from Frege, 125-136, p.136, who points out that in inference we need ‘transportable’ conclusions. E.g. the inference (my example) ‘a is F, therefore it is F’ is valid, but the conclusion is not ‘transportable’, i.e. cannot be placed in other contexts where the antecedent is missing.
    So what happens when you transpose the premises in your (A*)?

    this same individual called ‘Mark is an evangelist, some Greek is called ‘Mark’

    Ergo a demonstrative like ‘this same individual’ is not transportable.
    I see a few people here saying that in standard textbook logic the order doesn’t matter. Then they need to explain why it wouldn’t. Geach discusses the problem in various places, though he offers no coherent theory.

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  7. The light ostrich Avatar
    The light ostrich

    I agree of course that

    Some Greek is called ‘Mark’, every person called ‘Mark’ is an evangelist, therefore some Greek is an evangelist

    is valid and does not depend on order of premises. However ‘every person called ‘Mark’ is an evangelist’ does not mean the same as ‘Mark is an evangelist’. Clearly not.

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  8. The London Ostrich Avatar
    The London Ostrich

    And let’s be clear about the argument that this supports.
    1. Changing the order of the sentences (if you like) renders the argument invalid.
    2. It follows that the same sentences have a different meaning when transposed.
    3. Furthermore, no sentence replacing the first sentence in (B) can possibly have the same meaning as the second sentence of (A).
    4. Ergo the meaning of the second sentence of (A) is non-transportable.
    >>It is interesting that our thoughts, which must be temporally sequential, are able to grasp atemporal logical relations such as entailment, consistency, and inconsistency.
    That’s not the point at all. The point is that the thought expressed by the second sentence of (A) cannot be expressed before the first sentence is expressed. The proper name is semantically dependent upon the indefinite antecedent ‘some Greek called ..’.
    Does that make more sense?

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  9. BV Avatar
    BV

    >>I borrow the term from Sainsbury (‘Fregean Sense’, in Departing from Frege, 125-136, p.136, who points out that in inference we need ‘transportable’ conclusions. E.g. the inference (my example) ‘a is F, therefore it is F’ is valid, but the conclusion is not ‘transportable’, i.e. cannot be placed in other contexts where the antecedent is missing.<< For example, 'Al is fat, therefore he is fat.' Yes, the argument is valid. But the conclusion is not *he is fat,* which is not a proposition but a propositional function, but *Al is fat.* This is an invalid form: a is F, ergo x is F. So the conclusion is transportable. Are you thinking of arguments as arrays of sentences? I would say that an argument is an array of propositions (that's not a df. but specifies a nec. cond.) and that one does not infer a sentence from a sentence, but the content of a sentence from the content of a sentence. Inference requires understanding, and what we understand are not sentence-tokens but their senses, or perhaps them as expressing propositional senses. This whole discussion is murky as hell because there is no clear distinction between propositions and sentences used to express them.

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  10. The London Ostrich Avatar
    The London Ostrich

    >>This whole discussion is murky as hell because there is no clear distinction between propositions and sentences used to express them.
    Not really murky. Once again:
    1. Changing the order of the sentences (if you like) renders the argument invalid.
    We clearly both agree with this.
    2. It follows that the same sentences have a different meaning when transposed.
    I can’t see how you could disagree. If they had the same meaning, then the argument would be valid.
    3. Furthermore, no sentence replacing the first sentence in (B) can possibly have the same meaning as the second sentence of (A).
    For if so, it would have to have the same meaning as the untransposed second sentence of (A), ‘Mark is an evangelist’. I.e. the singular term would have to somehow refer forward just as it refers back in the valid argument. But this is impossible. The whole point and purpose of an indefinite sentence ‘Some Greek is called ‘Mark’’ is its indifference to what want before. The Greek Mark may be the same as someone mentioned previously, or may not. The indefinite article leaves that open, and is meant to.
    4. Ergo the meaning of the second sentence of (A) is non-transportable.
    I.e. we can’t transport its meaning to an earlier part of the text.
    Nothing here is remotely murky, but it’s hard to get your head around. The idea of a thought that could not possibly be entertained or expressed until some sentence had been uttered is a monstrosity. But a true monstrosity.

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  11. BV Avatar
    BV

    >>1. Changing the order of the sentences (if you like) renders the argument invalid.<< This is very tricky. You speak of THE argument. You are assuming that there is one argument. Not clear. I suggest that changing the order of the sentences gives rise to a second, invalid, argument. So there are two arguments, with different premises, one argument valid the other invalid. >>2. It follows that the same sentences have a different meaning when transposed.<< Yes, they express different propositions. >>3. Furthermore, no sentence replacing the first sentence in (B) can possibly have the same meaning as the second sentence of (A).<< Why? I could replace 'Mark is an evangelist' in (B) with 'Mark is a Greek evangelist.'This sentence has the same meaning as the second sentence in (A). >>4. Ergo the meaning of the second sentence of (A) is non-transportable.<< It could if you reformulate it. I am sorry, Ed, if I am being dense.

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  12. The London Ostrich Avatar
    The London Ostrich

    >>This is very tricky. You speak of THE argument. You are assuming that there is one argument.
    OK, let’s say say that (A) is three sentences which constitute a valid argument, whereas the rearrangement of those sentences does not constitute a valid argument.
    >>I could replace ‘Mark is an evangelist’ in (B) with ‘Mark is a Greek evangelist.’This sentence has the same meaning as the second sentence in (A).
    The second sentence in (A) is ‘Mark is an evangelist’ clearly does not have the same meaning as ‘Mark is a Greek evangelist’ ! Of course, the argument is now valid, but the conclusion follows from ‘Mark is a Greek evangelist.’ alone, whereas in (A) it only follows from both sentences taken together.

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  13. BV Avatar
    BV

    >> ‘Mark is an evangelist’ clearly does not have the same meaning as ‘Mark is a Greek evangelist’ !<< In one sense you are right, and in that sense what I said is preposterous. But an argument, I say, consists of propositions, not sentence-meanings. There is a sense in which the meaning of a sentence is its propositional SINN. For example, what is the meaning of 'I am hungry' uttered by me? On one way of looking at it, it is the same as the meaning it has when uttered by you. The sentence meaning is the same. But when I say it it expresses a different proposition than when you say it. *BV is hungry* versus *EB is hungry.*

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  14. The London Ostrich Avatar
    The London Ostrich

    >> In one sense you are right [‘Mark is an evangelist’ clearly does not have the same meaning as ‘Mark is a Greek evangelist’], and in that sense what I said is preposterous.
    What is the sense in which the Ostrich is not right?

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  15. BV Avatar
    BV

    You have to read the rest and think a little.

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  16. The London Ostrich Avatar
    The London Ostrich

    >>You have to read the rest and think a little.
    And so I did but I still don’t get it. I accept that the same sentence can have the same sense (‘I am hungry’) but also a different sense, as you say. But here is the problem:

    (B) [REQUIRED SENTENCE], some Greek is called ‘Mark’

    The requisite sentence must have the following properties. (a) It must express a singular proposition identifying a referent. (b) its sense or the proposition it expresses must allow us to infer ‘some Greek is an evangelist’ (c) it must be such that we cannot infer this by means of that sentence (or its sense) alone. (d) by the same token it must be clear that its referent is identical with whatever person the second sentence is true of. E.g. suppose the sentence is ‘Mark is a Greek evangelist.’ as you suggest above. This licenses the inference (although it violates (c) above), but perhaps the ‘some Greek’ mentioned in the second sentence could be a different Greek. By contrast, the original ordering (A) above can be true of only one individual, whereas your version could be true of two, i.e. two Greeks called ‘Mark’, one of whom is an evangelist.

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  17. BV Avatar
    BV

    It might help to simplify the problem.
    C. Some Greek is called ‘Mark.’ Mark is an evangelist.
    D. Mark is an evangelist. Some Greek is called ‘Mark.’
    In canonical English, the proposition expressed by (C) is:
    P. There is an x such that: x is called ‘Mark’ & x is an evangelist.
    Note that both occurrences of the individual variable ‘x’ are bound by the particular/existential quantifier.
    (D), however, does not express (P). For (D) leaves it open whether or not the Greek called ‘Mark’ is an evangelist. This is not left open by (C).
    So that is the linguistic datum, and a very interesting datum it is. We can straightaway infer from the datum that in some cases the display order of SENTENCES affects which PROPOSITION is expressed.
    Do you agree?

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  18. The London Ostrich Avatar
    The London Ostrich

    I think I agree with this, although containerists would disagree that (C) expresses P. For C contains a referring term ‘Mark’, whereas P does not. ‘x’ is a bound variable, not a referring term.
    But I am not a containerist.

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