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Bridge Principle – Surveying the Options

Arguably, the viability of the claim that logical consequence imposessystematic normative constraints upon our belief system hinges onwhether there is a defensible bridge principle. But how are we todecide whether a bridge principle is up to the job? One crucial testconsists in how a bridge principle fares in the face of the objectionspresented in§3.1. In addition, MacFarlane (MF2004: 12) advances the followingdesiderata:

• The StrictnessTest: At least when it comes to ordinary, readilyrecognizable logical implications leading to conclusions that theagent has reason to consider, there is something amiss about an agentwho endorses the premises but fails to believe the conclusion. Such anagent is “not entirely as she ought to be”. The StrictnessTest delivers a prima facie strike against reason-based bridgeprinciples. According to such bridge principles, my reasons for beinglogically coherent may be overridden by other reasons. In suchsituations I violate Strictness. (MacFarlane’s criterion isinspired by Broome 2000: 85.)

• The Priority Question: The attitudinalvariants have a distinctive advantage when it comes to dealing withExcessive Demands worries. But relativizing one’s logical obligations to, say,one’s logical beliefs or one’s logical knowledge invitesproblems of its own, according to MacFarlane.^[S1] The problem according to MacFarlane is that

  we seek logical knowledge so that we will know how we ought to reviseour beliefs: not just how we will be obligated to revise them when weacquire this logical knowledge, but how we are obligated to revisethem even now, in our state of ignorance. (MF2004: 12)

  In other words, according to this intuition it is the facts aboutlogical consequence that constrain our doxastic attitudes regardlessof whether we are able to recognize them.

• Logical Obtuseness: Suppose someoneprofesses to believe \(A\) and \(B\), but refuses to take a stand on(neither believes nor disbelieves) the conjunction \(A \land B\).Intuitively, such a person is liable to criticism. Principles withnegative polarity allow for breaches of Obtuseness. So long as theagent does not activelydisbelieve \(A \land B\), ournegative bridge principles find no fault with cases like these. Ifthis intuition carries any weight, negative principles may ultimately prove to be too weak.

As is already becoming apparent, our desiderata are at times intension with one another. For instance, theStrictness Test favors ought-based principles, whereasExcessive Demands and the Preface Paradox tell against them and hence pull in theopposite direction. As things stand, there can thus be no one bridgeprinciple that satisfies all of them. Consequently, in evaluating ourbridge principles against these criteria, we will need to decide onthe relative importance of our various desiderata. Each principle willbe assessed according to how well it performs when judged against theappropriately weighted desiderata. The principle (or group ofprinciples) that performs best across the board takes the prize. Atleast this is the route taken by MacFarlane (MF2004) and, with respectto a slightly different set of desiderata, Field (2009a).

Let us takeIMP (aka Co+) as our point of departure again. As we have alreadyobserved,IMP falls victim both to theNo Guidance Objection and theBootstrapping Objection. A natural reaction in light of these failings is to retreat to theweaker reason operator, thus giving rise to the following twoprinciples (and their various attitudinally constrainedcounterparts):

• (Cr+) If\(A_{1}, \dots, A_{n}\models C\), then if \(S\) believes all the\(A_{i}\), \(S\)has (defeasible) reason to believe\(C\).
• (Cr−)If \(A_{1}, \dots, A_{n}\models C\), then if \(S\) believes all the\(A_{i}\), \(S\)has (defeasible) reason not to disbelieve\(C\).

Though this move does show some initial promise, this group of narrowscope principles does not ultimately succeed either. While, arguably,the (Cr)s are immune to theNo Guidance Objection, things look less promising when it comes to Bootstrapping. Accordingto the (Cr)s, believing \(A\) automatically gives rise to a reason tobelieve \(A\). Many will view this kind of (partial)auto-justification with some suspicion.^[S2] What is more, almost by definition, the (Cr)s violate theStrictness Test.

Let us turn to the Bs. The Bs, recall, are characterized by the factthat deontic operators act both on the antecedent and on theconsequent of the embedded conditional. But these two deonticoperators are generally underwritten by norms stemming from differentsources. For simplicity, let us focus on (Bo+) (the discussion tofollow generalizes to the remaining Bs). In the embedded conditional“if \(S\) ought to believe all the \(A_{i}\), \(S\) ought tobelieve \(C\)”, theought in the antecedent has itssource in whatever epistemic norms make it the case that \(S\) oughtto believe the \(A_{i}\) (we may assume that the \(A_{i}\) are notthemselves acquired by logical inference and so the norms in questionwill not themselves be logical or logic-induced). It is in virtue oftheir compliance with this norm that the premises enjoy a positiveepistemic status. \(C\)’s positive epistemic status, bycontrast, may only be derivative; it may be inherited by dint of\(C\)’s logical relation to the \(A_{i}\). Thus, on this picturethe normative force of logical consequence resides in its ability topropagate whatever epistemic goodness the premises might enjoy totheir logical consequences. For example, the \(B\)-principle

• (Bp+) If\(A_{1}, \dots, A_{n}\models C\), then if \(S\) may believe all the\(A_{i}\), then \(S\) may believe \(C\).

could be based on an evidential norm, which might constitute the source of theagent’s permission to believe the \(A_{i}\):

• (EN)\(S\) maybelieve \(A\) if only if \(A\) is sufficiently likely in light of theevidence.

Thus, the reason why \(S\) may believe the \(A_{i}\) may be given by(EN): it is because all of the \(A_{i}\) enjoy the necessary evidentialsupport.(Bp+) then states that one may believe the logical consequences of thepropositions one is permitted to believe on account of their evidential support.

As a result of this, the Bs, unlike the Cs, are immune to theNo Guidance worry as well as to theBootstrapping Objection. That is not to say, however, that the Bs do not face potentialdifficulties of their own. The chief drawback, according toMacFarlane, is that the Bs are too weak: “according to the Bs,logical consequence is a channel through which existing norms ofbelief (obligation, permission, reason) can beextended” (MF2004: 10, the emphasis is the author’s). (Bo+),for instance, says thatif you ought to believe the\(A_{i}\), you ought to believe their (joint) entailments. But if itis not the case that you ought to believe the premises, it provideslittle guidance as to what you should do. And therein lies the sourceof MacFarlane’s worry:

according to the Bs, then, logic is only normative for those whosebeliefs are already in order—that is, for those who believe whatthey ought to believe (or may believe, or have reason to believe). Tothe unfortunate others, logical norms simply do not apply. (MF2004: 10)

Now, the Bs do offersome guidance. After all, according to(Bo+), for instance, upon discovering that an unacceptable proposition\(C\) follows from my belief that \(A\), I can immediately concludethat it is not the case that I ought to believe \(A\). Nevertheless,it may be felt that one should be bound by logic even when one hasbeliefs one ought not to have. That is, it might bethought that it would constitute anadditional strike againstan agent whose beliefs are not in good order, if he failed to belogically coherent. It is a legitimate question, however, whetherlogic really is normative in this way. We will return to this questionin§5.1.

This brings us to the Ws. The Ws have considerable intuitive upside,which have earned them a number of followers. Broome (2013) appears togo in for a weak attitudinal restriction of(Wo+). Sainsbury (2002) seems to advocate an attitudinal restriction of(Wr−), while Streumer (2007) adopts a version of unrestricted(Wr−). MacFarlane (MF2004)himself settles on a combination of (Wo−) and (Wr+). However,not all of the \(W\)s are contenders. The (Wp)s, for one, are out ofcontention for being too weak. The (Wp)s imply that I have permissionto see to it that if I believe \(A\) and \(B\), I also believe \(A\land B\), but this does not even so much as provide me with a reasonto believe the latter when I believe the former. As MacFarlane putsit, “the difference between the (Wp)s and the position thatthere are no logical norms for belief seems slim indeed”(MF2004: 10). Let us therefore focus on the remaining Ws. Forsimplicity, let us begin by considering(Wo+):

• (Wo+) If\(A_{1}, \dots, A_{n}\models C\), then \(S\) ought to (if \(S\)believes all the \(A_{i}\), \(S\) believes \(C\)).

(Wo+) elegantly dodges both theNo Guidance and theBootstrapping Objection. As for the former, the wide-scope reading provides just the wiggleroom needed to neutralize Harman’s objection: suppose I believe\(A\) and \(A\supset B\). According to(Wo+), I may meet my logical obligations in one of two ways: by eitherretaining my beliefs and also coming to believe \(B\),or byditching at least one of my antecedent beliefs in \(A\) and \(A\supset B\) so as to absolve me from the obligation to believe \(B\).When \(B\) turns out to be untenable, the latter course of actionrecommends itself: we revise our beliefs in light of theirunpalatable consequences. This not only meets Harman’schallenge; it paints an attractive picture of the interplay betweenlogic and other epistemic norms: logic, on the whole, does not requireus to have any particular beliefs (except logical truths, perhaps).Rather it prohibits certain constellations of doxastic attitudes. Whenwe reason we must negotiate these global constraints on our system ofbeliefs with other epistemic norms—local norms that guide us indetermining whether a given proposition is to be believed. Thiselement of negotiation is well captured by the wide-scope reading.

Moreover, provided one is willing to accept that deontic modals arebest read as propositional operators,(Wo+) is untroubled by Broome’s reflexivity worries. It yields that,for any proposition \(A\) that I happen to believe, I ought to(believe \(A\) or not to believe \(A\)).

Things look less rosy when we consider(Wo+)’s performance with respect to some of the other desiderata. It requiresus to have an attitude (namely the attitude of believing) towardsany consequence of the beliefs I retain, thus placingExcessive Demands on us. Moving to (Wo−) avoids not only that problem, but alsothat ofClutter Avoidance. It does not, however, stave off the Preface problem. What is more, asa principle of negative polarity (Wo−) faces theLogical Obtuseness problem.

We could alternatively try our luck with an attitudinal principle. Tofix ideas, let us opt for

• (Wo+b) If\(S\) believes that \(A_{1}, \dots, A_{n}\models C\), then \(S\) oughtto (if \(S\) believes all the \(A_{i}\), \(S\) believes \(C\)).

The upside is that(Wo+b) takes care of the problem ofExcessive Demands. It does not, however, take care ofClutter Avoidance. At least not as it stands. To fix this, we could further try thefollowingaddendum:

• (Wo+b*)If \(S\) believes that \(A_{1}, \dots, A_{n}\models C\) and \(S\)considers \(C\) or has independent reason to consider \(C\), then \(S\) ought to(if \(S\) believes all the \(A_{i}\), \(S\) believes \(C\)).^[S3]

However, even if the amended principle successfully wards offClutter Avoidance, it remains vulnerable to Preface Paradox-type considerations.

To deal with it, we could weaken our wide scope principle by replacingthe strictought operator by the defeasiblereasonoperator. The thought is that one’s reasons for being logicallycoherent may be trumped in Preface-like situations. We will return tothis point in the next section. Let us note already, however, thatsuch a non-strict principle arguably solves theExcessive Demands problems. To deal withClutter Avoidance, the principle can be modified along the lines of(Wo+b*), to yield the analogous (Wr+b*). However, as we noted, principles ofthis type fail to meet theStrictness Test. But perhaps thelesson to be learned just is that the normative force of logic isnon-strict. This bitter pill would be considerably sweetenedifreason-based principles really did offer a way around thePreface Paradox.