Normativity of Logic and the Preface Paradox

normativity logic

I recently came across a surprising claim: that logic is normative. That is, it is in some sense wrong to deny logic. The claim isn’t surprising because it’s controversial; instead, it’s so obvious that it’s initially jarring to see it spelled out explicitly. What is controversial is the claim that followed: that logic’s normativity isn’t universal. In other words, that it is sometimes rational to accept a deductive argument as valid, accept all its premises as true, and yet still deny its conclusion. Let’s see why this might be and whether it holds up.

The Preface

A well-known paradox, the Preface, goes as follows: I assert each thing I state in this post. After all, if there were something here I did not wish to assert, I would not state it. However, I also assert that I’m wrong about at least one thing I say here. Write anything long enough, and chances are, no matter how thoroughly you check yourself, you’ll get at least one thing wrong. (This post isn’t very long, but as an amateur writing on a complex topic, the post needn’t be very long for me to feel confident that there’s at least one mistake in it.)

So far so good. Here’s the trouble. Let’s label my assertions in this post p0, p1, p2, … pn. I’m apparently asserting that each of those is true, but also denying that their conjunction – (p0 & p1 & p2 & … & pn) – is true. According to classical logic and most non-classical logics – as well as common sense – if some group of propositions is each true, then their conjunction must be true.

So what can I do? I stand by each of my propositions. I also stand by the principle that in a post of this length, on this topic, I have almost definitely made a mistake somewhere. I, therefore, stand by each proposition but do not stand by their conjunction. Do I then deny the argument from truth of each conjunct to truth of conjunction? To do so would seem to fly in the face of the simple meaning of the word “and”.

Solution by degrees

There’s an obvious solution that ought to be addressed off the bat. Belief isn’t an all-or-nothing phenomenon. There are degrees of belief. I believe that my birthday is on September 10th, and I believe that if David Hume were alive today, he and I would be drinking buddies. But I’m more confident of the former than of the latter.

Here’s the proposed solution then: for each pi in this post, my belief isn’t total. Maybe I’m 83% confident in p0, 92% confident in p1, and so on. So, while I’m confident enough in each proposition to assert it, once the uncertainty from each is factored in, it makes perfect sense that the uncertainty of the conjunction will be huge.

This solution works as far as it goes, but there’s a sense in which it changes the subject. Maybe belief does work in degrees but assertion does not. Assertion is an act – you either take it or you don’t.

Or, well, so the intuition goes. Maybe the right way out is to deny that assertions are all-or-nothing phenomena. I don’t have a strong opinion as to the best solution to the paradox. But we now have the necessary background to evaluate Jc Beall and Greg Restall’s rather surprising claim, from their book Logical Pluralism, that the Preface paradox serves as an exception to the normativity of logic.

Beall and Restall: denying that normativity is universal

It’s worth quoting Beall and Restall verbatim (p.26):

…even though we accept each of the conjuncts p1, …, pn, we do not accept the conjunction. On the contrary, we explicitly reject it…. we accept all of the premises of a valid argument and we reject the conclusion. Furthermore, this is not an argument whose validity we dispute. No, we acknowledge the argument to be valid. We accept the premises. Nonetheless, we reject the conclusion. We have knowingly violated the norm we have espoused: the norm forbidding us to accept the premises of a valid argument while simultaneously rejecting the conclusion.

In response, it is reasonable to ask: what does it mean that they “acknowledge the argument to be valid”? Doesn’t it mean they agree that in all cases in which the argument’s premises are true, the conclusion must be true? If so, haven’t they demonstrated that they don’t, in fact, hold the argument to be valid?

It’s worth noting that Beall and Restall reject normativity in this case so as to avoid having to assert a contradiction. Since they assert the negation of the conjunction, they cannot also assert the conjunction because that would entail asserting a contradiction. I gather this from the emphases in their explanation: “a logical consequence of this collection of commitments is the inconsistent conjunction…and we certainly do not believe this” (p.25). Their claim later on in the same discussion that “…we do not endorse any particular inconsistent proposition (we hope!) in the course of this book” (p. 26) suggests the same.

However, if that is the motivation, they’re in trouble – they implicitly assert a contradiction anyway, by denying normativity. To see why, first consider that they emphasize that universal necessity is part of what makes validity what it is (p.32, emphasis added):

A valid argument is one whose conclusion is true in every case in which all its premises are true.

So here’s the problem: they claim that the argument from truth of conjuncts to truth of conjunction is valid, and that the Preface case is an exception. But part of what they mean by valid is “without exception”. In other words, their proposed way out entails both “there is no case where these kinds of premises are true and this kind of conclusion false” and “there is a case where these kinds of premises are true and this kind of conclusion false”– a straightforward contradiction. So long as they agree that validity means truth preservation in all cases, they contradict themselves by asserting validity and yet claiming an exception case.

Their defense is to claim “if one has good grounds to reject [the] conjunction, and one has good grounds for each of the conjuncts, it seems that one has good grounds for having an incoherent collection of beliefs” (p. 26). That’s fair enough. The problem is that there are at least three obvious ways of dealing with this situation that don’t involve contradictory views of validity:

  1. They can deny the validity of the argument. They can say that while the move from conjuncts to conjunction is usually truth-preserving, it is not always so. Since validity requires truth in all cases, this means denying the validity of the argument.
  2. They can assert both the conjunction and the negation of the conjunction. This is obviously inconsistent. But then, so is their (implicit) view that valid arguments both have and don’t have exceptions. The conjunction inconsistency is, intuitively, easier to swallow.
  3. They can deny that assertion is an all-or-nothing act, or find some other epistemic or praxiological analysis that dissolves the paradox by denying one of its premises.

“3” is a tall order – it begs the question that such an (accurate) analysis exists. But even if “3” isn’t a possibility, it seems fairly clear that “1” and certainly “2” are less drastic than denying the universality of normativity.

What would denying the universality of normativity mean?

I can think of only one way of defending the Beall/Restall position: to claim that denying normativity is in some sense to deny the applicability of logic for the case in question. In other words, rather than claiming that valid arguments both do and do not have exceptions, they might be claiming that some cases fall outside the domain of what we may apply logic to.

Thus, the rule that from the truth of some given propositions follows the truth of their conjunction is universal – within the scope of cases the argument reaches, so to speak. The case of the Preface just falls outside of that scope.

This is, of course, incredibly ad hoc. Beall and Restall could accept this, and simply say that that’s just the corner the Preface backs us into.

Maybe this sounds plausible. We all know of cases in which we’re willing to allow, say, our emotions to govern over our rationality. Here’s an example:

Premise) If I want to do well on the test tomorrow, I should go to sleep early tonight.

Premise) I want to do well on the test tomorrow.

Conclusion) I should go to sleep early tonight.

I might accept this argument as valid and the premises as true and still get trashed tonight like it’s my birthday and Hume’s visiting. There’s nothing particularly troubling about this. Humans are notoriously irrational creatures.

So maybe it’s reasonable to say: look, logic tells me to assert the conjunction, but given other considerations, I respectfully decline.

The problem with this is that in the Preface case, nothing extra-logical is at play. All kinds of things factor into whether I decide to go to bed early tonight, of which arguments are only one. In fact, to the extent that I consider the above argument valid and the premises true, I do accept the conclusion – I should go to sleep early tonight. The break happens when I decide to do something I agree I shouldn’t – in other words, in the realm of action, not argumentation.

In the case of the Preface, however, the only action at stake is assertion. There don’t seem to any other factors at play in the willingness to assert other than belief in the proposition in question. Belief, in turn, is wholly determined by perceived truth. And truth is what’s at stake in the matter of validity. In other words, there is nothing other than the perceived soundness of the arguments involved that is relevant to my willingness to assert the Preface conjunction. Assertion, while it is an act, does not depend on anything extra-logical in the cases under discussion. (Side note: Of course, we can imagine a case where I might choose not to assert the truth because it could land someone in trouble or lead to some other unwanted consequence – this would be a case where willingness to assert does depend on something extra-logical. Nothing of the sort is at play here).

It’s quite true that there’s more to human motivation than truth and argumentation. However, within the domain of argumentation, there is nothing outside the scope of logic’s applicability. This is so simply by definition. Logic just is the study of argumentative consequence – Beall and Restall say as much themselves. Any argument that factors into our willingness to assert something will not fall outside logic’s reach. To the extent that Beall and Restall argue for denying the normativity of logic in the case of the Preface, their claims are governed by logic.

If the Preface scenario seems to yield contradiction, then we must either deny the law of non-contradiction, deny the argument from truth of conjuncts to truth of conjunction, or find some problem in the premises of the setup. We cannot simply hide the problem in some praxiological cabinet out of logic’s reach.

Interested in paradoxes? Check out this primer on the Sorites Paradox and its solutions.

1 thought on “Normativity of Logic and the Preface Paradox”

  1. I want to propose another way to explain why we sometimes agree with all the propositions but deny the conjunction.
    The fault in the example you make with going to sleep should not be attributed to people often being irrational. As you say, logic is how we reason and when we act contrary to the apparent arguments, I believe it is more often the case that the arguments are not representative of the situation rather than the person acting without judgment.
    So an add to make your example reflect reality could be
    Premise) I prefer partying over certanty of doing well on the test tomorrow.
    This addition will remove any contradiction between the premises and a conclusion of partying over sleeping.
    This goes under the solution “finding some problem with the premises”, and I believe it to be the solution to most or all cases where one want to accept all propositions but not the conjunction of them.
    Another example:
    P1) I want to sit down
    P2) I have time to sit down
    C) I should sit down
    This example is missing the premises
    P3) Im outside and it is raining
    P4) I prefer getting inside over sitting in the rain.
    I havent read the paper you are refering to, so sorry if this kind of thing is already discussed properly there. But this is at least a better explanation to why people seem to act irrational often, than just claiming people to be truly irrational.

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