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A critique of Cartesian knowability.

1. The Cartesian Knowability Conflict Thus Far

According to the anti-realist all truths are knowable by cognitive agents like us (or with finite extensions of our cognitive powers). Where K denotes the property of being known such an agent at some time, [for all][phi] ([phi]9 [right arrow] [??]K[phi]). But Frederick Fitch has seemingly shown that given two rather apparent epistemic assumptions (lines 2 and 3 below) this principle entails that all truths are known. [Fitch 1963: 135-142]
1.) [phi] [right arrow]           Anti-Realism
  [??]K[phi]

2.) K[phi] [right arrow] [phi]    KIT, "Knowledge Implies
                                    Truth"

3.) K([phi] & [phi]) [right       DOK, "Distributivity of
  arrow] (K[phi] & K[phi])          Knowledge"

4.) K([phi] & [logical not]K      Instance of 2
  [phi]) [right arrow] ([phi]
  & [logical not]K[phi])

5.) K([phi] & [logical not]K      From 4 by Propositional
  [phi]) [right arrow] [logical     Logic
  not]K[phi]

6.) K([phi] & [logical not]K      Instance of 3
  [phi]) [right arrow] (K[phi]
  & K[logical not]K[phi])

7.) K([phi] & [logical not]K      From 6 by Propositional
  [phi]) [right arrow] K[phi]       Logic

8.) [logical not]K([phi] &        From 5 and 7, by Reductio
  [logical not]K[phi])

9.) [logical not][??]K([phi] &    By the necessity of 2 and 3
  [logical not]K[phi])              and derivation of 4-8

10.) ([phi] & [logical not]K9)    Substitution of ([phi] &
  [right arrow] [??]K([phi] &       [logical not]K[phi])
  [logical not]K[phi])              into 1

11.) [logical not]([phi] &        From 9, 10 MT
  [logical not]K[phi])

12.) [phi] [right arrow]          From 11, by an equivalence
  K[phi]                            in classical logic.


In intuitionistic logical systems typically favored by antirealists, 12 does not follow from 11. However, one could infer that "All unknown propositions are false."
13.) [logical not]K[phi] [right   11, by implication in
  arrow] [logical not][phi]         intuitionistic logic


And either result is devastating. Notice that on the assumption that [there exists][phi] ([logical not]K[phi] & [logical not]K[logical not][phi]) contradictions can be derived on either 12 or 13.

Tennant [1997] proposed that the anti-realist principle be restricted to truths that are "Cartesian," that is, propositions that may be known without inconsistency: [[phi] & [logical not](K [phi] [??] [perpendicular to])][right arrow][??]K[phi]. The turnstile [??] stands for "logico-mathematical deducibility." The absurdum [perpendicular to] applies to three cases: those K[phi] where [phi] is itself inconsistent, cases in which inconsistency results from iterations of K operators (e.g. Fitch's Paradox) and existentially inconsistent propositions such as "No knowers exist," that is, propositions that cannot become the objects of propositional attitudes of any agent. The third of these apparently explains the "Cartesian" nomenclature. Step 10 above fails since knowing that ([phi] & [logical not]K[phi]) proves [perpendicular to] as was shown by line 9. He elsewhere elaborates that he neither regards this as the only recourse for the antirealist nor regards the Fitch proof as a crisis for antirealism. [Tennant 1997: 260-261] [Tennant 2001: 108-109]

Tennant's revision has not enjoyed the warmest reception. Williamson opines that "Tennant's rule looks desperately ad hoc." [2000: 109] And Hand and Kvanvig argue that it is "obviously ad hoc" because it revises a theory of truth, not on the basis of an antirealist insight into the nature of truth thus conceived, but merely in order to dodge a paradox. [1999: 422-423] It seems akin to claiming "...antirealism holds for all propositions except the counter-examples." [1999: 425] But Tennant replies that such restrictive dodges are not uncommon to philosophy, that he has offered other considerations in favor of the restriction and perhaps most importantly that the principle is not a trivialization of the realist/anti-realist debate.
   The realist is willing to assert that even among the Cartesian
   propositions, there could well be propositions that are true, but
   whose truth might in principle elude detection, whether by us or by
   any finite extensions of ourselves. This is a modal claim
   (adverting to possible existence), and of course it conflicts with
   the correspondingly modalized claim of the anti-realist: that it is
   necessary that all propositions whose being know is not provably
   inconsistent will, if true, be knowable. [2001a: 112]


Williamson, on the other hand, worries that Tennant's revision fails to solve Fitch's Paradox. He urges that 9 & (K[phi][right arrow]En) is Cartesian where [phi] stands for the proposition "There is a fragment of Roman pottery at that spot" (assuming a suitable context), E is the predicate "is even" and n rigidly designates the number of books currently on his table. Does a story involving K[[phi] & (K[phi][right arrow]En)] contain an inconsistency?
   ...the story is obviously consistent. Of course, n may in fact be
   odd, in which case, since that number could not have been even, the
   story expresses an impossible state of affairs. Nevertheless, the
   story itself is still consistent; one cannot discern by reason
   alone that the description which fixes the reference of 'n' picks
   out an odd number. Someone who asserts En because he failed to see
   one of the books is not guilty of an inconsistency. A more liberal
   interpretation of inconsistency might trivialize... (Tennant's
   principle). [2000: 110]


Since Tennant's favored logical system allows that [logical not][alpha] [??] [alpha][right arrow][beta], we may thus deduce...
1.) [phi] & [logical not]K[phi]       Assump.

2.) [phi] & (K[phi][right arrow]En)   1, by theorem [logical not]
                                        [alpha] [??] [alpha][right
                                        arrow][beta]

3.) [logical not]{K[[phi] & (K[phi]   Cartesianhood of 2
  [right arrow]En)] [??]
  [perpendicular to]}

4.) [MATHEMATICAL EXPRESSION NOT      3 and Tennant antirealism
  REPRODUCIBLE IN ASCII]

5.) K[[phi] & (K[phi][right arrow]    By DOK and KIT
  En)] [??] K[phi]

6.) K[[phi] & (K[phi][right arrow]    By DOK and KIT
  En)] [??] K[phi][right arrow]En

7.) K[[phi] & (K[phi][right arrow]    6, 7 and Modus Ponens
  En)] [??] En

8.) [MATHEMATICAL EXPRESSION NOT      7, possibility closes over
  REPRODUCIBLE IN ASCII]                entailment

9.) [MATHEMATICAL EXPRESSION NOT      Proof of 2 from 1, and 4 and 8
  REPRODUCIBLE IN ASCII]

10.) [phi] & [logical not]K[phi]      9, by necessity of number
  [??] En                               properties


However, Williamson notes that he could just as well have used the predicate [logical not]E throughout this argument in the place of E. So an analogous argument would prove...
11.) [phi] & [logical not]K[phi]      Proof substituting [logical
  [??][logical not]En                   not]En for En 1-10


And thus...
12.) [logical not]([phi] &            10, 11 Reductio
  [logical not]K[phi])


Hence, Tennant's antirealism conflicts the assumption of an unknown truth in line 1.

Tennant complains that Williamson's argument turns upon two assumptions of Cartesianhood. Neither K[[phi] & (K[phi][right arrow]En)] nor K[[phi] & (K[phi][right arrow][logical not]En)] seem to entail [perpendicular to]. Yet at least one of these two propositions will be non-Cartesian. Given the rigid designation of a number n, En and [logical not]En are polar propositions; that is, they are such that if possibly true, they are necessarily true and if possibly false, they are necessarily false.
   Any provable or refutable mathematical proposition is polar. So
   too--if we countenance metaphysical necessity and
   impossibility--are propositions such as 'Water is [H.sub.2]O' and
   'Water is XYZ.' Not all polar propositions are effectively
   decidable. Moreover, if we countenance metaphysical necessity and
   impossibility, then not all polar propositions' truth values will
   be determined a priori... Which propositions are Cartesian in the
   actual world will depend, in general, on which polar propositions
   are true. Recall that i|/ is Cartesian just in case [logical
   not](K[psi][??][perpendicular to]). To say that absurdity is not
   derivable from K[psi] is equivalent to saying that absurdity is not
   derivable from K[psi] in conjunction with any set X of necessarily
   true propositions. Whether this definition calls for the
   consideration of only sets X of all whose members are knowable a
   priori, or calls for the consideration also of sets X some of whose
   members might be knowable only a posteriori, is an issue of
   principle on which we are not at present forced to take a stand.
   [2001b: 269]

      For polar [theta], it is only when [theta] is true that [psi] &
   (K[psi] [right arrow] [theta]) is Cartesian. [2001b: 271]

      It should be clear to anyone with a sympathetic understanding of
   the spirit of the proposed restriction that for a proposition to be
   Cartesian one ought to be unable to derive absurdity from it modulo
   any necessarily true propositions. It is a logical convention of
   long standing that mention of theorems as premises can be
   suppressed. [2001b: 264]


Thus, Williamson cannot instantiate to both line 3 of the reductio above and to its proposed analogue because one will inevitably violate this "polarity" restriction.

Williamson responds that he was entitled to each of these lines in his proof; neither generates a logico-mathematically inconsistency. He invites the reader to "Just try!" to derive a contradiction. [2010: 199] He marshals the quotes above to show that "Cartesianhood" is being revised and that the revision trivializes Tennant's principle; for surely one of the two claims K[phi] & (K[phi][right arrow]En)] or K [[phi] & (K[phi][right arrow][logical not]En)] conflicts with a metaphysically necessary truth. Apparently now the turnstile [??] in the account of Cartesianhood refers to derivability in a system where every necessary truth is treated as a theorem (this seems explicit in the quotes above). Roughly, Tennant means that [phi] is Cartesian if and only if [therefore][PHI][right arrow][logical not][([PHI] & K[phi]) [??] [perpendicular to]]. But in classical logic, all propositions with this property are knowable. Suppose 9 is Cartesian. Substituting [logical not]K [phi] for [PHI] the forgoing formalization of Cartesianhood will quickly prove [logical not][therefore][logical not]K [phi]. So if [phi] is Cartesian, [logical not][therefore][logical not]K [phi]. And in classical modal systems this is equivalent to [??] K [phi]. The fact that all Cartesian [phi] are knowable is a theorem of logic. This "trivializes" anti-realism by rendering it tautological (classically). [see Williamson 2010: 202]

Tennant's response takes aim at Williamson's "concocted" number n. If n is a rigid designator and neither story harbors a mathematical inconsistency, then Williamson is committed to the claim that there exists some natural number (of books on his desk) that is both odd and even and thus a conflict with arithmetic. "Apparently Williamson did not take seriously enough his claim that n was the name of a number." [Tennant 2010: 17] Second, Tennant complains that Williamson's reinterpretation of Cartesianhood of a proposition [phi] as [therefore][PHI][right arrow][logical not] [[PHI] & K [phi]) [??] [perpendicular to]] as was the result of an uncharitable reading. He hightlights portions of one of the earlier quotes for purposes of clarification. Although in the quotes above Tennant explicitly states that "for a proposition to be Cartesian one ought to be unable to derive absurdity from it modulo any necessarily true propositions" he clarifies this point by stressing other portions of that quote.
   It is a logical convention of long standing that mention of
   logico-mathematical theorems as premises can be suppressed when
   exploring the logical relations that hold among contingent
   statements. [2010: 18]


Accordingly, the "necessarily true propositions" to which he was referring were propositions such as "No natural number is both odd and even" and the contingent statements in question are those such as "The number of planets is 9" and "The number of planets is even." A similar point applies where the number in question is a rigidly designated number of books on Williamson's desk.

These clarifications aside, however, Tennant concedes that Williamson's proof would have fared better had he used an identity claim of the form x = y in place of En to have avoided the relevance of these clarifications. But, Tennant adds, this would merely force him to define logico-mathematical consistency in terms of logical systems that include the Kripkean theorem that [for all] x [for all] y ([therefore] x = y [disjunction] [therefore] [logical not]x = y). Again, if Williamson asserts the Cartesianhood of both line 3 of his proof above modified in terms of identity claims and its analogue, then he will be committed to a contradiction with Kripkean modal logic.

I will pause here to assess the debate so far. First, of all, if Cartesianhood needed to be redefined in terms of consistency with all metaphysically necessary truths in order to avoid Williamson's challenges (or arguments like them), Tennant nonetheless maintains that the resulting anti-realist principle still holds interest [2010: 19]. In light of the derivability of [therefore][PHI][right arrow] [logical not] [([PHI]& K [phi]) [??] [perpendicular to]] in classical logic, it is surely hard to see how this could be the case. Yet Tennant thinks that it is "obvious" that the anti-realist should invite this revision if necessary. [2010: 19] So the first item that I will reexamine is whether Tennant will be forced to such a potentially trivializing reformulation in order to block Williamson-style challenges. Further, Tennant thinks that "the anti-realist will not have to take a stand" on this due to the availability of a new principle of Cartesian knowability [2010: 20, 23]. So secondly I will examine this new principle and see if it is "Fitch-free" as advertised or whether Williamson's objection regarding trivialization can be resurrected there as well. The new restrictive principle on knowability shares with the old an emphasis on logico-mathematical consistency. There is, then, initial reason to wonder if a broader form of consistency, consistency with all metaphysical truths, will have to be invoked there as well. Let us take each of two these issues in turn.

2. Tennant, Truth and Tautologies

Tennant concedes that if Williamson had couched his reductio in terms of identity claims then it "might have some purchase" and that it is "puzzling in retrospect" that he appealed to numbers and number properties instead. [2010: 21 and 20 respectively] Still, as we saw, he thinks such an alternative maneuver can be blocked. I think a revised form of Williamson's proof will have more purchase than Tennant has noted. I will offer two brief arguments for the conclusion that Tennant's Cartesian restriction will require the revision proposed by Williamson. First, the Kripkean theorem [for all]x[for all]y ([therefore] x = y [disjunction] [therefore] x [not equal to] y) blocks an analogue to Williamson's original attack. But there are other worries. Consider a true identity claim a = b. By simply making any true but otherwise arbitrary a = b a premise, we can argue that...
1.) [phi] & [logical not]K[phi]       Assump.

2.) a = b                             Assump.

3.) [phi] & (K[phi][right arrow]      1, by theorem [logical not]
  a [not equal to] b)                   [alpha] [??] [alpha][right
                                        arrow][beta]

4.) [logical not]{K[[phi] &           Cartesianhood of 3
  (K[phi][right arrow] a [not
  equal to] b)] [??]
  [perpendicular to]}

5.) [MATHEMATICAL EXPRESSION NOT      4 and Tennant antirealism
  REPRODUCIBLE IN ASCII]

6.) K[[phi] & (K[phi][right arrow]    By DOK and KIT
  a [not equal to] b)] [??] K[phi]

7.) K[[phi] & (K[phi][righ arrow]     By DOK and KIT
  a [not equal to] b)] [??] K[phi]
  [right arrow] a [not equal to] b

8.) K[[phi] & (K[phi] [right          7, 8 and Modus Ponens
  arrow] a [not equal to] b)] [??]
  a [not equal to] b

9.) [MATHEMATICAL EXPRESSION NOT      8, possibility closes over
  REPRODUCIBLE IN ASCII]                entailment

10.) [MATHEMATICAL EXPRESSION NOT     Proof of 3 from 1, 5 and 9
  REPRODUCIBLE IN ASCII]

11.) [phi] & [logical not]K[phi]      10, by necessity of identity
  [??] a [not equal to] b

12.) a [not equal to] b               1 and 11

13.) CONTRADICTION, LINES 2 AND 11


It would seem to me that the fact that Tennant's revised anti-realism is consistent with the claim that there exists an unknown truth would be a Pyrrhic victory if it is nonetheless inconsistent with the conjunction "There is an unknown truth and a true identity claim." Apparently Tennant will need to block line 4. And Williamson's reformulation of "Cartesianhood" seems to be the only way to do it.

Technically, the forgoing reductio does not resurrect Fitch's paradox; that paradox showed that antirealism conflicted with only the first conjunct. Can this ambitious project be conducted with Tennant's revised anti-realist principle? I think that it can be. We simply need to recall that identity claims are not the only metaphysically necessary truths beyond the domain of logic and mathematics. Claims, for example, about essential natures are necessary if true at all. And antirealism itself is one such claim. Hand and Kvanvig note that "...antirealism is committed to more than the mere claim that truth happens to be epistemic; it is essential to the nature of truth that it be epistemic." [1999: 424]. So let's make the following readjustments on the derivation just presented.

Step 1: replace "a = b" and a [not equal to] b" in the previous proof with "TAR" and "[logical not]TAR" respectively where "TAR" denotes "Tennant Antirealism."

Step 2: change the justification of line 11 to "necessity of the nature of truth."

Given Hand and Kvanvig's point, TAR is "polar" in Tennant's sense of the term and the move from line 10 to 11 so adjusted is valid. So the revised anti-realism is by itself incompatible with the existence of an unknown truth.

Thus Tennant's anti-realism has to be restricted so as to avoid challenges involving metaphysically necessary truths known a posteriori (identity claims, for example) and metaphysically necessary truths known a priori (such as theories of truth). Tennant tends to only address the former. Williamson's "trivializing" restriction seems to be the only one that will handle all of these cases. The necessary resort to defining knowability in terms of consistency with all necessary truths threatens Cartesian knowability with trivialization. Let us now turn to investigate whether the same is true for Tennant's new principle of knowability.

3. Is the Globally Restricted Knowability Principle Trivial?

Tennant is seemingly very optimistic about a new "globally restricted knowability principle." "The Globally Restricted Knowability Principle.renders Williamson's quibbles moot--whether he raises them with reference to En, or with reference to a=b (where a and b are rigid designators)." [2010b, p. 23] The essential alteration in Tennant's new account is that OK9 now requires that one's grounds for 9 be consistently knowable. Accordingly...

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this formulation, [DELTA] is a set of assumptions {[[delta].sub.1] - [[delta].sub.n]} upon which [phi] is based via a proof [PI], K[DELTA] is defined in a Frobenian way as (K[delta] | [delta] [member of] [DELTA]} and [perpendicular to] is apparently understood in terms of the three part account presented at the beginning of this article. Thus, [DELTA] (as opposed to [phi]!) is "Cartesian" iff [logical not](K[DELTA] [??] [perpendicular to]). [2010a: 224] In response to Fitch, Tennant explains that one's grounds for knowing a proposition of the form ([phi] & [logical not]K[phi]) would be a set of assumptions that included the proposition 9 and the proposition [logical not]K[phi] [2010a: 226]. But {[phi], [logical not]K[phi]} is a non-Cartesian set, so the existence of an unknown truth is no threat to the new antirealist principle. Further, Williamson's [[phi] & (K[phi][right arrow]En)] was derived from ([phi] & [logical not]K[phi]) by an appeal to the theorem [logical not][alpha] [??] [alpha][right arrow][beta], and thus also grounded ultimately on a non-Cartesian A, again {[phi],[logical not]-K[phi]} [2010b: 20-23].
   For now Williamson's attempted applications of (the globally
   restricted principle) patently violate the restriction that the set
   [DELTA] (at that stage) be Cartesian, even when Cartesian status is
   construed by reference to logical, not metaphysical, consistency.
   For the set in question is precisely {[phi],[logical not]K[phi]},
   as the reader can easily verify by inspection. [2010: p. 22,
   emphasis in the original]

      The Globally Restricted Knowability Principle therefore renders
   Williamson's quibbles moot-whether he raises them with reference to
   En, or with reference to a = b (where a and b are rigid
   designators). [2010: 23]


And, though Tennant does not stress this point, pace the objections of Hand and Kvanvig it appears that this restriction has some principled connection to anti-realist conceptions of truth.
   ...if our grounds for [phi] are indeed [DELTA], then the inferred
   possibility of knowing that [phi] surely presupposes the
   possibility of knowing that [DELTA]. Indeed, if it were impossible
   to know the joint truth of the assumptions in [DELTA], how could
   one be confident in inferring from the intermediate conclusion
   [phi] to the knowability claim [??]K[phi]? [2010a: 224]


But is it true that Tennant's reformulation avoids Fitch and Williamson "even when Cartesian status is construed by reference to logical, not metaphysical, consistency?" That is the topic for investigation in this section. If this were true, I think it would be an impressive victory for antirealism. But I think the antirealist will have to settle for far less and will once again have to resort to formulations of knowability involving (trivializing) restrictions to consistency with metaphysical truths.

I wish to make a charitable concession (or emendation) from the outset. The question is "Should we strengthen the notion of "consistent knowability," [logical not](K[DELTA] [??] [perpendicular to]), with a closure principle? Tennant's Frobenian account does not subject the members of [DELTA] to closure of knowledge over known entailments. But consider a case in which [DELTA] ={q, q [right arrow] p, [logical not]Kp}. I see no reason why an agent could not possess each of the elements of this set within the domain of his reasons. But there is a straightforward proof n from this set to the unknowable [phi] that p&[logical not]Kp. But without a closure requirement for consistency of [DELTA] this set cannot be disqualified on grounds of being "non-Cartesian." We need an account of K[DELTA] that is stronger than the one Tennant offers and minimally that account should insist on consistency where knowledge is closed under known entailments. In what follows, I will allow Tennant this stricter notion of K[DELTA].

Should we further stipulate that the set [DELTA] include only true reasons as members? It is strange that nowhere in his account does Tennant address this crucial question. In both "Revamping the Restriction Strategy" and "Williamson's Woes" the elements of [DELTA] are consistently referred to as "assumptions" not as "truths." The emphasis is on consistency of knowing one's reasons [DELTA] rather than on truth. Keeping this point in mind, let us proceed to assess some potential counterexamples to the global knowability principle. Throughout this exercise, let us assume the truth of an identity claim a = b and bear in mind again that Tennant's preferred form of intuitionistic logic includes the theorem [logical not][alpha] [??] [alpha][right arrow] [beta].

Let's first consider a counterexample with an obvious flaw. Suppose [DELTA] = {a [not equal to] b}. Using the theorem just mentioned, it is possible then for an agent to construct a proof [PI] for the claim [phi] that a = b[right arrow]a [not equal to] b. But if it is true that a = b, then clearly it is not possible to know this [phi]. It might be objected that this does not show that [DELTA] must be restricted so as to include only true elements as members (or at least not include denials of metaphysical truths), but rather that merely illustrates that the antirealist should insist that their rule applies only where [phi] is true. And it is reasonable to assume that Tennant already intended this restriction though the formal account of his revamped view does not make it entirely explicit (the anti-realist insists, after all, that all truths are knowable) and it was explicit in his account of previous formulations of his principle of knowability.

Here is a suspicious attempt to amend the counterexample. Suppose [DELTA] = {a [not equal to] b, p}. It is possible then for an agent to construct a proof [PI] for the claim [phi] that Kp[right arrow] a [not equal to] b, but the agent may press the proof further my means of a simple conjunction to prove the further 9, specifically, (Kp[right arrow] a [not equal to] b) & p. Again, if it is true that a = b, then clearly it is not possible to know this [phi]. Could it be true that (Kp[right arrow] a [not equal to] b) & p? Assuming that there is some p that is unknown, it would appear that we must answer in the affirmative. Although it is easy to see that where our consistently knowable set of reasons is a [DELTA] such that [DELTA] = {a [not equal to] b, p} an agent is in a position to make a derivation of a falsehood from this set, namely that a [not equal to] b, that does not alter the fact that the proposition which we in fact derived, (Kp[right arrow] a [not equal to] b) & p, could be true and on the theorem that [logical not][alpha] [??] [alpha][right arrow][beta] this proposition would be true just in case there were such a thing as an unknown truth p.

I suspect that a sensible antirealist would object to the proof [PI] in this "counterexample" on grounds that it is intuitionistically invalid. When our agent moves from the claim that a [not equal to] b to the claim that Kp[right arrow] a [not equal to] b they are employing an inference commonly referred to as "weakening" which is valid in classical propositional logic, but not in intuitionistic logic (the move fails to take into account the importance of relevance in for intuitionists). If this objection is the right one, then we have found a further restriction, or clarification, upon the revamped account of antirealism pertaining to which types of proof [PI] satisfy the formulation. Accordingly, [PI] must be intuitionistically acceptable. But I suspect, as we saw earlier, this restriction was implicit all along. Let us assume this to be the case here as well and endeavor instead to amend the counterexample accordingly.

In the foregoing counterexample, I used an intuitionistically invalid move, but the same problems can arise from the same set of reasons {a [not equal to] b, p} by means of only intuitionistically acceptable inferences. By use of the theorem [logical not][alpha][??] [alpha][right arrow][beta], our agent might infer that a = b[right arrow][logical not]Kp. By an intuitionistically valid instance of contraposition, they may infer that [logical not][logical not]Kp[right arrow] a [not equal to] b. Thus our original set proves that ([logical not][logical not]Kp[rigth arrow] a [not equal to] b) & p. And this is not knowable. For if it were to be known, it would have to be the case that Kp and intuitionistically this entails [logical not][logical not]Kp. Thus, we will again reach the conclusion that [??]K a [not equal to] b and thus [??]a [not equal to] b pace our assumption that a = b was a necessary truth.

So it seems that in addition to our reasonable restrictions on [phi] and [PI] we will need a stronger restriction on [DELTA] than merely that of "consistent knowability" as defined logico-mathematical terms above. We must further insist that it not be the case that [DELTA] contains a metaphysical falsehood as an element. And, obviously, even this restriction is too weak for there will be cases in which [DELTA] does not include such a falsehood as a member but merely entails a metaphysical falsehood. Consider a case in which [DELTA] = {p, p[right arrow]q, q[right arrow][a [not equal to] b} where p and q are contingently true. This will resurrect the very same issues addressed in the previous paragraph. And even this restriction will not work in the absence of our previous insistence that K[DELTA] require that knowledge is closed under entailment. Consider the case in which [DELTA] = {p, p[right arrow]q, a = b[right arrow][logical not]Kq}. From this set an agent could swiftly prove the unknowable [phi] that q & (a = b[right arrow][logical not]Kq). Could this be a true [phi]? I don't see why not.

In summary, in order to avoid Fitch-style paradoxes Tennant will need to revamp, or more precisely "severely strengthen," his criteria for K[DELTA]. It must be the case that for [DELTA] to qualify as a Cartesian set two criterion must be met.

1.) No contradictions follow logico-mathematically when each element of the set is known and that knowledge closes under entailment.

2.) Every element of the set is metaphysically possible, ie, no element of the set contradicts a metaphysical truth, and no contradiction of a metaphysical truth can be derived from the set even under the criterion laid out in 1 above.

This will be a very weak principle of knowability. But is it trivial? It would appear so. The two criterion jointly require that every element of and every entailment of set [DELTA] can be known without contradicting any metaphysical truth. So it seems obvious that if there is any proof [PI] showing the entailment of [phi] from this set then we already know that K[phi] is consistent with all metaphysical truths. Therefore it is contingent. Therefore it is possible. We are once again faced with a formulation of knowability that is merely tautologous.

In closing, the prospects for a non-trivial restriction on knowability principles appear daunting. I'm not sure that there is much hope in terms of Cartesian or "consistency" restrictions here, but Tennant has done quite a bit to lighten the way towards such a solution if there is any. I hope to have demonstrated that there is still much to be explored and, more to the point, much that remains to be if such strategies are to pay off in the end.

REFERENCES

Fitch, Frederick (1963), "A Logical Analysis of Some Value Concepts," The Journal of Symbolic Logic 28: 135-142.

Hand, Michael, and Kvanvig, Jonathan (1999), "Tennant on Knowability," Australasian Journal of Philosophy 77: 422-423.

Kripke, Saul (1980), Naming and Necessity. Cambridge: Harvard University Press.

Tennant, Neil (1997), The Taming of the True. Oxford: Oxford University Press.

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Williamson, Timothy (2010), "Tennant's Troubles," in Joe Salerno (ed.), New Essays on the Knowability Paradox. Oxford: Oxford University Press, 183-204.

[c] Troy Nunley

TROY NUNLEY

nunley.troy@gmail.com

Independent Scholar
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Author:Nunley, Troy
Publication:Analysis and Metaphysics
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Geographic Code:1USA
Date:Jan 1, 2011
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