Abstract
Some authors have called into question the normativity of logic, using as an argument that the bridge principles for logical normativity (MacFarlane, In what sense (in any) is logic normative for thought, 2004)? are just by-products of general epistemic principles for belief. In this paper, I discuss that suggestion from a formal point of view. I show that some important bridge principles can be derived from usual norms for belief. I also describe some possible ways to block this derivation by modifying the epistemic norms or weakening the bridge principles. Finally, I discuss different philosophical interpretations of these results.
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Notes
In a recent paper, Labukt (2019, section 2) raises a similar point, and he claims that the normativity of logic (as expressed in bridge principles) is not ‘distinctive’.
For Harman, these were two objections against the principle we will call NS (he called it ‘The Logical Implication Principle’). MacFarlane framed these objections as desiderata.
Alternatively, X can be a set of propositions. In general, this does not make a big difference. However, in this context using sentences is more cautious, for we will describe cases where an agent believes \(\phi \) but does not believe a logically equivalent sentence \(\psi \). Using propositions would make these scenarios harder to represent.
Harman (1986, p. 11): ‘The Logical Implication Principle entails that, if one believes both P and if P then Q, that can be a reason to believe Q. But, clearly, that is not always a reason to believe Q, since sometimes when one believes P and also believes if P then Q, one should not come to believe Q’.
Harman (1986, p. 14): ‘One cannot be expected even implicitly to believe a logical consequence of one’s beliefs if a complex proof would be needed to see the implication’.
MacFarlane also discusses the Preface Paradox, which I assume the readers know. Yet, he finally adopts the strict principle \(Wo-\), according to which you are not permitted to have a Preface-like belief set. He takes this to be a case of ‘conflicting obligations’ (p. 14), where failing to comply the bridge principle does not always merit criticism. Unlike MacFarlane, Steinberger (2019a, p. 17) claims that the ought-principles are affected by the Preface Paradox, and adopts a reason-principle.
Field provides this probabilistic bridge principle, where P is a credence function (2009, p. 262): ‘Employing a logic L involves it being one’s practice that when simple inferences \(A_{1},\ldots ,A_{n} \vdash B\) licensed by the logic are brought to one’s attention, one will normally impose the constraint that P(B) is to be at least \(P(A_{1})+\cdots +P(A_{n})-(n-1)\)’.
See Horty (2001) for more on this topic.
Another strategy to avoid this result is to adopt a non-normal deontic logic, but it is unclear how it would work, for the commitment to logical omniscience comes with standard epistemic logic. See footnote 23 for more details about this.
All the proofs can be made in neighborhood semantics too. Now there are no abnormal worlds in the model, but the abnormality of the operators comes from a different kind of accessibility relation, where worlds can see many sets of worlds. The proofs become a bit more complicated and the models will have a more complex structure for the epistemic accessibility relation. Formally speaking, the only clear benefit is that we regain the closure of the epistemic operator under classical equivalence. But I don’t think this property is particularly important in this context. Similarly, the proofs can be made in a non-classical approch to non-normality. For example, the inconsistent abnormal worlds might be characterized by LP and the incomplete abnormal worlds might be characterized by K3. However, I believe that introducing those non-classical logics and their three-valued matrixes in this context is unnecessary and it could be distracting.
In this definition, worlds are taken as sets of formulas.
For simplicity, we do not include epistemic or deontic formulas in abnormal worlds, but for our results this does not matter: whatever epistemic or deontic formulas are true in abnormal worlds, it would not modify the results.
Seriality means that every world D-sees at least one world. In this context, D is serial over normal worlds: every normal world D-sees another normal world.
Both D and E might also be serial on abnormal worlds; for example, every abnormal world might D and E-see just itself. As a matter of fact, an abnormal world might D or E-see any set of worlds. In any case, the results in this paper would still follow. Since abnormal worlds are just epistemic representations, I don’t think that seriality is necessarily well-motivated, and this is why I don’t include it in the models.
Moorean anomalies can become stronger if standard principles such as K, D or 4 were adopted for the notion of belief. In that case one could obtain (by Reductio) a Fitchean paradox; i.e. a proof that every truth is believed in the actual world.
An anonymous referee observed that still, bridge principles might imply that (e.g.) if you believe that you believe p, then you ought to believe that you believe \(p \vee q\). I agree that there might be normative principles about iterations of the belief operator and logical implication. Indeed, in normal epistemic logic, these iterations are also closed under logical implication. However, that is not the point of bridge principles in our sense: they are not about what you should believe to believe. Iterated operators involve complex issues regarding epistemic transparency, which require independent treatment.
This principle will also be written in conditional form (assuming \(\rightarrow \) is the material conditional): \(O((B \phi _{1} \wedge \cdots \wedge \phi _{n}) \rightarrow B \psi )\). Given that we are using classical logic, and the operator O is normal, both forms are equivalent.
As a referee observed, the proof depends on \(w''\) being not closed under conjunction. However, it is worth-remarking that any other logical failure would allow a proof of the same fact with a similar structure. In other words, the proof depends on abnormality in general (i.e. on the fact that at least one logical principle fails), but not on a specific view about any connective in particular.
A defense of a falsity norm can be found in Raleigh (2013, p. 249), although he does not propose only \(F _{\rightarrow }\) but a stronger biconditional norm \(F_{\leftrightarrow }\). Observe that the dialetheists would oppose this norm, for they might believe some falsities (as far as these falsities are also truths).
The reader may observe that the proof depends on a problematic assumption of deontic logic, which is the introduction of the disjunction under the deontic operator. This rule is responsible for Ross’ paradox. It is worth remarking that some contemporary approaches to the failure of deontic introduction of the disjunction, such as Cariani (2013), claim that in order for \(O(a \vee b)\) to be true, then both a and b should be ‘above a benchmark’ of permissibility. This idea is not compatible with WS as a principle. For suppose \(\top \) is an obvious tautology. By reflexivity \(\top \vDash \top \), and this together with WS implies \(O(\lnot B\top \vee B\top )\). However \(\lnot B\top \) is clearly not above a benchmark of permissibility. In other words, this problematic reading of the disjunction is pervasive in the discussion on bridge principles. Adopting another kind of disjunction would require developing entirely different bridge principles.
The argument can be reconstructed like this: If reason forbids p, then if you believe p and not believe q you are acting against reason (for you believe p); while if reason requires p, it also requires q (because of evidence transmission under consequence), so believing p and not believing q would go against reason (for you do not believe q). Since reason either forbids p or requires p (Epistemic Strictness), believing p and not believing q goes against reason.
In the literature about bridge principles, another common objetion is Clutter Avoidance: unrestricted principles require you to believe many irrelevant sentences which are implied by your beliefs. However, there is no need to discuss this objection here, for the solutions to Excessive Demands that I describe in the paper (for example, adopting a negative polarity, or replacing ought by having a reason) can also solve the problem of Clutter Avoidance.
My reconstruction involves a simplification: MacFarlane (2004) takes disbelief as a primitive notion, but I define disbelief as \(B \lnot \). In MacFarlane’s terms (p. 9), the principle I offer is not exactly \(Wo-\) but \(Wo\lnot \). Of course, we can modify the framework and introduce a disbelief operator. This could be more appropiate from dialetheists, who claim that you can believe a sentence and its negation. But it would make the syntax and the semantics unnecessarily complex (we would need another modal operator for disbelief with a specific characterization). Some proofs should be modified too. See the next footnote.
An anonymous referee suggested to replace \(F_{ \rightarrow }\) by a disbelief principle (D): ‘If \(\phi \) is true, then you ought not disbelieve \(\phi \)’. Formally, \(\phi \rightarrow O\lnot D \phi \). Clearly, D together with \(F_{\rightarrow }\) will imply the original \(Wo-\) of MacFarlane: ‘If \(\phi _{1},\ldots ,\phi _{n}\) imply \(\psi \), then \(O(\lnot B \phi _{1}, \cdots \vee \lnot B\phi _{n} \vee \lnot D \psi \))’, following Fact 2. However, D alone will not imply our \(Wo-\) or the original version with disbelief, at least not with the methods we provided before (a simple case: D alone cannot prove that you ought not to believe a contradiction). Additional principles relating belief and disbelief will be needed.
This is connected to the deontic/evaluative distinction. From a purely evaluative point of view, maybe one should know the name of every Russian citizen. But as I clarified in Sect. 3.1., I am not assuming any specific reading of the deontic operator.
The fact that WS cannot be derived from \(T_{ \leftarrow }\) is just a collorary of Fact 1. The inclusion of the negative counterpart \(F_{ \leftarrow }\) makes this result stronger and more interesting. Observe that \(F_{\leftarrow }\) would be rejected by paracomplete logicians, for they claim that some sentences should not be believed because they are indeterminate (not false).
A very similar proof would show that \(T _{\leftarrow }\) and \(F _{\leftarrow }\) also do not imply \(Wo-\). We would have to modify the abnormal world; for example, instead of lacking \(p \vee q\), it could have \(\lnot (p \vee q)\).
A reader might be interested in narrow-scoped epistemic norms, such as \(B \phi \rightarrow OB \phi \) and \(\lnot B \phi \rightarrow O \lnot B \phi \). There norms are problematic: they are affected by the Bootstrapping objection (Broome 1999), since they imply that you ought to believe everything that you actually believe. From a formal point of view, it is easy to show that these norms do not imply NS. Suppose that we have a model with one normal world w, and wDw, but where the beliefs in w are logically incomplete (say, the agent believes p but does not believe \(p \vee q\)). Given the structure of the model, both epistemic norms are satisfied (indeed, if \(\phi \) then \(O\phi \)). However, NS fails: Bp holds but \(OB(p \vee q)\) does not.
Unlike ought, here is no standard modal logic for reasons. There are more sophisticated approaches to reasons, mostly using weights, defaults or scales (Horty 2012), but this would introduce some unnecessary complications in the formalism, which are not particularly illuminating for this specific case (which is rather abstract and does not involve deontic conflicts). Taking reasons as some kind of modal accessibility is a simplified version of these approaches (in particular, Nair 2016): reasons do not satisfy AND, they satisfy one-premise validity, and they are defeasible. On the other hand, for the positive results to hold, we only need the reason operator to be closed under one-premise validity. And we take this assumption as plausible (for a similar discussion, see Rosa (2019, fn. 19)). The introduction of the disjunction under the reasons operator might be controversial (as it is with the ought operator), but as I mentioned in footnote 23, this worry also applies to the wide-scoped bridge principles.
See Broome (2014) for a developed treatment of reasons and norms on this approach.
The reader can find a defense of similar epistemic principles in Whiting (2010, p. 216). Whiting proposes a modification of \(T \leftrightarrow \) were ought is replaced by epistemic permissibility.
In a recent paper, Rosa (2019) proposed a bridge principle for agnosticism: he claims that if \(\phi \) implies \(\psi \), then you have a reason against believing \(\phi \) and suspending judgment regarding \(\psi \); formally, \(R(B \phi \rightarrow \lnot S \psi )\). This bridge principle cannot be derived from \(T_\rightarrow \) and \(F_\rightarrow \) (for they do not involve the concept of suspension of judgment); however, it can be derived from \(F_\rightarrow \) together with \(T_{S}\) (i.e. if \(\phi \) is true, then \(R \lnot S \phi \)), mirroring the proof of Fact 2.
References
Boghossian, P. (2003). The normativity of content. Philosophical Issues, 13, 31–45.
Broome, J. (1999). Normative requirements. Ratio, 12, 398–419.
Broome, J. (2014). Rationality through reasoning. Oxford: Oxford University Press.
Bykvist, K., & Hattiangadi, A. (2007). Does thought imply ought? Analysis, 67(4), 277–285.
Cariani, F. (2013). Ought and resolution semantics. Nous, 47(3), 534–558.
Engel, P. (2004). Truth and the aim of belief. In D. Gillies (Ed.), Laws and models in science (pp. 77–97). London: King’s College Publications.
Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995). Reasoning about knowledge. Cambridge, MA: MIT Press.
Ferrari, F., & Moruzzi, S. (2020). Logical pluralism, indeterminacy, and the normativity of logic. Inquiry, 63(3–4), 323–346.
Field, H. (2009). What is the normative role of logic. Proceedings of the Aristotelian Society, Supplementary Volumes, 83, 251–268.
Field, H. (2015). What is logical validity? In Colin R. Caret & Ole T. Hjortland (Eds.), Foundations of logical consequence (pp. 33–70). Oxford: Oxford University Press.
Greenberg, A. (2020). Should I believe all the truths? Synthese, 197, 3279–3303.
Harman, G. (1986). Change in View. Cambridge, MA: MIT Press.
Horty, J. (2001). Agency and deontic logic. Oxford: Oxford University Press.
Horty, J. (2012). Reasons as defaults. Oxford: Oxford University Press.
Kolodny, N. (2007). How does coherence matter? Proceedings of the Aristotelian Society, 107, 229–63.
Kolodny, N. (2008). Why be disposed to be coherent? Ethics, 118, 437–63.
Labukt, I. (2019). Is logic distinctively normative? Erkenntnis. https://doi.org/10.1007/s10670-019-00142-1
MacFarlane, J. (2004). In what sense (if any) is logic normative for thought? Paper presented at the Central Division APA symposium on the normativity of logic.
Nair, S. (2016). Conflicting reasons, unconflicting ‘ought’s. Philosophical Studies, 173(3), 629–663.
Raleigh, W. (2013). Belief norms and blindspots. The Southern Journal of Philosophy, 51(2), 243–269.
Read, S. (2006). Monism: The one true logic. In D. Devidi & T. Kenyon (Eds.), A logical approach to philosophy (pp. 193–209). Berlin: Springer.
Rosa, L. (2019). Logical principles of agnosticism. Erkenntnis, 84, 1263–1283.
Russell, G. (2020). Logic isn’t normative. Inquiry, 63(3–4), 371–388.
Schroeder, M. (2009). Means-end coherence, stringency, and subjective reasons. Philosophical Studies, 143(2), 223–248.
Southwood, N. (2008). Vindicating the normativity of rationality. Ethics, 119(1), 9–30.
Stei, E. (2020). Non-normative logical pluralism and the revenge of the normativity objection. The Philosophical Quarterly, 70(278), 162–177.
Steinberger, F. (2019a). Consequence and normative guidance. Philosophy and Phenomenological Research, 98(2), 306–328.
Steinberger, F. (2019b). Three ways in which logic can be normative. Journal of Philosophy, 116(1), 5–31.
Streumer, B. (2007). Reasons and entailment. Erkenntnis, 66(3), 353–374.
Wedgwood, R. (2013). The right thing to believe. In T. Chan (Ed.), The aim of belief (pp. 123–139). Oxford: Oxford University Press.
Whiting, D. (2010). Should I believe the truth? Dialectica, 64(2), 213–224.
Acknowledgements
I would like to thank Eleonora Cresto, Lucas Rosenblatt, Ulf Hlobil and two anonymous referees of this journal for their valuable comments on earlier drafts of this paper. This paper was also greatly benefited from the discussion at the conferences Formal Methods in Philosophy (Munich, 2018) and VI Workshop on Philosophical Logic (Buenos Aires, 2018).
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Tajer, D. The Normative Autonomy of Logic. Erkenn 87, 2661–2684 (2022). https://doi.org/10.1007/s10670-020-00321-5
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DOI: https://doi.org/10.1007/s10670-020-00321-5