Stanford University

The Normative Status of Logic (Stanford Encyclopedia of Philosophy)

1. The normative status of what?

Before we can hope to make any headway with these questions a number
of clarifications are in order. First and foremost, in asking after
the normative status of logic, we had better get clearer on what we
mean by “logic”. For present purposes, I will take a logic
to be a specification of a relation of logical consequence on a set of
truth-bearers. Moreover, I will assume consequence relations to
necessarily preserve truth in virtue of logical form. For simplicity,
I will use “(models)” to denote such a consequence
relation. My default assumption will be to take the double turnstile
to denote the semantic consequence relation of the classical
first-order predicate calculus. But not much hangs on this. Partisans
of other types of non-classical consequence relations may read
“(models)” as referring to their preferred consequence
relation.

Presumably, if logic is normative for thinking or reasoning, its
normative force will stem, at least in part, from the fact that truth
bearers which act as the relata of our consequence relation and the
bearers of other logical properties are identical to (or at least are
very closely related in some other way) to the objects of thinking or
reasoning: the contents of one’s mental states or acts such as
the content of one’s beliefs or inferences, for example. For
present purposes I will assume the identity between truth-bearers and
the contents of our attitudes, and I will assume them to be
propositions.

1.1 Characterizing logical consequence in terms of its normative role

One may approach the question of the normativity of logic by
taking the notions of logical consequence and of validity to be settled
and to then investigate how these (and perhaps related) notions
constrain our attitudes towards the propositions standing in various
logical relations to one
another.[1]
An alternative approach has it that logic’s normative role in
thinking or reasoning may be partly definitive of what logic is.
Hartry Field, for one, advances an account of validity along the
latter lines. In his 2015, he argues that neither the standard model-
or proof-theoretic accounts of validity nor the notion of necessary
truth-preservation in virtue of logical form succeed in capturing the notion of validity. More
specifically, neither of these approaches is capable of capturing the
notion of validity in a way that does justice to logical disputes,
i.e., debates over which logical system is the correct one. On the
standard approaches to validity, such disputes are reduced to merely
verbal disputes inasmuch as “validity” is defined relative
to the system of logic in question. No one, of course, ever disputed
that a given classical argument form is valid relative to the notion
of validity-in-classical logic, where as an intuitionistically valid
argument is valid relative to the notion of validity-in-intuitionistic
logic. The problem is that there is no neutral notion of validity one
could appeal to that would enable one to make sense of logical
disputes as genuine debates, which, arguably, they are. What is needed
to capture the substantive nature of these disputes, therefore, is a
workable non-partisan notion of validity, one that is not internal to
any particular system of logic. The key to availing ourselves of such
an ecumenical notion of validity, Field claims, resides in its
conceptual role. The conceptual role of the notion of
validity, in turn, is identified with the way in which a
valid argument normatively constrains an agent’s doxastic
attitudes. Roughly, in the case of full belief, in accepting an
argument as valid, the agent is bound by the norm that she ought not
believe the argument’s premises while at the same time not
believing its conclusion. In other words, validity’s conceptual
role resides (at least in part) in the normative role played by valid
arguments in reasoning. Note that Field is not proposing
to define validity in terms of its normative role. The notion
of validity, Field contends, is best taken to be primitive. But even
once we take it to be primitive, it still stands in need of
clarification. It is this clarificatory work that is done by a
characterization of validity’s conceptual role. And it is in
this sense that the normative role of logic is supposed to
characterize the very nature of validity (understood as a notion
shared by various distinct logics).

In a similar vein, John MacFarlane (2004, Other Internet Resources;
henceforth cited as MF2004) contends that a fuller understanding of
how logical consequence normatively constrains reasoning may help us
settle long-standing issues in the philosophy of logic, debates surrounding the
very nature of validity. Attempts at resolving such questions have
been thwarted because of their suspect methodology: they relied on the
unreliable (because theory-laden) testimony of our intuitions about
validity. Appealing to the normative role of logic, MacFarlane hopes,
would give us a new angle of attack and hence a potentially better
handle on these vexed questions. MacFarlane, too, may therefore be
read as suggesting that a proper account of logic’s normative
role in reasoning will ultimately enable us to hone in on the correct
conception of logical consequence. As examples MacFarlane considers
the dispute between advocates of relevantist restrictions of the
notion validity and those who reject such restrictions (see entry
relevance logic),
and the question of the formality of logical validity (see entry
logical consequence). The
hope, in other words, is that an account of the normative role of
logic, will help us pin down the correct concept of validity. In this
respect, then, MacFarlane’s project may be thought to be more
ambitious than Field’s whose aim is to provide a logic-neutral
core concept of validity in terms of its normative role. For
MacFarlane a correct account of the normativity of logic would
constitute a potential avenue through which logical disputes may be
decided; for Field such an account merely renders such disputes
intelligible and so serves as a starting point for their
resolution.[2]

A potential problem with approaches like Field’s and
MacFarlane’s is that logical consequence does not appear to have
a unique normative profile that sets it apart from other, non-logical
consequence relations. For instance, that one ought not believe each
member of a set of premises while at the same time not believing (or
disbelieving) its conclusion, is a feature that logical consequence
seems to share with strict implication. At least in one sense of
“ought”, I ought to believe that this is colored, if I
believe it to be red, just as much as I ought to believe (A), if I
believe (A land B). If the general principles characterizing
logic’s normative role fail to discriminate logical
consequence among other types of consequence, we cannot identify the
conceptual role of validity with its normative role as Field
proposes. We cannot do so, at least, unless we impose further
conditions to demarcate properly logical consequence (see entry
logical constants).
The problem discussed here was raised (albeit in a different context)
by Harman (1986: 17–20) when he argues that logic is not
“specially relevant to reasoning”. One response, of
course, is simply to concede the point and so to simply broaden the
scope of the inquiry: Instead of asking how logic (narrowly construed)
normatively constrains us, we might ask how strict implication
(Streumer 2007) or perhaps a priori implication
does.[3]
A further response is that neither Field nor MacFarlane are committed
to demarcating logic or carving out any “special” role for
it. Their principles are left-to-right conditionals: the existence of
a logical entailment gives rise to a normative constraint on doxastic
attitudes. One can thus question the operative notion of entailment by
questioning the normative constraint. This, it might reasonably be
maintained, is all that Field and MacFarlane need for their
purposes.

1.2 Logical pluralism

We said that not much of our discussion below hinges on the choice of
one’s logic. However, while we countenanced the possibility of
disagreement over which logic is correct, we have simply presupposed
that there must be a unique correct logic. And this latter assumption
does seem to bear on our question in potentially significant ways. The
issue has yet to be explored more fully. Here I offer a number of preliminary distinctions
and observations.

Logical pluralists maintain that there is more than one correct logic
(see entry on
logical pluralism).
Now, there are perfectly uncontroversial senses in which several distinct
logical systems might be thought to be equally legitimate: different
logical formalisms might lend themselves more or less well to
different applications, e.g., classical propositional logic may be
used to model electric circuits, the Lambek calculus naturally models
phrase structure grammars, and so forth. If “correct” or
“legitimate” is merely understood as “having a
useful application”, monists should have no complaints about
such anodyne forms of “pluralism”. The monist may happily admit that there is a vast number of systems of logic that make for
worthy objects of study, many of which will have useful
applications. What monists must have in mind, then, is a more demanding sense of “correctness”.
According to Priest, the monist takes there to be, over and above
questions of local applicability, a core or “canonical”
(Priest 2006: 196) application of logic. The canonical use of logic
consists in determining “what follows from what—what
premises support what conclusion—and why” (idem). It is
only when the question is framed in these terms, that the full force
of the opposition between monists and pluralists can be appreciated.
The monist maintains that there is but one logic fit to play this core
role; the pluralist insists that several logics have an equally good
claim to playing it.

One consequence of pluralism, then, is that in a dispute between
advocates of different logics both of which lay claim to being the
correct logic in this sense—say, a dispute between a classical
logician and an advocate of intuitionistic logic—neither party
to the dispute needs to be at fault. Each logic may be equally
legitimate. For this to be possible, it must be the case that even the
canonical application of logic can be realized in multiple ways.
Pluralists differ in the accounts they offer of this multiple
realizability. One
influential such account has been advanced by J.C. Beall and Greg
Restall (2005). According to their account, several logics may be
equally qualified to fulfill the core function of logic, because “logical consequence” admits of several distinct interpretations (within a specified range). Roughly, (A) is a logical consequence of a set of formulas (Gamma) if and
only if, in every case in which all of the members of (Gamma) are
true, so is (A). Depending on how we understand “case”
in our definition—e.g., as Tarskian models (classical logic),
stages (intuitionistic logic), situations (relevant logic),
etc.—we arrive at different concepts of logical consequence.

What does this mean for the question of logic’s normative
status? It follows that it is only once we choose to disambiguate
“logical consequence” in a particular way—as classical or
intuitionistic, say—that the normative import of that particular
conception of consequence makes itself felt. After all, on the
pluralist picture a given conception of consequence cannot be
normatively binding in virtue of being (uniquely) correct, i.e., in
virtue of being descriptively adequate with respect to the entailment
facts, as it were. Hence, if I opt for an intuitionistic conception of
consequence and you go in for a classical one, I have no grounds for
criticizing your move from, say, (neg neg A) to (A), save
perhaps pragmatic ones. To be sure, such a move would be impermissible
according to my preferred notion of consequence, but it is perfectly
acceptable according to yours. According to the pluralist, then, there
exists no absolute sense, but only system-relative senses, in which a
logic can normatively bind us. Pluralism about logic thus seems to give rise
to pluralism about logical normativity: If there are several equally
legitimate consequence relations, there are also several equally legitimate sets of logical norms. Consequently, it is hard to see prima
facie
how substantive normative conflicts can arise. If the consequence
relations of classical and intuitionistic logic are equally
legitimate, there is little to disagree about when it comes to the
norms they induce. The classicist and the intuitionist simply have
opted to play by different rules.

This line of thought leads to a potential worry, however. For logical
norms do not merely bind us in the way that the rules of a game bind
us. I hold myself to be answerable to the rules governing a game
(of chess, say) so long as I wish to participate in it. However, the
normativity of logic does not seem to be optional in the same way. The
norms of logic are themselves responsible to our broader epistemic
aims (and would thus need to be coordinated with other epistemic
norms). Hence, if my epistemic aim is, say, acquiring true beliefs
(and avoiding false ones), this may give me a reason to prefer one set
of logical norms over another. For imagine I could choose either of
two logics (L_{1}) and (L_{2}). Suppose, moreover, that (A) is
true and that (A models_{L_{1}} B) but (A not models_{L_{2}}
B), for some relevant proposition (B). Even according to Beall and
Restall, not all logics are equal. To pass muster, a logic must
satisfy certain core conditions. In particular, it must be
truth-preserving. Assuming that both (L_{1}) and (L_{2}) are
truth-preserving, it follows that (B) is true. But then it would
seem that there is a clear sense in which (L_{1}) outperforms
(L_{2}) in terms of the guidance it affords us. According to the
norms (L_{2}) gives rise to, there is presumably nothing amiss about
an agent who believes (A) but not (B); according to (L_{1})
there presumably is. Hence, (L_{1}) is more conducive to our
epistemic aims. It follows that whenever two putatively equally
correct logics, (L_{1}) and (L_{2}) are such that (L_{2}) is a
proper sub-system of (L_{1}), one would appear to have reason to opt
for (L_{1}) since it is more conducive to one’s aim of having
true beliefs. In cases where we are dealing with two logics (L_{1})
and (L_{2}) such that (models_{L_{1}}) and (models_{L_{2}})
are not sub-relations of one another, things may be more complicated.
All the same, even in such cases, it may well be that our overarching
epistemic aims and norms give us reason to prefer one logic over
another, and hence that from the standpoint of these epistemic aims,
the logics are not equally good after all. The aim of these
considerations is not to undermine forms of logical pluralism like the
one advanced by Beall and Restall, but merely to point out that once
we take the normative dimension of logic into account, we must also
reckon with the broader epistemic goals, which the norms of logic
may be thought to be subservient to.

Field (2009b) argues for a different form of logical pluralism, one
which leaves more room for normative conflict. Logical pluralism is
not, for Field, the result of ambiguity in our notion of logical
consequence. Rather, it has its source in non-factualism of epistemic
norms. His non-factualism, in turn, is fueled partly by general
concerns, partly by the nature of how we choose such norms. Among the
general concerns Field (2009c) mentions are Hume-style worries about
the impossibility of integrating irreducible normative facts into a
naturalistic world view, Benacerraf-style worries about our ability to
gain epistemic access to such facts, and Mackie-style worries about
the “queerness” of such facts (i.e., that they not only
appear to have no room within our scientific picture of the world, but
that, furthermore, they are supposed to have a somewhat mysterious
motivational pull to them). The latter issue of norm selection amounts
to this. Given a set of epistemic goals, we evaluate candidate norms
as better or worse depending how well they promote those goals.
According to Field, we have no reason to assume that there should be a
fact of the matter as to which choice of logic is the uniquely correct
one; there will typically not be a unique system that best optimizes
our (often competing) constraints.

That being said, it seems that we can sensibly engage in rational
debates over which logic to adopt in the light of various issues
(vagueness, the semantic paradoxes, etc.). Consequently, there is a
clear sense in which normative conflicts do arise. Now, since Field
takes it to be an essential component of the notion of logical
consequence that it should induce norms (Field 2009a,b, 2015), we
choose a logic by finding out which logical norms it makes most sense
for us to adopt. But because Field does not take there to be a fact of
the matter as to which set of norms is correct and since the question
as to which of the norms best promotes our epistemic ends is often
underdetermined, we may expect there to be several candidate sets of
logical norms all of which are equally well-motivated. We are thus
left with a (more modest) form of logical pluralism on our hands.

What both of these types of pluralism have in common from the
point of view of the question of the normativity of logic, though, is
their rejection of the view that logical norms might impose themselves
upon us simply as a result of the correctness of the corresponding
logical principles. As such, pluralist views stand diametrically
opposed to realist forms of monism such as the one championed by Gila
Sher (2011). According to Sher, logical principles are grounded,
ultimately, in “formal laws” and so in reality. It is
these formal laws that ultimately also ground the corresponding
logical
norms.[4]

2. Normative for what?

Next let us ask what it is that logic is normative for, if indeed it
is normative. The paradigmatic objects of normative appraisal are
actions, behaviors or practices. What, then, is the activity or
practice that logical norms apply to?

2.1 Logic as normative for reasoning

One response—perhaps the most common one—is that logic
sets forth norms for (theoretical) reasoning. Unlike
thinking, which might consist merely of disconnected sequences of
conceptual activity, reasoning is presumably a connected, usually
goal-directed, process by which we form, reinstate or revise doxastic
attitudes (and perhaps other types of states) through inference.
Consider the following two examples of how logic might give rise to
norms. First, suppose I am trying to find Ann and that I can be sure
that Ann is either in the museum or at the concert. I am now reliably
informed that she is not in the museum. Using logic, I conclude that
Ann is at the concert. Thus, by inferring in conformity with the valid
(by the standards of classical logic) logical principle of disjunctive
syllogism, I have arrived at a true belief about Ann’s
whereabouts. Second, if I believe that Ann is either at the concert or
the museum, while at the same time disbelieving both of the disjuncts,
it would seem that there is a tension in my belief set, which I have
reason to rectify by revising my beliefs appropriately. Logic may thus
be thought to normatively constrain the ways we form and revise
doxastic attitudes. And it does so, presumably, in our everyday
cognitive lives (as in our example), as well as in the context of more
self-conscious forms of theoretical inquiry, as in mathematics, the
sciences, law, philosophy and so on, where its normative grip on us
would seem to be even
tighter.[5]

2.2 Logic as constitutively normative for thought

Other philosophers have taken the normativity of logic to kick in at
an even more fundamental level. According to them, the normative force
of logic does not merely constrain reasoning, it applies to all
thinking. The thesis deserves our attention both because of its
historical interest—it has been attributed in various ways to
Kant, Frege and
Carnap[6]—and
because of its connections to contemporary views in epistemology and
the philosophy of mind (see Cherniak 1986: §2.5; Goldman 1986:
Ch. 13; Milne 2009; as well as the references below).

To get a better handle on the thesis in question, let us agree to
understand “thought” broadly as conceptual
activity.[7]
Judging, believing, inferring, for example, are all instances of
thinking in this sense. It may seem puzzling at first how logic is to
get a normative grip on thinking: Why merely by engaging in
conceptual activity should one automatically be answerable to the
strictures of
logic?[8]
After all, at least on the picture of thought we are currently
considering, any disconnected stream-of-consciousness of imaginings
qualifies as thinking. One answer is that logic is thought to put
forth norms that are constitutive for thinking. That is, in order for
a mental episode to count as an episode of thinking at all, it must,
in a sense to be made precise, be “assessable in light of the
laws of logic” (MacFarlane 2002: 37). Underlying this thesis is
a distinction between two types of rules or norms: constitutive ones
and regulative ones.

The distinction between regulative and constitutive norms is Kantian
at root (KRV A179/B222). Here, however, I refer primarily to a related
distinction due to John Searle. According to Searle, regulative norms
“regulate antecedently or independently existing forms of
behavior”, such as rules of etiquette or traffic laws.
Constitutive norms, by contrast

create or define new forms of behavior. The rules of football or
chess, for example, do not merely regulate playing football or chess
but as it were they create the very possibility of playing such games.
(Searle 1969: 33–34; see also Searle 2010: 97)

Take the case of traffic
rules.[9]
While I ought to abide by traffic rules in normal circumstances, I
can choose to ignore them. Of course, rowdy driving in violation of
the traffic code might well get me in trouble. Yet no matter how
cavalier my attitude towards traffic laws is, my activity still counts
as driving. Contrast this with the rules governing the game of chess.
I cannot in the same way opt out of conforming to the rules of chess
while continuing to count as playing chess; in systematically
violating the rules of chess and persisting in doing so even in the
face of criticism, I forfeit my right to count as partaking in the
activity of playing chess. Unless one’s moves are appropriately
assessable in light of the rules of chess, one’s activity
does not qualify as playing chess.

According to the constitutive conception of logic’s normativity
the principles of logic are to thought what the rules of chess are to
the game of chess: I cannot persistently fail to acknowledge that the
laws of logic set standards of correctness for my thinking without
thereby jeopardizing my status as a thinker (i.e., someone presently
engaged in the act of thinking).

Two important clarifications are in order. For one, on its most
plausible reading, the thesis of the constitutive normativity of logic
for thought must be understood so as to leave room for the possibility
of logical error: an agent’s mental activity may continue to
count as thinking, despite his committing logical
blunders.[10]

That is, although one may at times (perhaps even frequently and
systematically) stray from the path prescribed by logic in one’s
thinking, one nevertheless counts as a thinker provided one
appropriately acknowledges logic’s normative authority over
one’s thinking. Consider again the game of chess. In violating
the rules of chess, deliberately or out of ignorance, I can plausibly
still be said to count as playing chess, so long, at least, as I
acknowledge that my activity is answerable to the rules; for example,
by being disposed to correct myself when an illegal move is brought to
my
attention.[11]
Similarly, all that is necessary to count as a thinker is to be
sensitive to the fact that my practice of judging, inferring,
believing, etc., is normatively constrained by the laws of logic. It
is not easy to specify, in any detail, what the requisite
acknowledgment or sensitivity consists in. A reasonable starting
point, however, is provided by William Taschek who, in his
interpretation of Frege, proposes that acknowledging

the categorical authority of logic will involve one’s possessing
a capacity to recognize—when being sincere and reflective, and
possibly with appropriate prompting—logical mistakes both in
one’s own judgmental and inferential practice and that of others.
(Taschek 2008: 384)

A second point of clarification is that the agent need not be able to
explicitly represent to herself the logical norms by which she is
bound. For instance, it may be that my reasoning ought to conform to
disjunctive syllogism in appropriate ways. I may be able to display
the right kind of sensitivity to the principle by which I am bound
(with the right prompting if need be), without my having to possess
the conceptual resources to entertain the metalogical proposition that
(neg A, Alor B models B). Nor must I otherwise explicitly
represent that proposition and the normative constraint to which it
gives rise.

With these clarifications in place, let us turn to a central
presupposition of the approach I have been sketching. What is being
presupposed, of course, is a conception of thinking that does not
reduce to brute psychological or neurophysiological processes or
events. If this naturalistic level of description were the only one
available, the constitutive account of the normativity of logic would
be a non-starter. What is being presupposed, therefore, is the
permissibility of irreducibly normative levels of descriptions of our
mental lives. In particular, it is assumed that the boundary between
the kinds of mental activity that constitute thinking and other kinds
of mental activity (non-conceptual activity like being in pain, for
instance) is a boundary best characterizable in normative terms. This
is not to deny that much can be learned about mental phenomena through
descriptions that operate at different, non-normative levels—the
“symbolic” or the neurological level of description,
say—the claim is merely that if we are interested in demarcating
conceptual activity from other types of mental phenomena, we should
look to the constitutive norms governing it. Davidson (1980, 1984),
Dennett (1987), and Millar (2004) all hold views according to which
having concepts and hence thinking requires that the agent be
interpretable as at least minimally sensitive to logical norms. Also,
certain contemporary “normativist approaches” according to which
accounts of certain intentional states involve ineliminable appeals to
normative concepts may advocate the constitutive conception of logic’s normativity
(e.g., Wedgwood 2007, 2009; Zangwill 2005).

2.3 Logic as normative for public practices

So far the answers to the question “What is logic normative
for?” we considered had in common that the “activities” in
question—reasoning and thinking—are internal, mental
processes of individual agents. But logic also seems to exert
normative force on the external manifestations of these
processes—for instance, it codifies the standards to which we
hold ourselves in our practices of assertion, rational dialogue and
the like. While much of the literature on the normativity of logic
focuses on internal processes of individuals, some authors have
instead emphasized logic’s role as a purveyor of public standards for normatively regulated practices.

Take the practice of asserting. Assertion is often said to “aim
at truth” (or knowledge, Williamson 2000: Ch. 11) as well as
being a “matter of putting forward propositions for others to
use as evidence in the furtherance of their epistemic projects”
(Milne 2009: 282). Since I take the asserted propositions to be true
and since truths entail further truths, I am “committed to
standing by” the logical consequences of my assertions or else
to retract them if I am unable to meet challenges to my assertion or
its consequences. Similarly, if the set of propositions I assert is
inconsistent at least one of my assertions must fall short of being
true and the set as a whole cannot be regarded as part of my evidence.
Plausibly, therefore, logic does have a normative role to play in
governing the practice of assertion.

Peter Milne takes an interest in assertion mainly in order to
“work back” from there to how logic constrains belief. He
concludes that logic exerts normative force at least on the stock of
beliefs that constitute the agent’s evidence (Milne 2009: 286).
Other authors explicitly prioritize the external dimension of
reasoning, conceived of as a social, inter-personal phenomenon.
According to them, it is reasoning in this external sense (as opposed
to intra-personal processes of belief revision, etc.) that is the
primary locus of logical normativity (MacKenzie 1989). The norms
govern our rational interactions with our peers. For instance, they
might be thought to codify the permissions and obligations governing
certain kinds of dialogues. Viewed from this perspective,
logic’s normative impact on the intra-personal activity of
reasoning is merely derivative, arrived at through a process of
interiorization. A view along these lines has been advanced by
Catarina Dutilh Novaes (2015). In a similar vein Sinan Dogramaci
(2012, 2015) has proposed a view he calls “epistemic
communism”. According to epistemic communism our use of
“rational” applied to certain deductive rules has a
specific functional role. Its role is to coordinate our epistemic
rules with a view to maximizing the efficiency of our communal
epistemic practices. On the basis of this view, he then elaborates an
argument for the pessimistic conclusion that no general theory of
rationality is to be had.

We will here follow the bulk of the literature in asking after the
normative role logic might play in reasoning understood as an
intra-personal activity. Yet, much of the discussion to follow applies
mutatis mutandis to the other approaches.

3. Harman’s challenge

Despite its venerable pedigree and its intuitive force, the thesis that
logic should have a normative role to play in reasoning has not gone
unchallenged. Gilbert Harman’s criticisms have been particularly
influential. Harman’s skeptical challenge is rooted in a
diagnosis: our deep-seated intuition that logic has a special
normative connection with reasoning is rooted in a confusion. We have
conflated two very different kinds of enterprises, viz. that of
formulating a theory of deductive logic, on the one hand, and what
Harman calls “a theory of reasoning” (Harman 2002) on the other. Begin
with the latter. A theory of reasoning is a normative account about
how ordinary agents should go about forming, revising and maintaining
their beliefs. Its aim is to formulate general guidelines as to which
mental actions (judgments and inferences) to perform in which
circumstances and which beliefs to adopt or to abandon (Harman 2009:
333). As such, the subject matter of a theory of reasoning are the
dynamic “psychological events or processes” that
constitute reasoning. In contrast, “the sort of implication and
argument studied in deductive logic have to do with [static,
non-psychological] relations among propositions” (idem). Consequently,

logical principles are not directly rules of belief revision.
They are not particularly about belief [or the other mental states and
acts that constitute reasoning] at all. (Harman 1984: 107)

Once we disabuse ourselves of this confusion, Harman maintains, it is
hard to see how the resulting gap between logic and reasoning can be
bridged. This is Harman’s challenge.

At least two lines of response come to mind. One reaction to
Harman’s skeptical challenge is to take issue with his way of
setting up the problem. In particular, we might reject his explanation of the provenance
of our intuitions to the effect that logic has a normative role to
play in reasoning as stemming from a mistaken identification of
deductive logic and theories of reasoning. It might be thought, for
instance, that Harman is led to exaggerate the gulf between deductive
logic and theories of reasoning as a result of a
contestable—because overly narrow—conception of either
logic or reasoning, or both. Advocates of broadly logical accounts of
belief revision (belief revision theories, non-monotonic logics,
dynamic doxastic logic, etc.) may feel that Harman is driven to his
skepticism out of a failure to consider more sophisticated logical
tools. Unlike standard first-order classical logic, some of these
formalisms do make explicit mention of beliefs (and possibly other
mental states). Some formalisms do seek to capture the dynamic
character of reasoning in which beliefs are not merely accumulated but
may also be revised. Harman’s response, it would seem (Harman
1986: 6), is that such formalisms either tacitly rely on mistaken
assumptions about the normative role of logic or else fall short of
their objectives in other ways. But even if one disagrees with
Harman’s assessment, one can still agree that such formal
models of belief revision do not obviate the need for a philosophical
account of the normativity of logic. That is because such models do typically tacitly rely on assumptions concerning the normative role of logic. An account of the normativity of logic would thus afford us a fuller
understanding of the presuppositions that undergird such theories.

On the other hand, some philosophers—externalists of various
stripes, for instance—may find fault with the epistemological
presuppositions underlying Harman’s conception of a theory of
reasoning. Harman views the aim of epistemology as closely linked to
his project of providing a theory of reasoning. According to
Harman’s “general conservatism”, central
epistemological notions, like that of justification are approached
from the first-personal standpoint: “general conservatism is a
methodological principle, offering methodological advice of a sort a
person can take” (Harman 2010: 154). As such Harman’s approach contrasts with much of
contemporary epistemology which, unconcerned with direct epistemic
advice, is mainly in the business of seeking to lay down explanatorily
illuminating necessary and sufficient conditions for epistemic
justification.[12]
Summarizing the first line of response, then, Harman’s skepticism is partly premised on particular conceptions of logic and of epistemological methodology both of which may be called into question.

The second line of response is to (largely) accept Harman’s
assumptions regarding the natures of deductive logic and of epistemology
but to try to meet his challenge by showing that there is a
interesting normative link between the two after all. In what follows, I focus
primarily on this second line of response.

Of course, saying that deductive logic and theories of reasoning are
distinct is one thing, affirming that there could not be an
interesting normative connection between them is quite another. As a
first stab at articulating such a connection, we might try the
following simple line of thought: theoretical reasoning aims to
provide an accurate representation of the world. We accurately
represent the world by having true (or perhaps knowledgeable) beliefs
and by avoiding false ones. But our doxastic states have
contents—propositions—and these contents stand in certain
logical relations to one another. Having an awareness of these logical
relations would appear to be conducive to the end of having true
beliefs and so is relevant to theoretical reasoning. In particular,
the logical notions of consequence and consistency seem to be relevant.
If I believe truly, the truth of my belief will carry over to its
logical consequences. Conversely, if my belief entails a falsehood it
cannot be true. Similarly, if the set of propositions I believe (in
general or in a particular domain) is inconsistent, they cannot
possibly afford an accurate representation of the world; at least one
of my beliefs must be false. Harman may be able to agree with all of this. His
skepticism pertains also (and perhaps primarily) to the question whether logic has a privileged role to play in reasoning; that the principles of logic are relevant to reasoning in a way that principles of other sciences are not (Harman 1986: 20). However, I want to set this further issue to one side for now.

Notice that this simple reflection on the connection between logic and
norms of reasoning leads us right back to the basic intuitions at the beginning of this entry: that there is something wrong with us when
we hold inconsistent beliefs or when we fail to endorse the logical
consequences of our beliefs (at least when we can be expected to be
aware of them). Let us spell these intuitions out by way of the following two principles. Let (S) be an agent and (P)
a
proposition.[13]

  • Logical implication principle (IMP): If (S)’s beliefs
    logically imply (A), then (S) ought to believe that
    (A).

  • Logical consistency principle (CON): (S) ought to avoid having
    logically inconsistent beliefs.

Notice that on the face of it
IMP
and
CON
are distinct.
IMP,
in and of itself, does not prohibit inconsistent or even
contradictory beliefs, all it requires is that my beliefs be closed
under logical consequence.
CON,
on the other hand, does not require that I believe the consequences
of the propositions I believe, it merely demands that the set of
propositions I believe be consistent. However, given certain
assumptions,
IMP
does entail
CON.
Against the background of classical logic, the entailment obtains
provided we allow the following two assumptions: (i) one cannot (and,
via the principle that “ought” implies “can”,
ought not) both believe and disbelieve one and the same
proposition simultaneously; and (ii) that disbelieving a proposition is
tantamount to believing its
negation.[14]
For let ({ A_{1}, dots, A_{n}}) be (S)’s inconsistent
belief set. By classical logic, we have (A_{1}, dots, A_{n-1}models
neg A_{n}). Since (S)’s beliefs are closed under logical
consequence, (S) believes (neg A_{n}) and hence, by (ii), disbelieves (A_{n}). So, (S) both believes and
disbelieves (A_{n}).

3.1 The objections

IMP
and
CON
are thus a first—if rather flatfooted—attempt at pinning
down the elusive normative connection between logic and norms of
reasoning. Harman considers these responses and responds in turn. The
following four objections against our provisional principles can, in
large part, be found in the writings of Harman.

(1) Suppose I believe (p) and (p supset q) (as well as Modus
Ponens). The mere fact that I have these beliefs and that I recognize
them to jointly entail (q) does not normatively compel any
particular attitude towards (q) on my part. In particular, it is not
the case in general that I ought to come to believe (q) as
IMP
would have it. After all, (q) may be at odds with my evidence in
which case it may be unreasonable for me to slavishly follow Modus
Ponens and to form a belief in (q). The rational course of
“action”, rather, when (q) is untenable, is for me to
relinquish my belief in at least one of my antecedent beliefs (p)
and (p supset q) on account of their unpalatable implications.
Thus, logical principles do not invariably offer reliable guidance in
deciding what to believe (at least, when the relation between logical
principles and our practices of belief-formation are understood along
the lines of
IMP).
Let us therefore call this the Objection from Belief Revision.

John Broome (2000: 85) offers a closely related objection, which
nevertheless deserves separate mention. Broome observes that any
proposition trivially entails itself. From
IMP
it thus follows that I ought to believe any proposition I in
fact believe. But this seems patently false: I might hold any number
of irresponsibly acquired beliefs. The fact that, by mere
happenstance, I hold these beliefs, in no way implies that I ought to
believe them. Call this variation of the Objection from Belief Revision, the
Bootstrapping Objection.

(2) A further worry is that a reasoner with limited cognitive
resources would be unreasonable to abide by
IMP
because she would be obligated to form countless utterly useless
beliefs. Any of the propositions I believe entails an
infinite number of propositions that are of no interest to me
whatsoever. Not only do I not care about, say, the disjunction
“I am wearing blue socks or pigs can fly” entailed by my
true belief that I am wearing blue socks, it would be positively
irrational of me to squander my scarce cognitive resources of time,
computational power and storage capacity in memory and so on, on idly deriving implications of
my beliefs when these are of no value to me. Harman fittingly dubs the
principle of reasoning in question Principle of Clutter
Avoidance
. Let us call the corresponding objection the
Objection from Clutter Avoidance.

(3) There is another sense in which both
principles—IMP
and
CON—place
excessive demands on agents whose resources are limited. Consider the
following example. Suppose I believe the axioms of Peano arithmetic.
Suppose further that a counterintuitive arithmetical proposition that
is of great interest to me is entailed by the axioms, but that its
shortest proof has more steps than there are protons in the visible
universe. According to
IMP,
I ought to believe the proposition in question. However, if the
logical “ought” implies “can” (relative to
capacities even remotely like our own),
IMP
cannot be correct. An analogous objection can be leveled at
CON.
An agent may harbor an inconsistent belief set, yet detecting the
inconsistency may be too difficult for any ordinary agent. We may summarize
these objections under the label Objection from Excessive Demands.

(4) Finally, I may find myself in epistemic circumstances in which
inconsistency is not merely excusable on account of my “finitary
predicament” (Cherniak 1986), but where inconsistency appears to
be rationally required. Arguably, the Preface Paradox constitutes such
a scenario (Makinson
1965).[15]
Here is one standard way of presenting it. Suppose I author a
meticulously researched non-fiction book. My book is composed of a
large set of non-trivial propositions (p_{1},dots, p_{n}). Seeing
that all of my claims are the product of scrupulous research, I have
every reason firmly to believe each of the (p_{i}) individually. But
I also have overwhelming inductive evidence for (q): that at least
one of my beliefs is in error. The (p_{i}) and (q) cannot be
jointly true since (q) is equivalent to the negation of the
conjunction of the (p_{i}). Yet, it would seem irrational to abandon
any of my beliefs for the sake of regaining consistency, at least in
the absence of any new evidence. The Preface Paradox thus may be
thought to tell against
CON:
arguably, I may be within my rational rights in holding inconsistent
beliefs (at least in certain contexts). However, it also seems to
constitute a direct counterexample to
IMP.
For in the Preface scenario I believe each of the (p_{i}) and yet
it looks as if I ought to disbelieve an obvious logical consequence
thereof: their conjunction (because (q) is transparently equivalent
to (neg (p_{1} land dots land p_{n}))).

So much for the objections to
IMP
and
CON.
The question raised by these considerations is whether these principles can be improved upon.

4. Bridge principles

Let us focus on
IMP
for now. Harman’s objections establish that
IMP—in
its current form, at least—is untenable. The question is
whether
IMP
can be improved upon in a way that is invulnerable to Harman’s
objections. In other words, the question is whether a tenable version
of what MacFarlane (MF2004) calls a bridge principle is to be
had. A bridge principle, in this context, is a general principle that
articulates a substantive relation between “facts” about
logical consequence (or perhaps an agent’s attitudes towards
such facts) on the one hand, and norms governing the agent’s
doxastic attitudes vis-à-vis the propositions standing
in these logical relations on the other.
IMP
is a bridge principle, albeit not a promising one.

Harman’s skepticism about the normativity of logic can thus be
understood as skepticism as to whether a serviceable bridge principle
is to be had. In order properly to adjudicate whether Harman’s
skepticism is justified, we need to know what “the options
are”. But how? John MacFarlane (MF2004) offers a helpful taxonomy of bridge principles which constitutes a very good
first approximation of the range of options. This section briefly
summarizes MacFarlane’s classification, as well as subsequent
developments in the literature.

Let us begin with a general blue print for constructing bridge
principles:[16]

  • ((star))
    If (A_{1},dots, A_{n} models C, textrm{ then } N(alpha(A_{1}),
    dots, alpha(A_{n}), beta(C))).

A bridge principle thus takes the form of a material conditional. The
conditional’s antecedent states “facts” about
logical consequence (or attitudes toward such “facts”).
Its consequent contains a (broadly) normative claim concerning the
agent’s doxastic attitudes towards the relevant propositions.
Doxastic attitudes, as I use the term, include belief, disbelief, and
degree of
belief.[17]
Here (alpha) may (but need not) represent the same attitude as
(beta). In fact, for principles with negative polarity, it may
represent the negation of an attitude: “do not disbelieve the
conclusion, if you believe the premises”.

In what ways, now, can we vary this schema so as to generate the space
of possible bridge principles? MacFarlane introduces three parameters
along which the schema may be varied. Each parameter allows for
multiple “discrete settings”. We can think of the logical
space of bridge principles as the range of possible combinations among
these parameter settings.

  1. In order to express the normative claims, we will need deontic
    vocabulary. Bridge principles may differ in the deontic operator they
    deploy: does the normative constraint take the form of an
    ought (o), a permission (p) or merely of having (defeasible)
    reason (r)?

  2. What is the polarity of the normative claim? Is it a
    positive obligation/permission/reason to believe a
    certain proposition given one’s belief in a number of premises (+)? Or rather is it a
    negative obligation/prohibition/reason not to
    disbelieve
    (−)?

  3. What is the scope of the deontic operator? Different bridge principles
    result from varying the scope of the deontic operator. Let (O) stand
    generically for one of the above deontic operators. Given that the
    consequent of a bridge principle will typically itself take the form
    of a conditional, the operator can take

    • narrow scope with respect to the consequent (C): (A
      supset O(B))
    • wide scope (W): (O(A supset
      B))
    • or it can govern both the antecedent and the consequent of the
      conditional
      (B):[18](O(A) supset O(B))

These three parameters admit of a total of eighteen combinations of
their settings and hence eighteen bridge principles. The symbols in
parentheses associated with each parameter setting combine to
determine a unique label for each of the principles: The first letter
indicates the scope of the deontic operator (C, W or B), the second
letter indicates the type of deontic operator (o[bligation],
p[ermissions], r[easons]) and the “+” or
“−” indicate positive and negative polarity
respectively.[19]
For example, the label “Co+” corresponds to our original
principle
IMP:

  • If (A_{1}, A_{2}, dots, A_{n} models C), then if (S) believes
    (A_{1}, A_{2}, dots, A_{n}), (S) ought to believe
    (C).

And “Wr−” designates:

  • If (A_{1}, A_{2}, dots, A_{n} models C), then (S) has reason to
    (believe (A_{1}, A_{2}, dots, A_{n}), only if (S) does not
    disbelieve (C)).

Many will regard the bridge principles we have presented thus far to
be problematic. They all relate “facts” about logical
entailment—assuming there are such things—to certain
normative constraints on the agent’s attitudes. The trouble, they will say, is
that these principles are not sensitive to the cognitive limitations
of ordinary agents. Agents, if they are even remotely like us, are not
apprised of all entailment “facts”. Consequently,
especially the “ought”-based principles (at least on some
understanding of “ought”) are therefore vulnerable to
Harman’s
Objection from Excessive Demands.

A natural response is to consider attitudinal bridge principles. I
call attitudinal bridge principles whose antecedents are
restricted to logical implications to which the agent bears an
attitude. For instance, to take the type of attitudinal principle
considered by MacFarlane, Co+ may be transformed into:

  • (Co+k) If
    (S) knows that (A_{1},dots, A_{n} models C), then if (S)
    believes the (A_{i}), (S) ought to believe (C).

According to (Co+k), the agent’s belief set ought to be closed
only under known logical consequence. Let us call this an
attitudinally constrained or, more specifically, the
epistemically constrained variant of Co+ (whence the
“k” in the label). Different authors may go in for
different types of attitudes. Knowledge, of course, is a factive
attitude. Some will wish to leave room for the possibility of
(systematic) logical error. For instance, an agent might mistakenly
comply with the principle (Asupset B, B models A). Perhaps even
someone with erroneous logical convictions such as this should, for
the sake of internal coherence, comply with the principles he deems
correct. An agent who sincerely took an erroneous principle to be
correct but failed to reason in accordance with it may be seen to
manifest a greater degree of irrationality than someone who at least
conformed to principles he endorses. But we can also imagine more
interesting cases of systematic error. Suppose I am impressed with an
argument for a particular non-classical logic as a means of parrying
the semantic paradoxes. I thus come to espouse the logic in question
and begin to manage my doxastic attitudes accordingly. But now suppose
in addition that unbeknownst to me the arguments that persuaded me are
not in fact sound. Again, it might be thought that though I am
mistaken in my adherence to the logic, so long as I had good reasons
to espouse it, it may nevertheless be proper for me to comply with its
principles. If logical error in either of these two senses is to be
accommodated, the appropriate attitude would have to be
non-factive.

A further issue is that ordinary agents are presumably normatively
bound by logical principles without being able to articulate or
represent those principles to themselves explicitly. Assuming
otherwise runs the risk of overly intellectualizing our ability to
conform to logical norms. The attitudes borne by such logically
untrained agents to the logical principles therefore presumably are
not belief-like. Perhaps such agents are better thought of as
exercising an ability or having a disposition to take certain forms of
entailment to be correct. See Corine Besson 2012 for a criticism of
dispositionalist accounts of logical competence, and Murzi &
Steinberger 2013 for a partial defense.

Having thus outlined the classificatory scheme, a number of additional
comments are in order. Notice that disbelieving (A) is to be
distinguished from not believing (A). One cannot rationally believe
and disbelieve the same proposition (although see
note 12).
Hence, I ought to ensure that when I disbelieve (A), I do not
believe (A). The converse, however, obviously does not hold since I
can fail to believe (A) without actively disbelieving it. I may, for
instance, choose to suspend my judgment as to whether (A) pending
further evidence, or I may simply never have considered whether (A).
Furthermore, I will remain neutral on the question as to whether the
attitude of disbelieving (A) should be identified with that of
believing (neg A).

Moreover, a note on deontic modals is in order. “You ought not
(Phi)” ((Oneg Phi)) is not the same as saying “It
is not the case that you ought to (Phi)” ((neg O Phi)).
But rather “You are forbidden from (Phi)ing”.
Consequently, “You ought not disbelieve (A)” should be
read as “disbelieving (A) would be a mistake”, as
opposed to “it is not the case that you ought to disbelieve
(A)”, which is compatible with the permissibility of
disbelieving (A).

Ought and may are understood to be strict notions. By contrast,
reason is a pro tanto or contributory notion. Having
reason to (Phi) is compatible with simultaneously having reason not
to (Phi) and indeed with it being the case that I ought not to
(Phi). Reasons, unlike oughts, may be weighed against each
other; the side that wins out determines what ought to be done.
Finally, I am here treating all deontic modals as propositional
operators. This too is not uncontroversial. Peter Geach (1982) and
more recently Mark Schroeder (2011) have argued that so-called
deliberative or practical oughts are best analyzed not as
operators acting on propositions but rather as expressing relations
between agents and actions. (Interestingly, MacFarlane (2014: Ch. 11)
has recently followed suit.) Nevertheless, I will assume without
argument that the operator-reading can be made to work even in the case of deliberative oughts. For defenses
of this position see e.g., Broome 2000, 2013; Chrisman 2012; and
Wedgwood 2006. We can capture the particular connection between an
agent and the obligation she has towards a proposition at a particular
time, by indexing the operator: (O_{S, t}). I will drop the indices
in what follows.

A last comment: MacFarlane is not explicit as to whether bridge
principles are to be understood as synchronic
norms—norms that instruct us which patterns of doxastic
attitudes are, in a specified sense, obligatory, permissible or
reasonable at a given point in time; or whether they are to provide
diachronic norms—norms that instruct us how an
agent’s doxastic state should or may evolve over time. To
illustrate the distinction, let us consider Co+ (aka
IMP)
once again. Understood synchronically, the principle should be
spelled out as follows.

  • If (A_{1}, A_{2}, dots, A_{n} models C), then if, at time (t),
    (S) believes (A_{1}, A_{2}, dots, A_{n}), then (S)
    ought to believe (C) at time (t).

In other words, the principle demands that one’s beliefs be, at
all times, closed under logical consequence. Alternatively, on might
interpret Co+ as a diachronic norm as follows:

  • If (A_{1}, A_{2}, dots, A_{n} models C), then if, at time (t),
    (S) believes (A_{1}, A_{2}, dots, A_{n}), then (S)
    ought to believe (C) at time (t’) (where (t) precedes
    (t’) suitably closely).

Different principles lend themselves more or less well to these two
readings. (C)- and (B)-type principles can be interpreted as
either synchronic or diachronic principles on account of the fact that
they make explicit claims as to what an agent ought, may or has reason
to believe or disbelieve given her other beliefs. The (W)s, by
contrast, are most plausibly read as synchronic principles. Such
principles do not, in and of themselves, instruct the subject which
inferences to make. Rather, they tend to proscribe certain patterns of
belief (and, perhaps, disbelief) or distributions of degrees of belief.

4.1 Evaluating bridge principles

With the logical terrain of bridge principles charted, the question
now arises as to which principles (if any) are philosophically viable.
This is discussed in the following supplementary document:

Bridge Principles – Surveying the Options

In that supplement we discuss a variety of desiderata that
have been put forward and consider candidate principles
with respect to those desiderata.

4.2 The Preface Paradox

Given that the Preface Paradox constitutes a major stumbling block for
many otherwise plausible principles, we do well to explore the ways in
which the Preface Paradox might be dealt with. One way, of course, of
dealing with the Preface Paradox is to deny it its force. That is,
one might try to outright solve, or in some way dissolve, the
paradox. Since it seems fair to say that no such approach has won the
day (see entry
epistemic paradoxes),
I will assume that the Preface Paradox intuitions are to be take
seriously.[20]

Alternatively, one might acknowledge the force of the Preface
intuitions while at the same time trying to hold on to a strict,
ought-based principle. But how? According to all such principles, I,
the author of a non-trivial non-fiction book (let us assume), ought to believe (or at
least not disbelieve) the conjunction of the propositions in my book,
given that I firmly endorse each conjunct individually.
MacFarlane’s response is that we must simply reconcile ourselves
to the irreconcilable: the existence of an ineliminable normative conflict. Our strict
logical obligations clash with other epistemic obligations, namely,
the obligation to believe that some of my beliefs must be mistaken.
Our agent becomes a tragic heroine. Through no fault of her own, she
finds herself in a situation in which, no matter what she does, she
will fall short of what, epistemically speaking, she ought to do.

It might be retorted that, as a matter of sound methodology, admitting
an irresolvable normative clash should only be our last resort. A
better approach (all other things being equal) would consist in
finding a way of reconciling the conflicting epistemic norms.

Among the qualitative principles we have been considering, the only
way out is via non-strict principles like (Wr+b*), which we considered
at the end of the previous section. On this principle, I, the author,
merely have reason (as opposed to having sufficient reason) for
believing the conjunction of the claims that make up the body of my
book, given that I believe each of the claims individually. The
crucial difference resides in the fact that this leaves open the
possibility that my reason for being logically coherent can be
overridden. In particular, it can be outweighed by reasons stemming
from other epistemic norms. In the case at hand, it might be thought
that our logical obligations are superseded by a norm of epistemic
modesty. This, of course, is not uncontroversial. Some maintain that
what the Preface Paradox shows is not merely that the normative grip
of logic does not take the form of a strict ought, but rather
that we in fact have no reason at all to believe in
multi-premise closure of belief under logical consequence: my reasons
for believing in the conjunction of my claims are not being trumped by
weightier reasons for disbelieving it; I have no logic-based reason
whatsoever to believe the conjunction in the first place.

So far, then, we have considered the following reactions to the
Preface Paradox: reject the Preface Paradox altogether; follow
MacFarlane and cling to the strict ought-based principle at the cost
of accepting an irresolvable normative clash; or opt for the weaker
reason operator and give up the intuition motivating the
Strictness Test.
But none of these proposals incorporates what is perhaps the most
natural response to the Preface Paradox outside of the debate
surrounding the normativity of logic. A standard response to the
Preface Paradox consists in appealing to graded credal states in lieu
of “full” (“qualitative”, “binary”
or “all-or-nothing”) beliefs. Such “credences”
or “degrees of belief” (I will use the two labels
interchangeably) are typically modeled by means of a (possibly
partial) credence function (which we will denote by
“(cr)”) that maps the set of propositions into the unit
interval. Probabilists maintain that an ideally rational
agent’s credence function ought to be (or at least ought to be
extendable to) a probability function (i.e., it ought to satisfy the
standard axioms of probability theory). In other words, an ideally
rational agent should have probabilistically coherent credences.

Probabilists have no trouble accounting for the Preface phenomena: the
subjective probability of a (large) conjunction may well be
low—even zero, as in the case of the Lottery Paradox (see entry
epistemic paradoxes)—even
if the probability assigned to each of the individual conjuncts is
very high (reflecting the high degree of confidence the author rightly
has in each of her claims).

A tempting strategy for formulating a bridge principle capable of
coping with the Preface Paradox is to incorporate these insights. This
might be done by going beyond MacFarlane’s classification and
devising instead a quantitative bridge principle: one in
which logical principles directly constrain the agent’s degrees
of belief (as opposed to constraining her full beliefs).

Hartry Field (2009a,b, 2015} proposes a bridge principles of just this
form. Here is a formulation of such a principle:

  • (DB) If
    (A_{1}, dots, A_{n} models C), then (S)’s degrees of
    belief ought to be such that: (cr(C) geq sum_{1 leq i leq n}
    cr(A_{i}) – (n – 1))

Note first that
DB
is a wide scope principle: it requires that our degrees of belief
respect the specified inequality, which can be achieved in one of two
ways: by suitably raising one’s degree of belief in the
conclusion or else by readjusting one’s degrees of belief in the
premises.

DB
is based on a well-known result in probability logic, which is
usually stated in terms of “uncertainties” (see Adams 1998
for more details; for a helpful overview, see Hájek 2001).
Define the uncertainty of a proposition (A), (u(A)) as
(u(A) = 1-cr(A)). Put in this way,
DB
says that the uncertainty of the conclusion must not exceed the sum of the uncertainties of the premises.
DB
can be seen to share a number of important features with standard
probability theory. Plug in (0) for (n) and you get that one
should assign (1) to any logical truth. Plug in (1) and you get
that one’s degree of belief in the premise of a valid
single-premise argument should not exceed your degree of belief in the
conclusion. The idea underlying
DB
is that uncertainties can add up and therefore need to be accounted
for when we are trying to determine how the logical relations between
our belief contents should affect our degrees of belief in those
contents. Even if my uncertainty about each of a large number of
premises is next to negligible when taken individually, the
uncertainty may accumulate so as to make the conclusion highly
(perhaps even maximally) uncertain. It is for this reason that
DB
gets us around the Preface Paradox; in the Preface case the number of
premises is sufficiently high for the conclusion to admit of a very
low credence.

5. Further challenges

5.1 Kolodny’s challenge

Logical norms are naturally regarded as a species of rational
requirements. If I believe a set of propositions and at the same time
disbelieve an obvious logical consequence thereof my set of beliefs
presumably exhibits a rational defect. Rational requirements are
characterized by their demand for coherence: they demand either a
particular kind of coherence among our attitudes or else coherence
between our attitudes and the evidence. Niko Kolodny has dubbed the
former “requirements of formal coherence as such” (Kolodny
2007: 229). They are formal in the sense that they concern
logical relationships between attitude contents or the arithmetical
relationships between the degrees of confidence we invest in those
contents. The qualification “as such” indicates that an
internal coherence among the attitudes is demanded to the exclusion of
other epistemologically relevant factors (evidential considerations,
for example). Requirements of this type, it has been argued (Broome
2000; Dancy 1977), take the form of wide scope principles. Hence, they
do not generally prescribe a particular attitude, but are satisfiable
in a number of ways. Or, to put it another way, they prohibit
particular constellations of attitudes. For instance, Wo−
proscribes states like the one just imagined, in which the agent
believes all of the premises of a valid argument while disbelieving its conclusion. It may
be satisfied, as we have seen, by either coming to believe the
conclusion or by abandoning some of the premises.

The status of logical norms as a species of rational requirement
raises weighty questions. For one, Kolodny (2005) has challenged the
seemingly natural assumption that rationality is normative at all.
That is, he has questioned whether we in fact have reason to
do what rational requirements require of us. It might be that
rationality makes certain demands on us, but that it is an open
question as to whether we should want to be rational. Here is not the
place to develop these ideas, let alone to try to resolve the
“normative question” for rationality (see Way 2010 for an
overview). In the absence of a convincing response to Kolodny’s
challenge, some might take umbrage at our talk of logical
norms. Strictly speaking, we should speak of them as
necessary conditions for rationality, leaving open whether we have
reason to be rational.

While it would take us too far afield to address the question of the
normativity of rationality, there is a related strand of Kolodny’s
argument that is more directly relevant to our discussion. The claim
in question, put forth in Kolodny 2007 & 2008, is that there
simply is no reason for postulating the existence of formal coherence
requirements as such at all. This may seem surprising. After all, to
take Kolodny’s simplest example, we certainly do have the
intuition that an agent who, at a given time, believes both (p) and
(neg p) is violating a requirement—a requirement, presumably,
of something like the following form:

  • (NC) (S)
    is required not to both believe (A) and (neg A) at (t) (for any
    time (t)).

If Kolodny is right that there are no pure formal coherence
requirements like
(NC),
how are we to explain our intuitions? Kolodny’s strategy is to
devise an error theory, thereby seeking to show how coherence (or near enough
coherence) in the relevant sense emerges as a by-product of our
compliance with other norms, norms that are not themselves pure formal
coherence requirements, thus obviating the need for postulating pure
formal coherence requirements.

Consider how this plays out in the case of
(NC).
Kolodny proposes an evidentialist response. Any violation of
(NC)
is indeed a violation of a norm, but the relevant norm being violated
is a (narrow scope) evidential norm: the norm, roughly, that one has
reason to believe a proposition only in so far as “the evidence
indicates, or makes likely, that” the proposition is true. A
norm, in other words, much like
(EN)
(in the supplement on Bridge Principles). The thought is that any
instance of my violating
(NC)
is eo ipso an instance in which my beliefs are out of whack
with the evidence. For when I hold contradictory beliefs, at least one
of the beliefs must be unsupported by the evidence. As Kolodny puts
it,

The attitudes that reason requires, in any given situation, are
formally coherent. Thus, if one has formally incoherent attitudes, it
follows that one must be violating some requirement of reason. The
problem is not, as the idea of requirements of formal coherence as
such suggests, that incoherent attitudes are at odds with each other.
It is instead that when attitudes are incoherent, it follows that one
of these attitudes is at odds with the reason for it—as it would
be even if it were not part of an incoherent set. (Kolodny 2007: 231)

Another way of making Kolodny’s point is to note the following.
Suppose I find myself believing both (p) and (neg p), but that
the evidence supports (p) (over its negation). If
(NC)
were the operative norm, I could satisfy it “against
reason”, i.e., by coming to believe (neg p). But adherence to
(NC)
contra the evidence seems like an unjustified
“fetish” for “psychic tidiness”. (Kolodny
proposes similar maneuvers for other types of putative formal
coherence norms, and for norms of logical coherence in
particular.)

What Kolodny assumes here is that there are, in Broome’s words,
“no optional pairs of beliefs” (Broome 2013: 85). That is,
it is never the case that belief in (A) and belief in (neg A) is
equally permissible in light of the evidence. As Broome points out,
Kolodny’s assumption is founded on a commitment to
evidentialism, which may cause some to get off the bus. Notice,
though, that even if we accept Kolodny’s argument along with its
evidentialist presuppositions, there may still be room for logical
norms. Such norms would not constrain beliefs directly, since only
evidence constrains our beliefs on Kolodny’s view. Yet, the
evidence itself would be structured by logic. For instance, if (A)
entails (B), then since (A) cannot be true without (B) being
true, any evidence that counts in favor of (A) should also count in
favor of (B). Logic would then still exert normative force. However,
its normative force would get only an indirect grip on the
agent’s doxastic attitudes by constraining the evidence. It is not clear how robust the distinction is, especially against the background of conceptions that take evidence to be constituted largely (or entirely) by one’s beliefs. Moreover, Alex Worsnip 2015 has argued that in cases of misleading higher-order evidence, failures of coherence cannot ultimately be explained in terms of failures to respond adequately to the evidence.

5.2 Consistency and coherence

At the outset we identified two logical properties as the two central
protagonists in any story about the normative status of logic:
consistency and logical consequence. So far our focus has been almost
exclusively on consequence. Let us now briefly turn to norms of
consistency.

The most natural and straightforward argument for consistency is that
the corresponding norm—something along the lines of
CON—is
entailed by the truth norm for belief:

  • (TN) For any
    proposition (A), if an agent (S) considers or has reason to consider
    (A), (S) ought to believe (A) if and only if (A) is
    true.[21]

The truth norm entails the consistency norm (given certain
assumptions):

  • (CN) For any
    agent (S), the set of propositions believed by (S) at any given
    time ought to be logically consistent.

For if the set of propositions I believe at a particular point in time
is inconsistent, they cannot all be true, which is to say that I am
violating the truth norm with respect to at least one of my
beliefs.

Some objections to the consistency norm are closely related to the
considerations of
Excessive Demands.
And even in cases where it would be within our powers to discover an
inconsistency given our resources of computational power, time and so
on, it may still be reasonable to prioritize other cognitive aims
rather than expending significant resources to resolve a minor
inconsistency (Harman 1986). However, many authors who invoke
(CN)
do so in a highly idealized context. They think of the norm not as
reason-giving or as a basis for attributing blame, but merely as an
evaluative norm: an agent with an inconsistent belief set is less than
perfectly
rational.[22]

Another reason for rejecting
CON
is dialetheism (see entry on
dialetheism).
Clearly, if there are true contradictions, there are special cases in which one ought to have inconsistent beliefs.

But there is a further worry about consistency borne out of less
controversial assumptions. It stems from the aforementioned fact that
we do not only evaluate our beliefs according to their truth status
but also in terms of their reasonableness in light of the evidence.
Accordingly, there would seem to be an epistemic norm, like
(EN)
in the supplement on Bridge Principles,
that one ought to (or may) believe a proposition only if that
proposition is likely to be true given the evidence. But if that is
so, the following well-known scenario may arise: it may be that, for a
set of propositions, I ought to (may) believe each of them in light of
the evidence, yet—because evidential support is not
factive—the resulting belief set turns out to be inconsistent.
Therefore, if rationality demands that I align my beliefs with the
evidence, rationality is no guarantee for logical consistency. Of
course, it is precisely this clash between our (local) evidential norm
and the (global) coherence norm of logical consistency that is
dramatized in the Preface and in the Lottery paradoxes.

In light of such considerations, no small number of
authors have come to reject the consistency norm (see inter
alia
Kyburg 1970 and Christensen 2004). A particularly
interesting positive alternative proposal was recently made by Branden
Fitelson and Kenny Easwaran (Fitelson and Easwaran 2015, Easwaran 2015). They advance
a range of sub-consistency coherence norms for full belief inspired by
Joyce-style accuracy-dominance arguments for probabilism as a norm for
credences (see Joyce 1998, 2009 and also the entry on
epistemic utility arguments for probabilism).
One important such norm is based on the following conception of
coherence. Roughly, a belief set is coherent just in case there is no
alternative belief set that outperforms it in terms of its lower
measure of inaccuracy across all possible worlds, i.e., just in case
it is not weakly dominated with respect to accuracy.

5.3 Logic vs. probability theory

Even if there is a plausible sense in which logic can be said to be
normative for thought or reasoning, there remains a worry about
competition. Logic-based norms usually target full beliefs. If that
is correct, a significant range of rationally assessable doxastic
phenomena fall outside of the purview of logic—most
significantly for present purposes, degrees of
belief.[23]
Degrees of belief, according to the popular probabilist picture, are
subject not to logical, but to probabilistic norms, in particular the
synchronic norm of probabilistic
coherence.[24]
Consequently, the normative reach of logic would seem (at best) to be limited;
it does not exhaust the range of doxastic phenomena.

Worse still, some philosophers maintain that degrees of belief are the
only doxastic attitudes that are, in some sense, “real”,
or at least the only ones that genuinely matter. According to them,
only degrees of belief are deserving of a place in our most promising
accounts of both theoretical (broadly Bayesian) and practical (broadly
decision-theoretic) accounts of rationality. Full belief talk is
either to be eliminated altogether (Jeffrey 1970), or reduced to talk
of degrees of belief (ontologically, explanatorily or otherwise).
Others still acknowledge that the concept of full belief plays an
indispensable role in our folk-psychological practices, but
nevertheless deem it to be too blunt an instrument to earn its keep in
respectable philosophical and scientific theorizing (Christensen
2004). Virtually all such “credence-first” approaches have
in common that they threaten to eliminate the normative role of logic,
which is superseded or “embedded” (Williams 2015) in
probabilism.

A number of replies might be envisaged. Here we mention but a few.
First, one may question the assumption that logical norms really have
no say when it comes to credences. Field’s quantitative bridge
principle is a case in point. As we have seen, it does directly
connect logical principles (or our attitudes towards them) with
constraints on the allowable ways of investing confidence in the
propositions in question. To this it might be retorted, however, that
Field’s proposal in effect presupposes some (possibly
non-classical) form of subjective probability theory. After all, in
order to align one’s credences with the demands of logic, one
must be capable of determining the numerical values of one’s
credences in logically complex propositions on the basis of
one’s degrees of belief in simple propositions. This is most
naturally done by appealing to probability
theory.[25]
But if so, it looks as if probability theory is really doing all of
the normative work and hence that logic would seem to be little more
than a redundant tag-along. Second, one might try to downplay the
importance of degrees of belief in our cognitive economy. In its
strongest form such a position amounts to a form of eliminativism or
reduction in the opposite direction: against credences and in favor of
full belief. Harman (1986), for instance, rejects the idea that
ordinary agents operate with anything like credences. Harman does not
deny that beliefs may come in varying degrees of strength. However, he
maintains that this feature can be explained wholly in terms of full
beliefs: either as belief in a proposition whose content is
probabilistic or else

as a kind of epiphenomenon resulting from the operation of rules of
revision [e.g., you believe (P) to a higher degree than (Q) iff it
is harder to stop believing (P) than to stop believing (Q)].
(Harman 1986: 22)

More moderate positions accord both graded and categorical beliefs
along with their respective attendant norms a firm place in our
cognitive economies, either by seeking to give a unified account of
both concepts (Foley 1993; Sturgeon 2008; Leitgeb 2013) or else by
reconciling themselves to what Christensen (2004) calls a
“bifurcation account”, i.e., the view that there is no
unifying account to be had and hence that both types of belief and
their attendant norms operate autonomously (Buchak 2014; Kaplan 1996;
Maher 1993; Stalnaker 1984). Summarizing, then, so long, at least, as
full belief continues to occupy an ineliminable theoretical role to in
our best theories, there still is a case to be made that it is to
logic that we should continue to look in seeking to articulate the
norms governing these qualitative doxastic states.

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