Abstract
Knowing that some state of affairs—expressed by a proposition, p—is possible (Ks◊p), and the possibility that one knows that p (◊Ksp) have, quite obviously, different meanings. This paper focuses only on their logical relationship—whether they entail one another. I will argue for the following three claims: (i) the basic verificationist principles of anti-realism, at least in their simplest forms, and in conjunction with some other, intuitively reasonable principles, do entail that these two concepts are substitutionally equivalent. (ii) Our pre-theoretical expectations question this outcome, as counterexamples can be manufactured. I will also argue that this substitutional equivalence has further, highly counter-intuitive implications. (iii) Finally, I will argue that some of the standard strategies to avoid the well-known paradoxes of anti-realism (as e.g. the Church–Fitch paradox) fail to (dis-)solve this new paradox, while others may be able to do that, but only at a considerable price. However, the introduction of a moderate anti-realist truth operator, the actual objective of this paper, does dissolve the paradox, and arguably at an affordable price.
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Notes
While the factivity of A-Rconv (├∀p(◊Kp → p)) was accepted and argued for in the recent past, this principle now seems to be abandoned. See, e.g., Tennant’s retraction, for the record (Tennant 2009, p. 225).
These concerns are, however, mitigated by the fact that the truth operator approach that will be introduced in Section 4 can also prevent the modal collapse in S4.
This claim warrants some caution: arguably, first-order possible-worlds semantics requires us to be able to track one and the same individual across other possible worlds. But our standard for the truth of a proposition—beyond its factuality—is really low: it requires only that the proposition could be known by someone at some point in time. Furthermore, our examples will focus on propositions that are factual, but not true, and so our main concern will be to show that no individual in any possible scenario could know the given factual proposition.
I don't find this point to be a serious restriction—other modalities, like metaphysical, nomological, etc. (if one would prefer to use any of them) could be defined within the accessibility relation on the set of possible worlds. Furthermore, as I will utilize this point later, the scope of the relevant possible worlds is restricted—if not in theory, then in practice—by the scope of the knowledge operator: only those worlds play any role in our considerations in which Kp is true of some propositions, p. But Kp can only be true in any possible world if there are epistemic agents, comparable to us, in that world.
Here is the formal proof:
(1)
K◊p → ◊p
factivity of K
(2)
p → ◊Kp
A-R
(3)
◊p → ◊◊Kp
◊-Elim, 2
(4)
◊◊Kp → ◊Kp
instance of S4
(5)
K◊p → ◊Kp
chain rule, 1, 3, 4
where ◊-Elim is, as follows: from ├ p → q derive ├ ◊p → ◊q. For this point, see e.g. Hughes and Cresswell (1984), p. 37.
Here is the formal proof for this direction:
(1)
◊Kp → p
A-Rconv
(2)
p → ◊p
T
(3)
◊p → ◊K◊p
A-R, using ◊p for p
(4)
◊Kp → ◊K◊p
chain rule, 1,2,3
(5)
K◊p → ◊KK◊p
A-R, using K◊p for p
(6)
◊K◊p → ◊◊KK◊p
◊-Elim, 5
(7)
◊◊KK◊p → ◊KK◊p
instance of S4
(8)
◊Kp → ◊KK◊p
chain rule, 4,6,7
(9)
◊KK◊p → K◊p
A-Rconv, using K◊p for p
(10)
◊Kp → K◊p
chain rule, 8,9
As we will consider later, accepting the converse of the anti-realist principle requires some restrictions on the scope of possibility; it is usually argued that ◊ should be interpreted in such a way that only the possible worlds in which p has the same truth value as in the actual world, w0, should count. But even if it is adopted, the above collapse is relevant: even if p is false in w0, it does not or should not exclude the possibility that ◊p is known in that world.
Or, that someone is in the appropriate (factive) mental state, if one prefers Williamson’s approach (2000).
In other words, p should be such that not only p, but also □p is true in w0.
And, according to our previous definition, 'not known' means here that no member of the epistemic community knows or will know that p.
It is often said that given all we know, the Goldbach conjecture can still be true. Doesn't that contradict our point? The answer is no; this kind of remark confuses epistemic and logical possibilities. Given all what we know, the Goldbach conjecture is (or might be) epistemically possible. But since it is a necessary statement, if it is false, then it is false in all possible worlds and so it cannot be possible—our knowledge (or more precisely the lack of it) is irrelevant for its (im-)possibility.
“Thus take the general statement ‘All men are mortal’. … But we could never arrange for a similarly conclusive answer to the mortality question, based similarly on direct observation. Even if we cared so much about the answer that we were prepared to increase our observations by resorting to direct experimentation, which I find a rather lugubrious expedient in this instance, still there would be a difficulty in checking all instances. We could not check on ourselves, and live to report an affirmative answer to the question.” (Quine 1976, p. 60).
James (1993).
Although this principle is far from being obvious—some, e.g. Nozick (1985), argue that knowledge is not closed under logical consequence (and thus under &-elimination)—it will not be argued for here.
Some anti-realists may disagree, arguing, e.g., that what is not known has no relevance for us. In other words, the possibility of unknown truths depends on our concept of truth. Radical anti-realism, represented for example by Hudson (2009), accepts the outcome of the Church-Fitch hypothesis and thus the implication that there are no unknown truths.
One may argue that u, i.e., ∃p(p& ~Kp), may be known in a roundabout way without knowing any actual instance of it. Since such a roundabout, existential proof would not depend on any particular propositions, it would assume the analyticity of u, and so u must be necessary in that case. But then this would generalize a counterexample in the opposite direction.
As I see it, only one of the 4 strategies considered here does or can actually utilize this option (the one that allows paraconsistent logic), so the worry here is somewhat academic. Alternatively, one may wonder, in what sense a theory that does not pose the question of the converse principle can be regarded anti-realist/verificationist.
It seems fairly obvious that VA (and VAconv) cannot prevent the collapse between ◊KAp and K◊Ap. But the real issue here is still the collapse between ◊Kp and K◊p. I take it for granted that p↔Ap and Kp↔KAp both hold in the actual world; their differences are relevant only in modal contexts. Accordingly, if ◊-Elim is still valid for these two biconditionals, then our previous proofs can easily be amended even if A-R and A-Rconv are replaced by VA and VAconv. On the other hand, if ◊-Elim does not apply to the biconditionals, then Edgington’s account comes with a hidden price-tag: rules like ◊-Elim are not a valid rule for sentences affixed with the actuality operator. It is not clear, however, what could replace those rules.
Such move explains the rather surprising outcome, mentioned above, that A-R and A-Rconv lead to the theorem that K◊p → p.
The words “relevant” and “close” are used here rather loosely. Closeness assumes some ordering or distance on the set of possible worlds and being close and being relevant may not overlap.
The introduction of the 2nd possibility operator does not originate in Tennant (2009). One of the anonymous reviewers suggested this formalization (along with the intro and elim rules). I’m grateful for the suggestion and glad to go along with it.
It should be noted, however, that the move from A-R to A-R* is far from obvious. First, A-R* makes the scope of the concept of truth quite narrow. Should truth really be restricted to the possibilities of “investigative outcomes, at future times, within the actual world”? Second, this shift from A-R to A-R* (or from ◊Kp to ♦Kp) makes the restriction to Cartesian sentences rather superfluous. Surely, non-Cartesian sentences cannot be in the scope of ♦Kp.
Obviously, MAR→ is an upgrade for A-R, while MAR← is for A-Rconv. More importantly, however, MAR→ and MAR← together provide a full, moderate anti-realist definition of truth:
- (MAR):
-
├ Tp ↔ (p & ◊Kp).
For more about this point, see Marton (2006).
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Acknowledgements
I thank the three anonymous reviewers of this journal for their valuable comments, suggestions and their provocative questions.
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Marton, P. Knowing Possibilities and the Possibility of Knowing: A Further Challenge for the Anti-Realist. Erkenn 86, 493–504 (2021). https://doi.org/10.1007/s10670-019-00115-4
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DOI: https://doi.org/10.1007/s10670-019-00115-4