Sorites (Σωρίτης), ISSN 1135-1349
Issue # 20 -- March 2008. Pp. 125-134
Knowing that p rather than q
Copyright © by Bjørn Jespersen and Sorites

Knowing that p rather than q

by Bjørn Jespersen


In a number of papers Jonathan Schaffer has outlined a programme of an epistemological position called contrastivism and ably defended it. The idea informing contrastivism is that there is no such thing as knowing something simpliciter. In the final analysis, knowledge is always knowledge standing out against a foil of eliminated relevant alternatives. Schaffer's phrase for knowing that something is true while its relevant alternatives are not is `knowing that p rather than q'. In Schaffer's words, «The `rather than'-clause is a mechanism of contrastivity. It explicitly articulates q.» (2005, p.11.) For instance, it is wrong to say that you know that it is snowing, but correct to say that you know that it is snowing rather than raining, in case the proposition that it is raining is one of the eliminated epistemic possibilities.

In this paper I wish to investigate the cornerstone of contrastivism, the notion of knowing that p rather than q. I shall not discuss, for instance, whether knowledge always presupposes the elimination of epistemic possibilities. Instead I shall address the following two questions:

∙ What is the notion of contrast that is central to knowing that p rather than q?

∙ What sort of object is the object of knowledge in the case of knowing that p rather than q?

Neither question has so far received sufficient attention so as to provide an exhaustive answer. Or put more pointedly, just as Schaffer objects that, «No contextualist has ever offered anything near a precise account of relevance» (2004, p.88), so it may be objected that no contrastivist has ever offered anything near a precise account of knowing that p rather than q. Below I broach (without advocating it) an account of knowing that p rather than q in order to get a firmer grip on the notion. The alternative I broach construes knowing that p rather than q as knowing the conjunction of p and the negation of q. As may have already transpired, I have misgivings about this alternative. My primary reservation about it is that it is debatable whether a mere conjunction succeeds in accommodating the contrast between p and its alternatives that contrastivism revolves around. Whatever its merits and flaws, however, the conjunction account is incompatible with a key constraint that Schaffer imposes on knowing that p rather than q and, therefore, cannot help advance Schaffer's contrastivist programme. (This is not to say, though, that the account might not turn out to suit an alternative epistemological theory based around the notion of knowing that p rather than q.)

First, I set out the bare bones of contrastivism, in particular its so-called ternicity constraint. Then I look into the notion of contrastivity, arguing why I find it problematic to reconcile with the ternicity constraint. Finally, I argue that neither Schaffer's propositions nor my conjunction p ∧ ¬q is appropriate as an object of knowledge due to hyperintensionality concerns.

1. Ternicity

Two strands come together in contrastivism. One is the contextualist tenet that the truth-conditions of sentences containing reference to knowledge, «... know ...», are context-dependent because, according to contextualism, `know' is an indexical term. The other is the notion of relevant alternative. Schaffer rejects that `know' is an indexical and intends instead to obtain context-sensitivity by means of q: any of a range of relevant alternatives to p may be assigned to q. Relevant alternatives, in turn, provide the backdrop against which p stands out as a known truth: p is true independently of q, but cannot be known without q.Foot note 1

Contrastivism brings the two strands together by construing knowledge as a ternary relation between a knower s, a proposition p that is known by s, and a so-called contrast proposition q, which is a relevant alternative to p that has been eliminated. Where there are multiple relevant alternatives, the contrast propositions are the conjunctions q1∧...∧qn. The contrastivist conception of knowledge is summed up in the ternicity constraint (2004, p. 77).Foot note 2

Ternicity: `Know' denotes a three-place relation Kspq.

The contrastivist semantics of `know' gives rise to the following schema of the truth-condition of instances of «... know ...».

Kspq iff

(i) p is true

(ii) s has conclusive evidence that p rather than q

(iii) s is certain that p rather than q on the basis of (ii).

Apart from the slippery notions of conclusive evidence and certainty, the truth-condition as it stands lends itself to at least two very diverse interpretations known from epistemological contextualism (see DeRose, 1992, pp. 918ff; Brendel and Jäger, 2004, §3). According to subject contextualism, s's context, or vantage point, determines the truth-condition of Ksp, such that s must know that p is true, s must have conclusive evidence for the truth of p, and s must be certain of the truth of p in virtue of that evidence. According to attributer contextualism, the attributer's context, or vantage point, determines the truth-condition, since it is the attributer who selects the range of alternatives to p. (Of course, the roles of subject and attributer may coincide in the same individual.) In both cases it falls to s to eliminate the relevant alternatives to p in order to get to know that p relative to the given range of alternatives.

Schaffer's contrastivism qualifies as attributer contrastivism, which is in turn a species of epistemological externalism. By selecting a particular range of contrast propositions q1∧...∧qn the attributer imposes stricter or laxer constraints on what qualifies as conclusive evidence for p rather than its contrastive contenders. Due to the flexibility as for the choice of relevant alternatives to p, it depends on the choice of values to `saturate' q (2004, p. 90) which particular truth-condition a given instance of the `Kspq' schema is.

The `Kspq' schema is just that -- a schema. If we are going to turn the formula into a sentence, ∀-binding the q variables would seem the obvious choice, since knowing that p rather than q means having eliminated all of the q's. The variable `s' may be substituted by the constant `a'. The variable `p' should be replaced by a constant `P', since we obviously wish to express, not that a knows just any old proposition p rather than its alternatives, but that a knows some particular proposition P rather than its alternatives. When the schema is transformed into a closed expression, the result is

Ternicity* : ∀q (KaPq).

We can use Ternicity* to express that a knows that P rather than all of its relevant alternatives (where `∀q' is shorthand for `q1∧...∧qn'). We also need universal quantification over q when `keeping score of the overall progress of inquiry' across contexts (2004, p. 84): all the relevant alternatives to P have been eliminated, all the relevant alternatives to P' have been eliminated; etc. We would need `∀q (...q...)', `∀q' (...q'...)', etc., to express this.

2. Contrastivity

The preposition `rather than' is, according to the Oxford Advanced Learner's Dictionary, ambiguous between (a) `in preference to' and (b) `instead of'. An example of (a) would be, «Ann prefers tea to coffee», with the implication that with no tea around Ann would be fine with coffee. An example of (b) would be, «Ann drinks tea instead of coffee», which I find equivalent to (though emphatically not synonymous with) the conjunction, «Ann drinks tea, and Ann does not drink coffee». This conjunction can be telescoped into, «Ann drinks tea and not coffee». Whether `instead of' or `and not', the idea is one at the expense of the other.

It is plain that the phrasing of contrastivism in English prose employs (b) only. Without an explanatory caption, however, the phrase `to know that p rather than q' might just as well mean that s knows that p is true, but does not know that, or whether, q is true; of p, q it is p that s knows. What needs to be made explicit is that q has been eliminated instead of being simply neglected. This might suggest that a rendering in natural language of this particular contrastivist tenet is not readily available. (Perhaps «s knows that p as opposed to q» might be an alternative formulation.) Thus, I take it to be a stipulation on Schaffer's behalf that the `rather than'-clause is a mechanism of contrastivity in the exacting sense of involving eliminated relevant alternatives. In the remainder of this paper I adopt the contrastivist notion of contrast as expressed by `rather than'.Foot note 3

A first step toward implementing the contrast between p, q is to require that the p, q that occur in Kspq be mutually exclusive, or incompatible (cf. Schaffer, 2005, fn. 5; 2005, p. 13). That is, `RA' abbreviating `relevant alternatives':

p → ¬q, for any qRA(p)

or equivalently

q → ¬p, for any qRA(p).

However, I can think of three reasons why this way of cashing out the contrastivist's ternicity constraint would be problematic. First, said requirement remains exterior to `Kspq', showing it to be a shallow logical formalisation. Second, the effect is that since s, p, q are all in the scope of K, the knower gets saddled with the knowledge that p, q are incompatible. This seems to me to exceed the boundaries of contrastivism, which, as I understand the position, does not require that the knower know about this relation between p and any of its relevant alternatives. If the contrastivist wishes to maintain that the propositions are incompatible without s knowing, the contrastivist needs to show how while preserving the basic ternary form `Kspq'. (Some form of the distinction between knowledge de dicto and knowledge de re might perhaps come in handy.) Third, and most importantly, as I understand the ternary Kspq, K needs to be a relation between a knower and two propositions that are, at the very least, ordered in a sequence. The first element of the sequence would be the selected proposition, the second element the eliminated proposition. What s knows is that p rather than q, not that q rather than p. Arranging p, q as an ordered pair makes it explicit that the order in which they occur matters. The result is

K<s, <p, q>>.

But the pairing of p, q makes K binary, since the arguments of K are now an individual and an ordered two-tuple. This is in flagrant breach of Ternicity. Since p, q need to be ordered as p, q, it would appear that the outcome of the quest for contrast between p, q inherent in knowing that p rather than q is a binary K rather than a ternary.

I am not sure how to square Ternicity with the required contrast between selected and eliminated propositions. Instead, I wish to suggest an alternative conception of contrastive knowledge. Consider the above example, «Ann drinks tea instead of coffee», which was unwrapped as, «Ann drinks tea, and Ann does not drink coffee». It will receive this formalisation in propositional logic.

p ∧ ¬q.

If s knows that Ann drinks tea rather than coffee, a first stab at an analysis would be

Ks (p ∧ ¬q).

That is, s knows that the conjunction of p and the negation of q is true.Foot note 4 To arrive at this attribution of knowledge to s, the attributer would have to carry out an instance of the following inferential schema.

. . .
[1] p → ¬q Assumption
[2] p Assumption
[3] ¬q 1, 2, MPP
[4] p ∧ ¬q 2, 3, ∧I
[m] Ks (p ∧ ¬q)..., KsI

KsI is an introduction rule for Ks validating the prefixing of `Ks' to line [4]. The details of the introduction rule need not detain us here, but (ii) and (iii) from Schaffer's statement of the truth-condition of Kspq would have to be in the mix. My counterpart of Ternicity* then is

qRA(P) (Ks (P ∧ ¬q)).

There are various ways of reading this formalism; for instance, «The conjunction of P and the negation of all of its relevant alternatives q1∧...∧qn is known by s», or «For any relevant alternative q to P, s knows that P is true and no q is». However, my concern is not so much the most accurate interpretation of the formalism as what it would mean to know that the conjunction of p and ¬q is true (for arbitrary p, q). For instance, does it mean to know of either of the conjuncts that it is true? If so, the following distribution qualifies as valid.

Ks (p ∧ ¬q) |= (KspKs¬q).

Knowing one thing, namely that the conjunction p ∧ ¬q is true, implies knowing two things, namely that p is true and that ¬q is true.Foot note 5 However, it is problematic that the consequent severs the relation between p and ¬q (or q) that is supposed to provide the contrast between the selected proposition and its rejected alternative(s). «KspKs¬q» just says that s knows two things, with no indication of their being related.

Does the antecedent also sever the relation between p and ¬q (or q)? Why, no; ∧ is, after all, a (binary) relation, its arguments being p and ¬q in the present case. But it is to no avail to explain the notion of knowing that p rather than q in terms of knowing p ∧ ¬q as long as it is not clear what it means to know a conjunction to be true or whether knowing that p rather than q can be adequately translated into knowing the conjunction of p and ¬q in the first place. In fact, what especially speaks against the conjunction interpretation of contrastivity is that the mere conjunction of two propositions lacks the biff of contrast between its conjuncts. The conjunction, if true, just records the fact that p is true and q is false.Foot note 6

Another contentious issue is the fact that elimination translates into negation. This translation probably captures the core idea of elimination, but records only the outcome, or exterior, of an act of elimination. The procedural, or interior, aspect of carrying out acts of elimination drops out of the story. This may well be unsatisfactory for a full statement of contrastivism, since processes of eliminating p's rivals conceptually precede s's knowledge of p. A full statement of contrastivism would then maintain that s sknows that p rather than q, because s (or s's evidence) has already eliminated q.

For these reasons I am hesitant to put `∀qRA(P) (Ks (P ∧ ¬q))' forward as an analysis of the notion of knowing something rather than something else.Foot note 7

It should be obvious why Schaffer could not possibly accept the conjunction analysis. Ks (p ∧ ¬q) violates Ternicity by being binary, and his q is my ¬q. What is more, should the implication Ks (p ∧ ¬q) |= (KspKs¬q) be deemed true, the right-hand conjunct Ks¬q of the consequent is too strong. It says that s actually knows something (namely that a certain negation is true) whereas Schaffer carefully settles for the less exacting notions of the subject having `conclusive evidence' for and being `certain' about p rather than q when stating his truth-condition for Kspq. This is a natural move, since knowing that ¬q would have to be spelt out as knowing that ¬q rather than some r. And ¬r would in turn have to be known rather than something else. And so on, with an infinite regress looming on the horizon that would turn contrastivism into an unmanageable theory of human knowledge.

3. Objects of knowledge

Even if a contrastivist should be prepared to flout Ternicity, there is an additional reason to be wary of

«∀qRA(p) (Ks (p ∧ ¬q))»

(for arbitrary p) as a workable analysis of knowledge á la contrastivism.

The reason is that, whether or not this analysis should be found satisfactory, the following problem needs to be addressed: is (p ∧ ¬q) appropriate as an object of knowledge? I think not. I am going to argue that neither Schaffer's propositions nor my conjunction (p ∧ ¬q) is appropriate as an object of knowledge.

First of all, what are p, q? Schaffer introduces p, q as propositional variables, construing propositions in the vein of possible-world semantics, to wit, as sets of possible worlds, or functions from the logical space of possible worlds to truth-values. The problem with this sort of propositions is that they are arguably both `too little' and `too much'. It is a thrice told tale that, and why, they are too little. They fail to accommodate a principle of individuation finer than logical equivalence. Cresswell in (1975) coins the phrase `hyperintensional' to characterize any two intensions that are logically equivalent yet distinct; Bealer speaks of a `conception 2 intension' as what is intended when entertaining an attitude (1982, p. 166). For instance, one thing is to know that Seoul is south of Pyongyang; quite another that Pyongyang is north of Seoul. Yet possible-world propositions cannot distinguish between inverse relations. Nor can they distinguish between any two necessarily true propositions, or between any two necessarily false (`impossible') propositions, since there is only one necessary proposition (the one true at all worlds) and only one impossible proposition (the one false at all worlds). Also, due to the classical definitions of the truth-functions, Ks (pq), for example, would come out identical to Kspq). Yet, intuitively, s may be innocent of the equivalence between pq and ¬pq, and so might know the former without knowing the latter. Schaffer notes that it is problematic using possible-world propositions as objects of knowledge, but proposes no remedy (2007, n. 28). The moral, I submit, is that it is not rewarding to construct either philosophical epistemology or epistemic logic around such propositions, because they are too crude to figure as objects of knowledge. The set-theoretic intensions of possible-worlds semantics are intensionality on the cheap.Foot note 8

But there is also a sense in which they are too much by being too fine-grained. Though extensionally individuated, they are still intensional entities and as such inappropriate as arguments for the truth-functions. For instance, in


the truth-function → does not operate on intensions. Instead it operates on the extensions of p, q, trading two truth-values for a third truth-value in accordance with its truth-table. In the truth-table the material implication pq is nothing other than a truth-value, which can figure as argument for other truth-functions, as in ¬(pq) ∨ r.

On the other hand, in


the arguments of the intensional relation of entailment, ⇒, are propositions, or truth-values-in-intension, i.e., p, q themselves. (Entailment takes two propositions, or a set of propositions and a proposition, and yields a truth-value.) Yet standard notation fails manifestly to flag the difference between intensions and their extensions.Foot note 9 This sin of omission catches up with us as soon as we prefix `Ks' to, e.g., `p ∧ ¬q' to form

«Ks (p ∧ ¬q)».

For if p ∧ ¬q is a truth-value then s knows that a certain truth-value is true. But truth-values cannot be known to be true, as little as a kettle of fish can. The standard escape is semantic contextualism: in such-and-such contexts `p', `q', «pq», etc., denote truth-values, while in such-and-such other contexts they denote propositions. Yet this makes the notation `p', `q', «pq» ambiguous, and the notion of ambiguous logical notation runs counter to the very idea of introducing logical notation in the first place.

All the same, let it be granted that, for instance, in «Ks (pq)» the complement `pq' denotes the intension of pq so that s knows that pq is a true proposition. This still will not do, though, exactly because of the crude individuation of p, q. The moral is that it is not going to be enough to sort out the contrast between p, q and then simply prefix some epistemic operator to whatever is the result. There is a leap from extensional and possible-world intensional contexts to hyperintensional contexts that cannot be neglected either conceptually or notationally.


Schaffer's contrastivist programme lacks as yet a theory of appropriate objects of knowledge. Whatever theory may be advanced to fill the lacuna, it seems nonnegotiable that the objects would need to be hyperpropositions. Another nonnegotiable requirement would seem to be that the internal structure of the contrastivist hyperproposition p_rather_than_q must accommodate a contrast between the selected p and the eliminated q. The fundamental problem, however, remains how to reconcile Ternicity with the fact that p_rather_than_q is one object rather than two.Foot note 10


Bjørn Jespersen
Section of Philosophy
Delft University of Technology
Delft, The Netherlands
B.T.F.Jespersen [at]

[Foot Note 1]

For a recent objection to the contextualist semantics of `to know', see Douven (2004). Roughly, Douven's argument is this. If `to know' is an indexical term, then it has context-sensitive comparatives and superlatives, as expressed by «Ann knows more than Berthold» and «Ann knows most of everybody». But it hasn't, so it isn't.

[Foot Note 2]

Schaffer adds that the K relation ought to be expanded to include a fourth variable t ranging over instants of time (2004, p.95, n.8), though leaving t out for convenience.

[Foot Note 3]

Schaffer argues that there is a linguistic analogy between `to prefer' and `to know', in that both verbs are, in the final analysis, ternary (2005, §3). According to Schaffer's analogy, just as the sentence «Ann prefers tea» is elliptical for «Ann prefers tea to F» (one substitution instance of which being, «Ann prefers tea to coffee»), the sentence «Ann knows that p» is elliptical for «Ann knows that p rather than q». Obviously, if Ann prefers tea to coffee and coffee to root beer then, by the transitivity of the preference relation, Ann prefers tea to root beer. There is a ranking with tea at the top, coffee in the middle and root beer at the bottom. But Schaffer's analogy between `to prefer' and `to know' is easily overstretched. If Ann knows that p rather than q we cannot introduce an r such that Ann knows that q rather than r, and therefore knows that p rather than r. Strained grammar aside, there is no such thing as knowing that p rather than q rather than r, with p at the top, q in the middle and r at the bottom.

[Foot Note 4]

Schaffer claims (personal communication) that an analysis in terms of Ks (p ∧ ¬q) results in what Keith DeRose calls an `abominable conjunction' in his (1995, pp. 27ff). In general, the conjuncts of an `abominable conjunction' are that some skeptical hypothesis is not known to be false and that some run-of-the-mill proposition is known. The following analysis translates the assumptions (1) and (2) into (1') and (2').
1. Moore knows that he has hands rather than stumps
2. Moore does not know that he has hands rather than vat images of hands
1'. Km(H ∧ ¬S) Assumption
2'. ¬Km(H ∧ ¬V) Assumption
3. KmHKm¬S 1', Distribution
4. ¬KmH ∨ ¬Km¬V 2', Distribution, DeMorgan
5. KmH 3, ∧E
6. ¬Km¬V 4, 5, Disjunctive syllogism
7. KmH ∧ ¬Km¬V 5, 6, ∧I.

The premises (1'), (2') are contrastive intuitions to the effect that (1) and (2) are true. The conclusion is the `abominable conclusion' that Moore does not know that he is not a (bodiless, hence handless) brain in a vat, but does know that he has hands. The rationale behind the transition from (2') to (4) would be that if you fail to know a given conjunction to be true it is because you fail to know at least one of the conjuncts to be true. However, the argument is in my opinion invalid, by deploying Distribution of K over ∧ twice to obtain two pairs of conjuncts each of which with a K at the head. K Distribution is, in my view, a clearly invalid rule in any properly restrictive epistemic logic. So no such `abominable conclusion' is validly inferable from the proposal presented above. (I am indebted to the Editor for discussion of this `abominable conjunction'.)

[Foot Note 5]

Evnine embraces this outcome, arguing that «being in a state of believing a conjunction simply is being in a state of believing its conjuncts. There is no state of believing p and q, distinct from the state of believing p and believing q.» (1999, p.215. I replaced `A', `B' by `p', `q'.) Unfortunately, Evnine does not reveal how to individuate such `states' (which I suppose to be the same for believing and knowing), so it remains unclear how strong or weak his claim is. Even so I would have to be convinced that this identity claim is true: Ks (pq) = KspKsq. I am not fond of ∧ flitting in and out of the scope of K. For one thing, it is inferable from KspKsq, via ∧E, that Ksp (or Ksq, if you wish), but not inferable from Ks (pq) without first factoring out the two conjuncts and placing them under K scope. And it is not logically necessary that knowers be capable of extracting individual conjuncts from a conjunction they know. (Indeed, in pragmatic terms it would be downright irrational for resource-bounded knowers to do so with each and every conjunction known to them.) So it is doubtful whether Ks (pq) is even equivalent to KspKsq.

[Foot Note 6]

This problem will be familiar from the translation of `but' into `∧' when used to conjoin propositions that contrast. For instance, if «Ann is poor but honest» (a telescoped phrasing of «Ann is poor, but Ann is honest») goes into «FaGa», the contrast between Ann being poor and Ann being honest conveyed by `but' is lost in translation. A consequence of the contrast is that «Ann is poor» and «Ann is honest» do not commute in «... but ...». See also (Blakemore, 2000).

[Foot Note 7]

Consider the following as an alternative interpretation which also articulates q: s knows p rather than q iff p and (s knows whether p or q). The idea underlying knowing whether p is to know which disjunct (possibly both) of pq is true. In rough notation (x ranging over propositions):

s knows whether p iff Ks ιx (x ∧ ((x = px = q) ∨ x = pq)).

ι is a function from sets of propositions to propositions that returns the only member of a singleton, and is otherwise undefined. Its value is the respective unique proposition that is known when it is known whether p or whether q or whether pq. This alternative is not sufficiently strong, however, since when p rather than q is known, q cannot be true. Thus, we need to add that not both disjuncts are true. Still, as with the alternative in terms of Ks (p ∧ ¬q), I fail to see how the K of p ∧ (Ks (whether p or q)) would meet the Ternicity constraint.

[Foot Note 8]

It is interesting to note that Fred Dretske -- contrastivist avant la lettre -- implicitly presupposes hyperintensional objects of knowledge (cf. 1970, pp. 1022-23). He says, «We have subtle ways of changing [...] contrasts and, hence, changing what a person is said to know without changing the sentence that we use to express what he knows.» If the neutral sentence is «Lefty killed Otto», the different things a person can be said to know can be spelt out by means of what is in effect topic/focus articulation: «It was Lefty who killed Otto», «It was Otto whom Lefty killed», and «What Lefty did to Otto was kill him». All three sentences are associated with `the fact that Lefty killed Otto', but Dretske maintains that «in knowing that Lefty killed Otto [...] we do not necessarily [...] know that Lefty killed Otto [...].» It is this differentiation that calls for hyperintensional objects of knowledge.

[Foot Note 9]

I suspect the historical culprit for this sort of notation must be the conception of modalities due to possible-world semantics, which treats `□', `◊' as being syntactically on a par with `¬'; both «¬p» and «□p» are well-formed formulae. This makes for handy notation, but it remains implicit that the argument of ¬ is a truth-value of p and the argument of □, p itself, i.e., the entire function. If `K' is introduced as a notational variant of `□' we get formulae like «Kp», and we are allowed to generate strings like, «¬pK¬p», where the extension/intension ambiguity of the notation is manifest. Moreover, if K is a hyperintensional operator, and □ an intensional operator, then we are in for three-way ambiguity as in, «(□pp) ∧ Kp».

[Foot Note 10]

The material presented herein is based on my reply to Jonathan Schaffer's talk `Contrastive closure and answers', which was delivered at Vrije Universitiet, Amsterdam, in October 2004 and published as Schaffer (2007). Thanks to Jonathan Schaffer for follow-up discussion via email.

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