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Disjunctive Propositions

In skepticism on 17/08/2011 at 11:54 pm

Question: do we attain certainty of at least some facts? We are certain, for example, that of two disjunctive propositions, one is true, the other false. We are certain about the truth of the principle of noncontradiction. While there is a trade-off between the utility of justifying informative claims and indubitably, for the principle of noncontradiction does at most say that there are at least some true sentences. Therefore, we know that there are at least some certainties about logic and there exists some true sentences. This is no argument, for it is exists outside the reach of argument: any criticism would, so the argument goes, have to assume the principle of noncontradiction, making it immune from argument. Therefore, the skeptical position is wrong about some state of affairs.

Is this true?

Two skeptical responses:

  1. We might note that a great deal of theory-laden cognition takes place when considering the principle of contradiction. We might, although it sounds silly at first blush, have made a simple mistake in our reasoning. Purported past cognitive state p is to person S actually a memory of past cognitive state p. We might misremember p, no matter how certain we are that we remembered p correctly; or be unaware that we have an improper justifier for p; or the justifier for p may be proper, but we may not be aware of it.
  2. The principle of noncontradiction is wholly supported on the structure of the argument itself, opening itself to the criticism that, while it is assumed as an axiom, logicians and mathematicians that one thought some axioms were obvious or indubitable have turned out to be, upon further reflection, far from obvious or indubitable. Just as Euclid’s fifth axiom could be rejected and still provide consistent non-Euclidean systems, the principle of noncontradiction can be rejected, producing paraconsistent logics.

Are we then certain of at least some facts?

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  1. This reminds me of the Duhem-Quine problem, where one doesn’t know if the premise or the conclusion is false, it’s a disjunction. But this does have a testable outcome. We may not be able to reason about it a priori, but we definitely have certainty that some newtonian predictions are false, such as the prediction of the planet vulcan between mercury and the sun. In my mind, even if we have some skeptical knowledge about what is true, we can often have a greater degree of certainty about what is false.

    And maybe certainty is to some extent not the right question, instead propositions should be understood as instrumental. In that case, we can often bypass certainty with utility. But I also have some personal leanings toward instrumentalism in science, I blame van Fraassen for that.

    • James,

      I feel the same way, but I would phrase it differently. When dealing with existential statements that refute scientific theories, it’s presently best to side with the existential statement iff it coheres with uncontroversial background assumptions.

      I put no stock in certainty either, but for different reasons: I’m not interested in the psychological states individual (or communities of) scientists hold; I’m interested in whether or not theories have a greater or lesser degree of verisimilitude, are true or false, or empirically adequate/empirically inadequate.

  2. Can the principle of noncontradiction be false?

    • Lee,

      I don’t know. I know very, very little of the philosophy of mathematics, so while I’ll put my money on logics that adopt the principle of noncontradiction being far more useful, far more conducive to criticism, than paraconsistent logics in a majority of situations, there are scenarios where, so I hear, paraconsistent logics are quite useful.

      In fact, I don’t know if such a statement as “The principle of noncontradiction is true” works at all, since it’s referring to a rule of operation. If so, it would be analogous to saying “The rule that cars in England drive on the left side of the road is true.” But then again, I’m just speculating; I don’t know.

  3. Is it true that all propositions are either true or false and not both?

    Paraconsistent logics may be useful, e.g. differentiating between alternative sets of contradictory premises — every contradiction has the same logical content in classical logic. However, “truth” in paraconsistent logic is not the same “truth” as in classical logic and the former certainly is not anything like correspondence to the facts. Also, when we discuss what paraconsistent logic is and isn’t, we do so in the context of classical logic. For example, it is either true, false, but not both, that the law of noncontradiction is not part of paraconsistent logic.

    More to the point, are you denying the existence of necessary truths, or are you just denying that we can know that a truth is necessary. There is a big and important difference, I think.

    • Lee,

      I’m of the mind that until I learn a bit more, I shouldn’t stick my foot in my mouth. That said, I will do my best to suck daintily on my toes. Yes, I deny the existence of necessary truths and I deny the existence of knowing that a truth is necessary. The second is trivial while the first is as radical as Quine’s position on necessary truths.

      Since I’m already ankle-deep down my throat, let me elaborate: some truths may be necessary under an axiomatic system, but are the axioms true? Of course we do not know, but are they true because its negation produces incoherence? Does the cheese stand alone? I’m unaware of any set of axioms such that its negation is incoherent …

  4. If there are no necessary truths, then are all propositions possibly false?

    I can criticise such a claim by appealing to the principle of noncontradiction — it generates paradoxes. Are you saying this is not a good criticism? If so, why not?

    • Lee,

      “If there are no necessary truths, then are all propositions possibly false?”

      I guess so, in the sense that the principle of noncontradiction is not grounded, justified, or foundational. Of course, it’s a good criticism of using alternatives, since we intuitively dislike paradoxes.

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