Validity, normativity and degrees of belief
Abstract
The assessment of an argument as valid appears to have normative import in relation
to how we should reason. Suppose we reject or question the standard definition of
validity as necessary preservation of truth. Can we illuminate or characterise what it is
for an argument to be valid by appealing to the distinctive normative role of valid
arguments? And might such considerations help us in choosing between alternative
logics, or at least in understanding what should guide such choices? I will ask how to
characterise the normative role of validity, in particular when we consider that many
of our beliefs are merely partial (in the sense we have some level of uncertainty in
them). I will focus in particular on two principles that Hartry Field advocates and ask
whether they can be used in characterisation of validity and/or to help illuminate
choices between different logics that people might advocate or employ. I will argue
that Field’s principles are not fit for this job and that no alternative principles can
retain the required neutrality either. The normative situation is derivative, complex
and can vary with context; it cannot provide the key to understanding validity.
1. Introduction
What is it for an argument to be valid? Suppose we reject or question the standard
definition of validity as necessary preservation of truth and seek to leave scope for
endorsing non-classical logics. One notable feature of validity is that the assessment of an
argument as valid is often taken to have normative import in relation to how we should
reason. Can we, then, illuminate or characterise what it is for an argument to be valid by
appealing to the distinctive normative role of valid arguments? And might such
considerations help us in choosing between alternative logics, or at least in understanding
what should guide such choices?
We should not simply assume the standard definition of validity, especially once we
countenance non-classical logics such as intuitionistic logic, paraconsistent logics, many-
1
valued logics, free logics etc. For one, different explanations of validity that are equivalent in
a classical, bivalent framework can come apart outside that framework. Consider, for
example, the account popular for teaching purposes, “valid iff not possible for the premises
to be true and the conclusion false”: unlike other characterisations, this would allow for a
valid argument with true premises and a conclusion that is neither true nor false. Within
non-classical frameworks there are often several account of validity in the running and
choosing between them will be a complex matter.
Here are three specific reasons to challenge the definition of validity as necessary truthpreservation. First, Hartry Field shows how assuming that validity is necessary preservation
of truth is problematic in the face of semantic paradoxes (e.g. 2009b, p.351, “it is
inconsistent to hold that the logic one accepts actually preserves truth”). Second, we may
want to allow for valid arguments involving imperatives or types of sentences that we do
not take to be true or false. Third, if we allow sentences that are indeterminately true – e.g.
because they are vague – or if we focus on claims we are uncertain of, then the idea of
necessary preservation of truth may not be general enough. For example, it is not clear that
valid arguments only need preserve absolute (or certain) truth – perhaps they should
impose constraints in uncertain or indeterminate cases too.
The question addressed in this paper is whether we could settle on the right
requirement for validity by appealing to the distinctively normative role of valid arguments,
where, of course, a central task will be explaining what this normative role is. An aim of this
kind is pursued by Hartry Field (e.g. 2009a), whose approach will be discussed in detail
below. And MacFarlane, in an unpublished but quite widely discussed piece delivered at the
APA, also hopes to make progress on questions of what validity amounts to by getting “a
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clear understanding of what the concept of logical validity is for.” (unpublished, p.2) I will
argue that this approach to illuminating validity will not succeed.
2. Normative requirements
Different logics can have different rules of inference. Might it then be that the normative
import is merely a matter of telling you how you should reason in order to reason correctly
as far as that logic is concerned? Not if we are to illuminate validity through normativity,
saying what is distinctive about valid arguments, partly with a view to aiding the choice of
logic. For, we can legitimately talk of how you should reason according to any system,
including weird and wonderful ones that we wouldn’t and shouldn’t accept as governing our
actual reasoning. Unless we have a grasp on how we should reason that is not relative to a
system, we will make no progress illuminating the idea of validity simpliciter or how one
logic can give a better account of validity than another.
So, it is important not to run together the recognition that logical systems provide
normative rules by-their-own-lights and the thought that what is distinctive about valid
arguments concerns their normativity. But, consider Beall and Restall, who defend a logical
pluralism, according to which there is more than one logical consequence relation. 1 On the
one hand, they count normativity among the “settled features associated with the role of
consequence” (2006, p.40) – or admissibility conditions – which each genuine logical
consequence relation must satisfy. Their logical pluralism aims to say more than that there
are multiple abstract logical systems, since it is committed to the claim that multiple such
systems – those also satisfying the admissibility conditions – provide logical consequence
I use the following notions interchangeably here: “the argument from  to C is valid”, “C is a logical
consequence of ” and “ entails C”.
1
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relations capturing what really does follow from what. And normativity looks to be the kind
of feature that will help to weed out the logical systems that don’t capture genuine logical
consequence: surely one way a system can be inappropriate is by it providing the wrong
rules by which we should reason. But when Beall and Restall come to apply the normativity
constraint to particular logics, it just looks like a matter of demonstrating normativity bythe-lights of that system. For they take it to be enough to show that there are “mistakes
that are deemed as such by the corresponding sense of ‘follows from’” (2006, p.55), without
showing that they are genuine mistakes. Surely using a syllogistic argument isn’t making a
mistake in reasoning, even if its propositional form ensures that it counts as such according
to propositional logic.
To assess whether some particular logic tracks the genuine norms governing us and our
reasoning, we need a clearer account of how consequence relations should affect our
beliefs. Here’s one principle that might be offered (which can be easily generalised for
arguments with more than two premises):
(BP) If A, B entail C then (you ought to see to it that if you believe A and you believe B,
you believe C)
So, if you believe the premises of a valid argument then you should also believe the
conclusion – a directive you can satisfy by giving up a belief in a premise, rather than always
taking on the new belief in the conclusion. BP is an abbreviation of “Broome’s Principle” to
acknowledge the similarity with Broome’s account of account of reasoning in terms of what
he describes as “normative requirements”, where, if p is a normative requirement for q, that
means that you ought to make it the case that if p, q (e.g., 2000, p.86).
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Broome describes normative requirements as “strict”: if you don’t fulfil them, you are
not “as you should be”. This contrasts with conditions that involve the subject having
reasons to do something, where those reasons can be trumped by reasons for doing
something else. For example, p might give reasons for q without it being true that that it
ought to be the case that if p, q, because reasons against q outweigh the reasons for q that
p gives. The claim is that if you believe a premise and the denial of a conclusion that follows
from it, then you are not entirely as you should be. That isn’t to say that you will know how
best to get into a better state – which belief to give up – or even that you should devote
mental energy to trying to find out; you are still not “as you should be”.
I suggest that, for the relation between normativity and validity that we are
investigating – whereby the former is used to characterise the latter – the normative
requirements do need to be strict. For the hope is that a thesis about the normativity of
logic will help characterise validity and thus distinguish logically valid inferences from other
kinds of inference. The normative relation then has to be strict, corresponding to the
distinctive strictness of validity. If the normative “hold” logic has on us is just a defeasible
reason for having certain beliefs etc., then it is not clear how it differs from other reasons
for beliefs or why, say, inductive reasoning doesn’t fit the mould.
A natural objection to Broome’s account is that it places us under too many obligations,
namely to believe all the consequences of our beliefs, including ones that can only be shown
to follow by some long, complex argument. One reply would qualify those cases in which
there is a normative requirement to believe q if you believe p. It could just hold in the cases
where the subject knows that q follows from p, for example, or where it is obvious that q
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follows from p or some such. Field, for example, takes the latter approach in his principles to
be discussed below.2
On the kind of view captured in (BP) or the many possible variants, facts about validity
deliver normative requirements. This suggests a picture according to which there is a
uniquely correct logic which dictates the normative requirements (though we needn’t
assume we know what that logic is). Someone convinced by a different logic will then simply
be led astray and count as irrational. This may seem to be an unwanted consequence, for
surely what logic you believe to be the correct one should impact on how you should reason
in order to count as rational. Field (2009a, p.261), for example, considers Bob who has made
a good case for weakening classical logic but then who “slips into classical reasoning that is
not licensed by his own theory”, and argues that in so slipping, Bob is “violating his rational
norms”.3
For the project we are considering in this paper, it would be better to proceed, instead,
in the other direction and try to start from an independent grasp of normative matters to
build up to an account of validity. We turn to this approach in the next section.
3. Getting a grip on how we should reason
Suppose that a) we ask “how should you reason?” and answer “using valid arguments” – a
seemingly reasonable, if partial, answer. But b) to “what are valid arguments?” we answer
2
Compare also the discussion of the “demandingness objection” in MacFarlane (unpublished). I will not here
tackle the problems Harman raises regarding the relations between logic and reasoning (1986). Field offers
replies to many of Harman’s objections and I will suppose that Field’s replies work – or at least are a promising
beginning – so that we needn’t assume that the project we are considering falls at the first hurdle. The devil
will be in the detail of the story and a full defence of any such account may require returning to Harman-like
objections. I will argue that the approach in question fails in any case.
3
We could take (what Field describes as) MacFarlane’s line in response and maintain that Bob has competing
obligations – both to reason according to the correct logic and to reason according to the logic he advocates.
But that would seem, again, to give up on the strictness requirement.
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something like “valid arguments are those that should govern your reasoning” (picking out
the normativity of validity as its key defining feature). That gives obviously circular answers
to “how should you reason?” and to “what arguments are valid?” which look unable to help
in illuminating either issue.
Since we are investigating the strategy of giving an answer like b), we need to say more
than is said in a) about how we should reason. A first approach could be to accept a
commitment to normative facts about how we should reason. Perhaps there are brute facts
governing correct reasoning and truths about validity are determined by those brute facts.
Field calls this “normative realism”. If we could appeal to a fact that we should reason using
modus ponens, for example – where this is not to be explained by saying, for example, that
arguments of that form preserve truth – then we would have a way into the circle and, if the
details could be worked out, an account of validity that explains it in normative terms. But
Field, among others, rejects such a commitment to objective irreducible normative facts
about reasoning, raising concerns about the metaphysical and epistemological
commitments as well as a “queerness” objection (2009b, p.354). Moreover, it is not clear
that commitment to brute facts over how we should reason which ground true claims of
validity has any advantage over brute claims about validity. This paper explores ways to do
without such commitments.
An alternative story about how normative matters determine facts about validity may
involve giving an account that reduces facts about how we should reason – and thus about
validity – to other facts, for example facts about actual (inferential) behaviour. Or there may
be a more complex story to be told avoiding normative facts and talking, instead, in terms of
(possibly variable) goals and normative demands relative to those goals.
7
Field suggests that questions about how we should reason should be answered by
consideration of our actual behaviour in reasoning and evaluating the beliefs of others
(2009a, p. 262 and 2009b, p.354). He calls his position “expressivist” in contrast with a
realist view on normative facts, suggesting that “it doesn’t make much sense to talk of a
norm being correct or incorrect”, though those norms can be better or worse relative to
different goals we may have (2009b, p.355). We can consider, alongside this view, the realist
alternative that normative facts are somehow reducible to facts about our actual epistemic
practices: it is subject to some of the same objections and may have some advantages over
the Fieldian anti-realist option.
Assume that, broadly speaking, we reason in ways that work for us. Then to understand
what arguments are valid, perhaps we should look to how we actually reason. At least, we
should see how we reason in the most part, idealise to allow for the fact that people make
mistakes and take the best theorised account of this to provide the details of how we should
reason and thereby what arguments count as valid.4
The (unavoidable) role of idealising leaves open the possibility that several different
systems could provide equally good models of our reasoning. For those different systems
might each capture most of the behaviour by taking different instances as exceptions to the
norms (so counted as mistakes according the system), and/or by taking different lines on
the question of how to balance off generality against simplicity. There may then be nothing
to choose between them and we may thus defend a pluralism – they are all equally good
(though some might be better for some purposes). Field’s describes his position as a
“relativist expressivist”: since he doesn’t endorse the logics reached by this method as
4
I put aside some general challenges that could be raised here. For example, there may not be sufficient
uniformity in our behaviour to generate a successful generalisation over it. Or fallacies that are sufficiently
prevalent could, by this method, be legitimated by their frequency, since they may figure in the generalisation
that best fits the facts about our reasoning.
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correct, he can accommodate multiple different systems as on a par. On the realist
alternative to Field’s position just described, the multiple different systems could be seen as
equally correct so that there can be different, conflicting answers to whether an argument is
valid and whether we should reason in a certain sort of way; whether such a pluralism is
acceptable is not something I can debate here. In contrast with these approaches, you may
regard the possibility of multiple equally good models as a mere theoretical possibility and
harbour an optimism that one system will be uniquely correct and thereby declared the
correct account of validity and how one should reason.5
To examine Field’s principles and to tackle one of the points I initially raised in
motivating a move away from just focusing on necessary truth-preservation, we next need
to introduce discussion of partial degrees of belief.
4. Introducing degrees of belief
If you think that p is probably true, it may be wrong to represent you as holding a full belief
that p: you may believe it to an intermediate degree. Similarly, if a book is on the borderline
between red and orange, you might believe to some intermediate degree that it is red. And
perhaps indeterminacy from other sources besides vagueness might reasonably lead to
intermediate degrees of belief. If beliefs can come in degrees, how does this impact on the
role that logical validity has on our reasoning and belief-formation?
If a valid argument has only one premise, a requirement such as “one should not
(believe the premise but not the conclusion)” – the one premise version of (BP) – has a
5
As a comparable view, consider Lewis 1994 who advocates a best-system account of laws of nature, which
identifies the laws of nature as those which provide the best systematic generalisation over the actual facts.
Lewis (p.479) speculates about whether “nature is kind to us” – meaning that one system is “robustly best” –
and regards that as a “reasonable hope”. Note that on Lewis’s system, the possibility of several systems arises
if there are different ways to systematise the facts, whereas with the logical position at issue here, there is
additional scope for different systematisations, since each one can clarify different cases as exceptions.
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natural extension to the requirement that one should believe the conclusion as much as (to
as a great a degree as) the premise. The generalisation to more than one premise can be
made in a number of different ways and we’ll focus on Field’s way below. There will be
further variations on this principle according to whether credence in premises and
conclusion is constrained for all valid arguments, or only those where one knows the
sequent is valid or where it is obvious that the sequent is valid etc., where the issues for this
degree-theoretic case will, at least to a large extent, mirror those questions raised in the
non-degree case mentioned in Section 2 above.
The move to a principle involving degrees of belief may also help us to deal with what
will otherwise look like problematic cases where it is not wrong to have contradictory
beliefs. The classic case is that of the Preface Paradox, in which an author believes every
assertion in her book, but in the preface sincerely asserts that not everything in the book is
true (based on knowledge of her own fallibility).6 Given that she knows exactly what is in the
book, this amounts to denying the conjunction of all the claims in the book, even though
that conjunction clearly follows (and is known by the author to follow) from the assertions
in the book taken together. A non-probabilistic normative principle will rule that she ought
not to have the combination of beliefs she has expressed. But the author could, it seems, be
perfectly rational and justified in holding them together. As Field points out (2009a, p.254)
similar situations can arise with physical theories, all of whose claims seem highly plausible
but which turn out to be inconsistent.
If you realise that you are committed to an absurdity given some of your beliefs but you
have good reasons to retain each of those beliefs, a sensible response is to admit that each
of your beliefs is merely probably true and thereby to acknowledge that you believe each to
6
See Makinson 1965.
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a partial (though high) degree. A normative principle governing your beliefs should
accommodate this option. What degree of belief you should have in each premise of a
problematic argument whose conclusion you do not believe will depend on the details of
the argument. For example, perhaps if there are lots of premises, you can believe each of
them to a high degree, whereas if there are just two premises, then you will not be
normatively OK if you believe them both to a high degree and reject the conclusion
completely: in such a case, you should drop your degrees of belief in the premises further
(or increase your degree of belief in the conclusion). We will turn to details of potential
principles in the next section.
So, by introducing degrees of belief, the hope is that we can reflect both how we should
respond to certain inconsistent sets of sentences (adopt certain patterns of partial belief in
them) and how we do, indeed, respond to them. I turn in the next section to assessing
Field’s degree-theoretic principles.
5. Field’s Principles
Field’s two most central principles concerning the normativity of logic are the following:
(D*) If it’s obvious that A1, ... An together entail B, then one ought to impose the
constraint that P(B) is to be at least P(A1) + ... + P(An) – (n-1), in any circumstance
where A1, ..., An and B are in question. (2009a, p.259)
(E)
Employing a logic L involves it being one’s practice that when simple inferences A 1, ...
An ⊢ 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(A1) + ... +P(An) – (n-1). (2009a, p.262)
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The first of these principles attempts to capture the normativity of logic by spelling out the
demands logic places on our beliefs in particular circumstances (when the inference is
obviously valid) – very roughly, that you shouldn’t believe the conclusion of a valid
argument less than you believe the premises, where the lack of confidence in each premise
can add up. The second principle may be hoped to bridge the gap between how we actually
behave and how we should behave and thus to allow us to avoid any apparent commitment
to normative realism. Here, the reference to “normally” may seem to compromise the
demand for strictness, but, in fact, it need not do so. The weakening is needed to allow for
mistakes in how people actually behave and it is compatible with strictness over what they
should do. People should follow the norms of the logic, but do not always do so, but
nonetheless the exceptions are sufficiently infrequent as to allow us to understand what the
norms are by considering how people normally reason.
Milne formulates Field’s (D*) as follows: “If it’s obvious that the premises of a valid
argument entail the conclusion then one ought to impose the constraint that the ...
uncertainty of the conclusion is never greater than the sum of the ... uncertainties of the
premises.” (2009, p.294).7 Here, one’s uncertainty of p is 1 minus one’s degree of belief in p.
Milne’s formulation brings out the similarity with a probabilistic result discovered by Ernest
Adams that the improbability of the conclusion of a valid argument does not exceed the
sum of the improbabilities of the premises.8 So, if you start with premises that are
uncertain, then the conclusion can inherit uncertainty from each of those premises and a
valid argument that has many very probable premises can have a very improbable
7
The elision here concerns Milne’s employment of a distinction between upper and lower uncertainties, which
are not relevant for our purposes. See also principle (VP) in Field (forthcoming).
8
Adams 1966. Edgington 1997 extends the model to apply it to vagueness where, she maintains, we can
reasonably have partial degrees of belief – or verities, as she calls them – in borderline case predications.
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conclusion. If we expect or require degrees of belief to track probabilities (or our beliefs
about probabilities) then the parallel principles may look appealing. Note that Adams
assumes classical logic and the identification of validity with necessary preservation of truth,
so, given that we are exploring the definition of validity and the choice between different
logics, we cannot use Adams’s result to justify Field’s principles (E) and (D*).
Field’s two principles are clearly closely related, and, indeed, Field offers (E) as a way to
“recast” (D*). We will largely focus on (E), which is most central to the project of
illuminating validity that we are interested in. Note that Field offers his specific principles
somewhat tentatively; but as well as criticising them as they stand I will show that the
problems are too general to allow those principles to be replaced by similar alternatives.
Before considering the details of Field’s principles, some clarification of their roles is
needed. In exploring Field’s position in relation to the project of defining validity via its
normative role, I may appear to be ignoring Field’s own appeal to primitiveness: “I don’t
mean that ‘implies’ is to be defined in terms of ‘ought’ or other uncontroversially normative
notions – it is better to view it as a primitive notion, not defined at all – but that we accept
norms that connect our beliefs about implication to our views of proper epistemic
behaviour” (Field 2009a, p.349). Field does hope to maintain the centrality of normativity to
the notion of validity, even if that notion is strictly primitive, for example, he says “our
notion of good argument is an essentially normative notion...” (Field 2009b, p.268).
It is not clear, however, how we should reconcile the normative “essence” of validity
with the primitiveness of the latter or what the claims about the centrality of normativity to
logical consequence amount to if we are not aiming to understand validity through its
normative role. The primitiveness claim also sits uneasily with reflections on our grasp of
the supposedly primitive notion. Validity isn’t the kind of notion that people (e.g. beginning
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logic students) grasp intuitively, and, in particular, it is not plausible to think we have a
primitive grasp on validity that will distinguish between the subtly different logical options
on offer in non-classical schemes. Even if people typically understand the difference
between ways we should argue and ways we should not, this won’t do to explain the grasp
of validity or how valid arguments differ from inductively good ones. Initial explanations of
the difference between valid arguments and inductively good arguments often use
metaphors about the former being “watertight” or “providing guarantees” (of the truth of
the conclusion given the truth of the premises), but these notions surely need spelling out –
if not in terms of necessary truth-preservation, then in some other terms – and they cannot
be relied on as they stand. Gesturing at some normative features of valid arguments will not
do unless those features are explained and distinguished from other normative roles that,
say, non-deductive arguments can have. This looks like it will lead to a characterisation of
validity in terms of normativity along the lines of the approach we are considering here, in
which case the appeal to primitiveness will have no role to play.
This may seem to overlook an option on which the normative principle doesn’t give a
full definition of validity but corresponds instead to some kind of necessary but not
sufficient condition. In his (forthcoming), Field refers to his account as “a primitivist (nondefinitional)” conceptual role account (p.26), which may suggest such a story. Perhaps, then,
to know that an argument is valid, you need to know that this will affect your degrees of
belief in the manner captured by his principle, even if that isn’t the whole story about
validity. But this would be enough of a role for the principle for the objections I raise below
to be problematic for Field. Those problems will equally take hold if his principles are
proposed as necessary conditions rather than as defining the concept of validity. If, for
example, you can advocate a many-valued logic and an account of validity without fitting
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Field’s (E) – as I argue in section 5d – then (E) isn’t a viable necessary condition. Whatever
exactly the role, the principle (E) needs to work and needs to work for different logics.
Rather than explaining exactly what the primitiveness or primitivist conceptual role
claims amount to, or what they entail or preclude, I return to focus on the project that seeks
to illuminate the notion of validity by its normative role. In particular, I turn to Field’s
principles with a view to their playing that role.
a) The “normally” qualification
According to Field’s principle (E), what is required for employing a particular rule is that one
normally imposes the relevant constraint on one’s degrees of belief. As discussed in Section
3, some “normally” qualification of this kind is necessary to allow for mistakes we will
inevitably sometimes make. The qualification threatens to open up various unwanted
possibilities and these difficulties are just as acute for a degree-theoretic principle like
Field’s. For example, the principle could count someone as employing several different
logics, and common errors threaten to get legitimised by their frequency. Also, we may
normally follow the kind of pattern Field identifies when we form and adjust our beliefs in
employing reasonable rules of inductive reasoning. For example, in many sorts of cases we
will normally believe Fxn on the basis of believing Fx1-Fxn-1, where our degrees of belief in
these propositions fit Field’s principle, which will then count such inductive principles as
part of the logic we are employing.
There may be ways to respond to some of these general issues, but the challenges
should not be dismissed lightly. I turn, however, to other more specific concerns and issues.
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b) Allowing for a subject’s lack of belief about something
It may sometimes be clearly right to represent a subject as having a particular degree of
belief in a proposition. In other cases, though – in particular when the subject hasn’t
considered the proposition in question – it’s less clear that there’s a specific degree of belief
in the proposition that it’s correct to ascribe. One line to take here would be to go ahead
and model our subject as having some degree of belief in each proposition anyway. After all,
the introduction of partial degrees of belief was partly made to allow for the subject’s
uncertainty in certain cases, and when the subject hasn’t considered the proposition in
question, perhaps we can see this as being equally uncertain between that proposition and
its negation, such that assigning each a degree of 0.5 is appropriate. Indeed, in a Bayesian
model of subjective probabilities and preferences, it is assumed that some probability
function represents the subject’s cognitive states. Someone’s subjective probability/degree
of belief in p is determined (very roughly) by what they would be prepared to bet for a
desirable outcome. We assume there is an answer to this, and if they have absolutely no
indication of whether p or not-p, then they can be modelled as having degree of belief 0.5.
If, though, they are betting on something that is clearly less likely than its negation, then we
might assume that the subject would recognise this, and, if asked to bet, would reveal a
degree of belief of less than 0.5. These assumptions are questionable in general, but are
particularly problematic for Field’s purposes.
Suppose our subject hasn’t considered the conjunction of two propositions, while
believing each conjunct to a high degree. We cannot model her as having a degree of belief
of 0.5 in that conjunction: her lack of consideration of it surely does not translate into
neutral uncertainty in this case. We can reasonably expect her to have a high (if slightly less
high) degree of belief in that conjunction, given her belief in the conjuncts. But on what
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basis do we make that attribution? If it based on the logical relations between conjuncts and
conjunction, then we are building the result about patterns of degrees of belief with valid
arguments into the data we are supposed to be using to determine what arguments are
taken to be valid and we cannot take it as evidence that the subject employs that rule. This
kind of worry is not shared by the Bayesian, who can acknowledge a higher degree of
idealisation than Field can and can make assumptions about the logic governing the
subject’s belief that Field cannot make.9 Field aims to determine what principles we follow
by looking at our patterns of degrees of belief; so we cannot assume we are following
certain principles to determine what our degrees of belief in certain compounds are without
begging the question.
In responding to the objection to a non-degree theoretic normative principle that it requires
you to believe all the consequences of your beliefs, it may be natural to distinguish between
those cases where you simply haven’t given any thought to some (perhaps complex)
proposition that follows from a set of your beliefs so believe neither it nor its negation and
those cases where you believe the negation of a consequence of those beliefs. To fault the
second and not the first of these situations we could formulate the principle in such a way
that belief in the premises and the negation of the conclusion is what is ruled out, so that it
does not speak to those cases where the subject has no beliefs either way as regards the
conclusion. Can a similar move be made in the degree-theoretic case?
Field recognises the need to allow for uncertainty without representing it with a
particular degree of belief and makes what looks to be the corresponding qualification: “a
9
Compare Jeffrey, The Logic of Decision, p.167: “Bayesian decision theory provides a set of norms for human
decision making; but it is far from being a true description of our behaviour. Similarly, deductive logic provides
a set of norms for human deductive reasoning, but cannot usefully be reinterpreted as a description of human
reasoning.”
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principle governing a person’s degrees of belief ought to be understood as having the tacit
assumption that the person has all the degrees of belief in question” (p. 255). Suppose, for
simplicity, we work with a restriction to the single-premise case, then the proposed
qualified principle would be:
“If you have a degree of belief in each the premise and the conclusion of a valid argument,
then you shouldn’t have a lower degree of belief in the conclusion than in its premise”.
When, though, do we count as having a degree of belief in a proposition? We surely
should not limit this to cases where we have explicitly formulated our degree of belief, for it
is very rare for us to make such degrees explicit to ourselves. Instead, we may assume there
are implicit degrees of belief indicated by how tentative the subject feels about the
proposition, how they would bet on it etc. But once we allow such extrapolating and
idealising, we return to the problem above of avoiding the presupposition of certain logical
principles in determining what the degrees of belief are. In other words, it is not obvious
how to occupy the middle-ground between allowing only explicit degrees of belief, of which
there will not be enough, and assigning degrees of belief on the basis of principles that
presuppose relations we are trying to establish. Will there be enough data about degrees of
belief to uncover behaviour pointing to particular principles followed?
c) Idealisations and numerical degrees
Gilbert Harman (1986) raises some far-reaching concerns about a strategy like Field’s that
appeals to degrees of belief.10 For example, he says, “People do not normally associate with
their beliefs degrees of confidence of a sort they can use in reasoning. It is too complicated
10
Field has in mind many of Harman’s earlier points against attempts to defend the normativity of logic when
defending his view, but, as Harman 2009 points out, he does not address Harman’s objections to the
employment of degrees of belief.
18
for them to do so. Degrees of belief are and have to be implicit rather than explicit, except
for a few special cases of beliefs that are explicitly beliefs about probabilities.” (p.22).
Field might argue that the employment of numerical degrees of belief only need be the
theorist’s tool for modelling subject’s degrees of belief: these can be based largely on
comparative judgements, where, for example, if the subject is more inclined to accept that P
than Q and more inclined to hang on to that judgement, that subject is attributed a higher
valued degree of belief in P than Q. This perhaps fits with Field’s recommendation that “the
focus shouldn’t in the end be on probability functions, but on certain probabilistic
constraints” (2009a, p.258). Representation theorems may then be hoped to show that if
our comparative judgements follow a certain sort of pattern, numerical values can be
assigned respecting the ordering, which might remove some of the worry that the
commitment to fine-grained value assignments goes well beyond the level of precision
within our own set of beliefs. But is it plausible to think that our comparative judgements do
follow the right kind of pattern? Will they yield an ordering, for example, rather than
including an array of cases where P and Q are very different sorts of propositions (e.g.
regarding relatively unlikely events of two very different types), for which the subject can
make no non-arbitrary comparison? The kind of representation theorem needed may be
unavailable because it requires doubtful assumptions, and a principle like (D*) that involves
summing values looks particularly problematic. 11
Let us return to the Preface Paradox to illustrate some of the problems for Field
hereabouts. The paradox is an instance of a phenomenon Harman emphasises – that it can
sometimes be rational to have beliefs even while knowing they are jointly inconsistent. As
11
Compare Keefe 2000, chapter 5, which considers degree theories of vagueness in relation to an objection
that specific values impose too much precision, arguing that theorists can’t rely purely on comparative
judgements at least when standard definitions of the connectives (involving addition or subtraction) are
employed.
19
we’ve seen, the appeal to degrees of belief seems to offer a way out of this problem. It is
only rational if your degrees of belief in two inconsistent propositions do not sum to more
than 1, or your degrees of belief in a set of n inconsistent beliefs does not sum to more than
n-1. But what are the grounds for thinking that our author does adopt the degrees of belief
fitting the desirable principle? If asked for a model of how probabilities might be distributed
to make sense of high probabilities for each of the assertions in the body of the book and
the negated conjunction corresponding to the preface claim, the pattern Field identifies
may be attractive. But that is not the task at hand: (E) appeals to the actual degrees of belief
our author has. We are to conclude that she still endorses conjunction introduction despite
her claim in the preface, because her degrees of belief fit the pattern captured in (E). But we
can’t distinguish between her genuinely following the principle (such that her uncertainty in
the conjunction of the claims is not greater than the sum of her uncertainties in those
various claims) and her just modifying her confidence in the individual claims in a more
modest (and less systematic) way.
Relatedly, we can ask whether the move to degrees of belief can really be the right way
to meet a natural generalisation of Harman’s problem to degree-theoretic beliefs. He
stresses that it can be rational to have inconsistent beliefs; the generalisation suggests that
you can have (independent) reasons to believe each of two inconsistent propositions to a
degree greater than 0.5. Is there anything more than wishful thinking to support the denial
of this?
To summarise these last two sub-sections, then. Appeal to degrees of belief opens up
interesting structural possibilities in relation to our dealings with arguments which may,
among other things, allow for a more subtle treatment of the Preface Paradox and other
20
cases where having inconsistent beliefs looks reasonable. But such an appeal requires heavy
machinery and strong assumptions going beyond what would otherwise be natural
descriptions of the states of our minds. There is insufficient reason to think that we (as
believers) work with such degrees of belief, adapting them in the light of arguments and
unwanted conclusions in the manner Field’s model would suggest. And we cannot see it as
an idealisation, where we just model our cognitive states as if they are degree-theoretic: the
aim was to illuminate the rules we employ and the degree-theoretic picture cannot do that
unless it is appealing to genuine features of our cognitive states.
These problems will face not just Field’s specific principle (E) but any other degreetheoretic attempt to determine what logic we endorse by assessing our actual degrees of
belief. And yet, as argued above, any principle used to determine what logic we endorse will
need to be degree-theoretic.
d) (E) and the employment of non-classical logics
Let us return to (E), presented here in a form that makes explicit the qualification to cases
where “the person has all the degrees of belief in question”:
(EQ) Employing a logic L involves it being one’s practice that when simple inferences A1, ...
An ⊢ 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(A1) + ... +P(An) – (n-1), if one has all the
degrees of belief in question.
One general form of worry is that S could reject a classical inference from  to C without (E)
revealing him as doing so, because rather than having lower degrees of belief in C than (E)
21
would predict (given his beliefs in ), he withholds any degree of belief in C instead. The last
clause of (EQ) ensures that S would then exhibit no patterns of belief that provide a
counterexample to him counting as employing that classical rule.
Consider, perhaps, someone who advocates (and reasons with) a relevant logic that
rejects disjunction introduction.12 Although she is not prepared to infer “P or Q” from P, it
need not be that she sometimes has a degree of belief in “P or Q” that is lower than her
degree of belief in P. It is surely better to represent her as having no degree of belief in Q or
in “P or Q” when Q is not relevant. But, this means that she will satisfy (EQ) for disjunction
introduction: when B is relevant, she happily draws the conclusion “P or Q”, when it isn’t,
she is silent on whether “P or Q” is true.
Next, I turn to a different class of logical positions. Many theorists have responded to
problems such as vagueness by rejecting Bivalence and introducing non-classical semantic
values, often continuum-many such values, corresponding to degrees of truth. Indeed, Field
himself employs a continuum-valued logic in the light of both semantic paradoxes and
vagueness (e.g. 2008, p.105, though intermediate values are not to be taken as degrees of
truth for Field). There are a wide range of such theories varying along a number of
dimensions including the number of semantic values, the definition of the connectives etc.
and, most importantly for our purposes, the account of validity and the logic endorsed.13
We should be able to treat all these as options that deliver their own normative demands,
as captured by (D*), and as options that count as being employed by their advocates as
determined by (E). But this turns out to be impossible.
12
See e.g. Anderson and Belnap, pp.430-432.
See, e.g., Keefe 2000, chapter 4, for a (critical) survey of such views and Smith 2008 for a recent book-length
defence of a view of this kind.
13
22
One option for the account of validity neatly mirrors the requirement on degrees of
truth in (D*); this declares an argument valid iff the degree of falsity (1 minus the degree of
truth) of the conclusion cannot exceed sum of degrees of falsity of the premises. 14 Field
does not include this definition among those alternatives he considers in his 2008. And,
indeed, it is not available as a definition on his theory. For it requires us to make sense of
the arithmetical operation it involves – in employing not just ordinal comparison between
the values of two different sentences, but values that can be manipulated in summing
operations etc. – and Field defends a Boolean-valued semantics in which the values are only
partially ordered (2008, p.105). Mere partial ordering is appealing, as, for example, we may
not be able to say which of “Joe is rich” and “Tim is thin” has the higher value when both are
borderline.15 But granting it would preclude the arithmetical operations required by this
definition of validity, so it blocks off a definition of validity which may be otherwise
appealing.
According to a popular alternative account of validity, an argument is valid iff whenever
the premises have value 1, so does the conclusion. After Field, I call this “weak validity”.16 To
consider such a definition of validity in relation to Field’s degree-theoretic normative
principles, we need to ask how intermediate semantic values relate to intermediate degrees
of truth. A natural assumption (though not Field’s) is that the degree of belief you should
have in an intermediately-valued statement is the corresponding intermediate degree of
14
This is the account of validity Edgington defends in her 1997.
See also Keefe 2012.
16
Field 2008 p.161: he gives reasons for counting weak validity as validity simpliciter (including the desirable
validating of the truth-rules – e.g. from p infer it is true that p). Peacocke 1981 offers such an account within a
theory of vagueness. A closely related alternative I won’t discuss here identifies validity with necessary
preservation of some wider set of “designated values”, such as those over 0.5. By contrast, an argument is
“strongly valid”, in Field’s sense, iff it is not possible for its conclusion to drop below its least true premise; see,
e.g., Machina 1976 and Forbes 1983 for this definition of validity within an account of vagueness.
15
23
truth.17 On this assumption, employment of weak validity is problematic. For although it
(appropriately) dictates that if some premises entail C, then we should not fully believe all
those premises without believing C, it appears to impose no constraints on one’s belief in C
when one’s belief in any of the premises is (even just slightly) less than one. Thus, it appears
to deviate from the rulings of Field’s principle (D*). Consider, for example, the following
argument: “p so Definitely p” (a Definitely operator is typically employed in frameworks
dealing with vagueness, allowing us to express that something is definitely true rather than
borderline). In many popular systems, this will be classed as weakly valid: if p is completely
true, so is definitely p. But suppose p is “Tim is thin”, when Tim is thin to degree 0.5; your
degree of belief in p should be 0.5, but your degree of belief in “Tim is definitely thin”
should be lower than that (perhaps zero – you recognise he’s a borderline case, which is to
say that he isn’t a definite case of thinness). This is to violate (D*), which would require that
you always have a degree of belief in the conclusion of a valid one-premise argument that is
at least as high as the degree of belief in the premise. Similarly, someone who endorses
such a logical system will not count as employing it if we judge that by Field’s principle (E).
Field’s principles thus fail to play the role we hoped they’d play, since they do not allow us
to make sense of someone following this reasonable logical option, and nor do they give a
suitable account of the normative demands on them.
The above argument doesn’t depend on the combination of views being correct, only
on them being views someone should be able to be counted as defending, which, as I
argued, (E) cannot allow. Discussion of the merits of the components here would take us too
far afield.18 Field’s normative principles do not maintain the required neutrality between the
17
For recent defences of versions of this position see Smith 2010 and MacFarlane 2010.
In brief, ways to resolve the tension include denying that “p therefore definitely p” is valid (for example, it is
not strongly valid) or denying, as Field does, that degrees of belief should mirror degrees of truth.
18
24
options on offer and thus cannot count subjects as employing various reasonable and
relatively popular logical systems. We cannot respond to this by preserving Field’s principles
and taking them to justify rejecting those options, for we have been given no independent
reason to defend the particular principles he has advocated. We cannot start from those
principles to decide these other controversial matters. Someone defending the combination
of views in a many-valued framework presented above could produce an alternative picture
of the normative situation, e.g. an alternative to (E) by which they would count as genuinely
advocating their position. Of course, that alternative would not be neutral either, and would
rule out alternatives that Field can better accommodate. We cannot decide between (E) and
such alternatives independently of deciding on the logical system we want to adopt. So
neither (E) nor its alternatives can be used to characterise validity through its normative role
without already presupposing some answers to questions that this principle should deliver.
Instead, I maintain, we should look at the options here in terms of the package overall,
in which the specific normative story for the system is part of the overall picture alongside
which arguments are declared valid, what definition of validity is endorsed etc. Field has not
just chosen the wrong general and neutral normative principles: what is suitable for one
system is not suitable for all. With a different sort of project in mind, we might allow for
different normative principles for different logics. But if we are interested in using it to
determine which of several logics someone employs, then the normative principle must be
neutral. Similarly if we seek to illuminate the notion of validity independently of a particular
logic, then a general principle is needed.
Different logics will need to be paired with different normative demands. We may agree
that for any choice of logical system, some corresponding normative principle can be
formulated, but this is not enough to help identify a core to the notion(s) of validity.
25
6. Conclusion
In the previous section, we raised a number of objections to Field’s principles concerning
the normativity of logic, in particular when used to illuminate the concept of validity.
Although some of my objections engaged with the specifics of the particular principles Field
offers, most undermine, more generally, principles (D*) and (E) and any alternatives which
might replace them while playing a similar role. For example, section 5a) considered the
“normally” qualification, where any candidate principle for the role (degree-theoretic or
otherwise) will need some such qualification. Similarly, sections 5b) and 5c) concerned
doubts about the employment of degrees of belief which didn’t rest on the specific
principles in which Field employs them, even though it was earlier argued that any such
principles would indeed need to be degree-theoretic. And 5d) both grappled with Field’s
specific principles and generalised it to illustrate the more general problem of finding
normative directives that are sufficiently neutral between alternative logics.
We can’t have, and don’t need, once-and-for-all principles dictating how the validity of
an argument should affect our behaviour. We should acknowledge that the normative
situation is derivative, complex and can vary with context and with details of controversial
philosophical views. The kind of project considered in this paper cannot succeed.
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