The speech act of denial and the propositional attitude of rejection have gained renewed importance in the philosophy of logic. One reason for this comes from the literature on truth. One initial reaction to the liar paradox is to say that the liar sentence,
(λ) This sentence (i.e. λ) is false,
is gappy, that is, neither true nor false. The solution works just fine for this particular case of the liar but when we consider the strengthened liar,
(λ∗) This sentence (i.e. λ∗) is not true,
we land right back in paradox, assuming the usual properties of truth, namely that
A (TI)
T"A#
T"A# (TE)
A
hold unrestrictedly (i.e. for arbitraryA), where"A# is a quote name or G¨odel number forA and consequence is classical.23 To see this, assume T"λ∗#. By
TE we have¬T"λ∗#, whence ¬T"λ∗# follows by¬I from no assumptions. It
follows byTI and the definition ofλ∗ thatT"λ∗#. By ¬I and double negation
23The rules are sometimes labeledT-IN andT-OUT respectively, but I see no reason to
break standard rule-naming conventions in this particular case even if an application ofTE results in a conclusion containing more occurrences ofT than occurs in the premise.
elimination we obtain T"λ∗# from no assumptions. Assuming any logic for
which EFQ holds in the formA,¬A2B, triviality ensues.
The gap theorist is still open to holding thatλ∗is gappy but if she does so she will have to adjust either her logic or else her truth principles accordingly in order to block triviality. One popular approach is to take up the former by weakening one’s logic, for example, by adopting a gappy logic, a move that seems natural for a gap theorist to make. However, this sort of approach, of which Kripke’s is a prime example, is open to aprima facie difficult challenge. For since the gap theorist admits thatλ∗is neither true nor false, whatλ∗says
is the case. The problem is that ‘is the case’ must mean something different from ‘is true’ and it is difficult to see what that might be. One reading is that it means ‘is assertible’. But then the gap theorist must hold thatλ∗ is assertible while knowing at the same time that it is not true. Such a concession flies in the face of the usualnon-pragmatic conception of assertibility according to whichAis unassertible bySif it is regarded bySto be untrue, a condition that is considerably weak as far as constraints on assertibility and unassertibility go. (Notice that this is not equivalent to what might at first glance appear to be its contrapositive, viz. “If A is regarded by S to be true, then A is assertible”. Indeed this would-be contrapositive is implausibly strong though not clearly on a non-pragmatic view of assertibility according to which that notion is constrained by rationality rather than pragmatic factors. For thatA is assertible does not imply that itought to be asserted.)
The gap theorist typically responds to this challenge by claiming that her utterance ofλ∗isnotan assertion of that sentence (or the proposition expressed
by that sentence, supposing that assertions act on propositions rather than sentences, but more on this just below), rather it is a denial of that sentence (or, again, the proposition expressed by the sentence). The gap theorist now has to take each assertion and denial as primitive, where the “Fregean” needs only one (and indeed, it does not matter which though Frege took assertion as the primitive speech act) by holding that a denial that A is an assertion
that ¬A.24 In other words, the gap theorist has to deny that there is a close connection between negation and denial.
Now positing two speech acts as primitive instead of just one of them is not necessarily a problem provided doing so is grounded in reasons independent of merely addressing the above challenge. Otherwise the positing of the additional primitive looksad hoc; there has to be some worthy additional payoff for the theoretical cost of positing the additional primitive. The gap theorist can avoid ad hoceryby endorsing the very same theory of assertion and denial the Fregean endorses under some intuitive notion ofsameness. For instance, each can hold that
(Assert) Ais assertible iffAis true;
(Deny) Ais deniable iffAis untrue.
But since a sentence is not untrue iff its negation is true according to the gap theorist, Deny can no longer be seen as a special instance of Assert and this explains why the two need to be taken as primitive for the gap theorist.
However, Deny is extremely implausible for a gap theorist to hold, at least when taken to hold for arbitrary sentences. One reason that a sentence may be untrue is because it is not truth-apt, for example, it might be a command such as ‘Take out the garbage’. If denials act on propositional contents which are themselves necessarily truth-apt, then it makes no sense to deny the content of ‘Take out the garbage’ since whatever that content is, it is not a proposition. Moreover, even if we restrict ourselves to truth-apt sentences, Deny still seems implausible according to some widely-defended gap theories as the following considerations bring to light.
Consider the gap theorist who denies thatλ∗ is either true or false because
it fails to express a proposition. Such a gap theorist cannot deny thatλ∗ since,
again,λ∗ fails to express a proposition and denials act on propositions. Where
24Frege famously held the view that the denial thatAjust is the assertion that¬A. See
there’s no content, there’s no denial. Unless denials act on something other than propositions or unless a denial of λ∗ is not a straightforward denial of
the content ofλ∗but the denial of some other proposition, e.g. the proposition thatλ∗ is true, the gap theorist will have to reject Deny in favor of something like the obvious variant:
(Deny#) Ais deniable iffAis false.
I think this is the most intuitive and plausible position for the gap theorist to take, though I consider the aforementioned options—that is, the idea that a denial ofAmay involve more than just the content ofA—in chapter 4, section 4.2 and find them wanting.25
With an endorsement of Deny#comes the need for a distinction between two notions of consequence, viz. consequence taken as (i) the preservation of truth from premises to conclusion and (ii) the preservation of falsity from premises to conclusion. In classical logic each is definable in terms of the latter for if the truth of each member of Γ implies the truth of A then at least one member of Γ is false if A is false. Likewise if the falsity of each member of Γ implies the falsity of A then the truth ofAimplies the truth of at least one member of Γ. In other words, classical logical truth preservational consequence anti- preserves falsity and its converse preserves falsity. However in a gappy logic (more precisely a gappy semantics, though we may think of logics as coming with a semantics, and we shall do so sometimes for convenience) each of the two notions of consequence is not definable in terms of the other. For consequence as truth preservation anti-preserves untruth, not falsity, the notion figuring centrally in Deny#.
As it turns out there are a variety of speech acts and propositional attitudes other than assertion, acceptance, denial and rejection that may be character- ized or rationally constrained by various notions of consequence distinct from 25One standard objection to thinking that the denial ofλ∗is the denial of the content of
T λ∗is that the attribution of different properties ofλ∗ are being denied in the respective denials ofλ∗andT λ∗—in the former it is untruth and in the latter it is truth. They must therefore be different acts.
consequence as truth preservation. One of those is doubt or dubitability. Con- sider a three-valued semantics where the values correspond to truth, falsity and some notion ofindeterminacy. Then one might think the following characteri- zation of dubitability ought to hold:
(Doubt) Ais dubitable iff Ais indeterminate
whereAmay be restricted to truth-apt sentences. This suggests the following principle concerning the rationality of doubting:
(DC) If each member of Γ is indeterminate implies that A is indeterminate thenAis dubitable if each member of Γ is.
Clearly the principle can be formulated only if we have at our disposal a notion of consequence as indeterminacy preservation. More importantly, we cannot hold people hostage to the principle if there are no rules they can follow which take them from indeterminate sentences (or whatever the truthbearers are, primarily or derivatively) to other indeterminate sentences. We should be able to hold someone as being rationally irresponsible for doubting each member of Γ on the grounds of their indeterminacy while believingAif the indeterminacy of each member of Γ implies the indeterminacy ofAjust as we would hold them account if they believed each member of Γ on the grounds of their being true while rejectingAeven though Γ impliesAin the truth-preservational sense.
With a marked separation of negation and falsity comes a need for distin- guishing various notions of consequence and the role they play in rationally constraining various speech acts and attitudes. Various of these relationships are explored in chapter 4 with a focus on consequence as indeterminacy preser- vation and its relation to the attitude of dubitability. A natural deduction system for indeterminacy and falsity preservation over Kleene’s matrixK3 is given and proved sound and complete with respect to the class ofK3matrices.