Mr. Magoo’s mistake

Abstract

Timothy Williamson has famously argued that the (KK) principle (roughly, that if one knows that p, then one knows that one knows that p) should be rejected. We analyze Williamson’s argument and show that its key premise is ambiguous, and that when it is properly stated this premise no longer supports the argument against (KK). After canvassing possible objections to our argument, we reflect upon some conclusions that suggest significant epistemological ramifications pertaining to the acquisition of knowledge from prior knowledge by deduction.

Appendix: what about (WP-dis)?

Is the disjunctively modified (WP) justifiable? Well, what was it that made (WP) appealing in the first place? Two features of Williamson’s presentation of the argument contribute to the intuitive plausibility of (WP) and obscure the matter at hand. One feature relates to the range for which the story is told and the other to the intervals of heights for which (WP) is stated. We take these points in turn.

First, Mr. Magoo’s story begins with his observation that the tree is not 0 inches tall. This makes the following steps intuitively compelling, since for normal trees it is reasonable that Mr. Magoo can correctly and directly judge that they are not 5, 6 or even 20 inches tall. But for this no proposition such as (WP-dis) is needed. It seems that we can easily go along with Williamson in the initial stages since we know that any reasonable evaluator of heights will not only know that the 666 inches tall tree, is not 0 inches tall, but will also know that the tree is not 20 inches tall. In normal conditions a person can know just by looking at a tree that it is not 20 inches tall, independent of his observations that it is not 3 or 4 inches tall. At these stages the appeal to (WP) is idle.

In fact, in stating the rationale for (WP) Williamson himself says that Mr. Magoo knows that the height of the tree is not 60 inches (p. 114). That is part of the reason why it seems evident that if Mr. Magoo knows that the tree is not 3 inches tall he can come to know that it is not 4 inches tall—he already knows it is taller than 4 inches. That much was already implicit in the setup. We easily play along with the first stages since we know already that Mr. Magoo knows that the tree is not 3, 4, or even 50 inches tall. But once we accept the first steps, the game seems to be over. We find no acceptable reason to stop the argument from proceeding past the limit that is directly discernable by a normal observer. To be more accurate, once the reader has accepted the first two steps, it is hard to see how a stopping rule can be devised. But what if we start the argument at a height that is at the limit of what Mr. Magoo can know by direct observation? For instance, let us say that 500 inches is the closest height to the actual height of the tree that Mr. Magoo can reliably judge that the tree is not.²³ Can Mr. Magoo proceed to infer that the tree is not 501 inches tall?

A negative answer to this question is in effect an unreasonable rejection of WP (and of (WP-KK) as well). It seems that if Mr. Magoo knows how poor his judgment of heights is, he may also legitimately infer from his knowing by looking at the tree that it is not 500 inches tall, that it is not 501 inches tall either. The point to think about, however, is the next step. Granted that Mr. Magoo knows that the tree is not 501 inches tall, can he then infer correctly from this knowledge and from his knowledge of his limited abilities to tell tree heights by looking, that the tree is not 502 inches tall? Here intuitions may diverge. It seems that when the case is laid open in this way, the intuition that he can so infer is no longer as compelling. To clothe this intuitive weakness in reasons, we suggest that this is so because the inference exceeds the evidence available to Mr. Magoo. No matter whether he knows that he knows the tree is not 501 inches tall or merely knows that it is not of that height, the evidence available to him does not support the claim that the tree is not 502 inches tall, at least not on the basis of this margin for error; and this brings us to the other point.

Both (WP) and (WP-dis), gain intuitive appeal from the density of the intervals chosen. The interval of 1 inch is very modest relative to the evaluative abilities of normal subjects. But, once again, this is not material to (WP). Clearly, the small interval of 1 inch, although it may very well be the source of our tendency to accept (WP) in the initial stages, is not essential to the proposition. (WP) should be true for any interval of which Mr. Magoo knows that he cannot discriminate just by looking. If (WP) is correct for intervals of 1 inch, it should be good for 5, or 10 inches as well (provided that Mr. Magoo cannot discriminate between heights differing by 10 inches). It seems, however, that when the interval is sufficiently large, only one or two applications of the principle are intuitively convincing.²⁴

Combining these two observations regarding the source of (WP)’s intuitive appeal, defeats repeated applications of the principle and therefore undermines the validity of the multiple-application-enabling (WP-dis). If we start the Magoo argument at the closest height that Mr. Magoo can know that the tree is not in inches and we take the interval to be the largest that Mr. Magoo can know that he cannot discern by mere looking, then, it seems to us, allowing for more than one application of (WP) is rather unintuitive. In any case, the intuitive appeal of (WP-dis), at least of its recurring application, is not maintained once the contingent features of the story are changed.

But, regardless of intuitive force, is there an argument to show that (WP-dis) is not acceptable?

There are two strategies to argue against the acceptability of (WP-dis). The first is a direct rebuttal, viz. showing that it leads to unwelcome consequences, or is in conflict with some other, well-based, principle. The second strategy, and the one we pursue here, is an indirect argument. Our rejection of (WP-dis) is motivated by an analysis of the faults of propositions similar in both form and function. The general strategy is to show that the unacceptable conclusion reached by Mr. Magoo’s argument can be reached without the (KK) principle. We introduce two propositions similar to (WP-dis), and articulate two Magoo-type arguments. We show how these new disjunctive propositions lead to the same contradiction as does (WP-dis) without assuming (KK). This shows that the appeal to knowledge (or to its known limits) is not essential to the argument. In other words, it seems that just as in these cases the contradiction is reached through the introduction of the disjunctive propositions and independently of (KK), in the Magoo argument too the disjunctive (WP-dis) is responsible for the mess and not necessarily (KK). So, if the argument is sound, there’s reason to suspect that disjunctive propositions like (WP-dis) should be rejected.

Assume that for a normal tree Mr. Magoo can tell that if he recognizes it from a distance of 100 yards, it must be taller than one foot. Sitting at one end of a football field, Mr. Magoo sees a tree at the other end of the field. He therefore knows that the tree at the far end of the field is not 12 inches tall. He also knows that it is not taller than 10,000 inches, since presumably, no tree is.²⁵

$$ K_M t_{12-10,000}\quad\quad(\hbox{Mr. Magoo knows that the tree's height in inches is between 12 and 10,000}) $$

(A)

Since knowledge entails belief (as will be assumed throughout) we may also suppose that:

$$ B_M t_{12-10,000}\quad\quad(\hbox{Mr. Magoo believes that the tree's height in inches is between 12 and 10,000}). $$

(B)

²⁶Now think of a principle quite similar to (WP-dis):

$$ K_M (B_M ^{looking\vee reflection}\sim t_i \supset \sim t_{i+1}) $$

(1)

The idea behind this principle is that, knowing the limitations of his own visual ability Mr. Magoo knows that if, upon looking and reflecting on his abilities alone, he has formed a belief that the tree is not of a certain height, then the tree must not be 1 inch taller than that. Now contrast (1) with:

$$ K_M (B_M^{looking} \sim t_i \supset \sim t_{i+1}) $$

(1’)

We will demonstrate that while the non-disjunctive (1’) does not lead to a contradiction, its disjunctive analogue, (1), does, and this without recourse to (KK).

Assume that after looking at the tree Mr. Magoo believes it is not 12 inches tall:

$$ B_M^{looking} \sim t_{12} $$

(2)

Now assume that Mr. Magoo performs a series of deductions, similar to those Williamson has him perform. To avoid possible failure of transparency of beliefs, assume that he is writing the steps of the deduction on paper, so that in this case:

$$ B_M p\supset K_M B_M p $$

(3)

From (2) and (3) it follows that Mr. Magoo knows that he believes the tree is not 12 inches tall:

$$ K_M B_M \sim t_{12} $$

(4)

Positing (2) as the antecedent in (1) we get:

$$ K_M (B_M \sim t_{12} \supset \sim t_{13}) $$

(5)

From (4) and (5) it follows by closure that Mr. Magoo knows that the tree is not 13 inches tall:

$$ K_M \sim t_{13} $$

(6)

Since knowledge entails belief, from (6) it follows that:

$$ B_M \sim t_{13} $$

(7)

At this point Mr. Magoo can posit (7) in (1) and get:

$$ K_M (B_M \sim t_{13} \supset \sim t_{14}) $$

(8)

Once again, since he is fully conscious and writing the steps of his deduction as he is going through them, Mr. Magoo knows that he believes that the tree is not 13 inches tall:

$$ K_M B_M \sim t_{13} $$

(9)

and therefore, again by closure we get:

$$ K_M \sim t_{14} $$

(10)

These iterations can be continued for the full expanse between 12 and 10,000 inches. Looking at his notes, Mr. Magoo concludes, and therefore knows, that the tree is not any height within this interval (or at least knows that he believes this):

$$ K_M \sim t_{12\hbox{--}10,000} $$

(11)

(11) clearly contradicts (A) thus ascribing to Mr. Magoo knowledge of contradictory propositions. Note that this conclusion amounts to possible inconsistency within Magoo’s cognitive system, regardless of the tree’s actual height (so no appeal to factivity is required). But more importantly, this time the blame cannot be laid on (KK), since the troubling result is reached without recourse to this principle. In the present argument, (KK) has been replaced with (3), which is, under the circumstances of the case, very plausible.

Now, the point to notice is that the knowledge in (6) is knowledge by direct perception and inference by reflection on Mr. Magoo’s visual abilities. The belief in (7) therefore, must also be modified as belief by looking and reflection. (7) is then plugged into (2) which—modified to accommodate belief by looking or reflection—allows the argument to proceed. So the full formulization of (7) should be:

$$ B_M^{looking\,\&\,reflection} \sim t_{13} $$

(7*)

Let us remind ourselves that our inquiry into this case has emerged from the question of the adequacy of the disjunctively modified (WP-dis). Our question was whether this disjunctive modification, allowing for knowledge gained by both mere looking and inference from reflective truths to appear as the antecedent in the conditional, is acceptable. Proposition (1) of the present argument has been modified in exactly the same way as (WP-dis). We now want to suggest that it is this liberal formulation which is the source of trouble. Think of the less permissive (1′) (structurally analogous to (WP-KK)):

$$ K_M (B_M^{looking} \sim t_i \supset \sim t_{i+1}) $$

(1′)

It is easy to see that (7*) cannot be posited in (1′). To put it explicitly, had the belief operator in (1) been modified to accommodate only belief by mere looking, it would have been wrong to posit (7) in (1) and the move to (8) and (11) would have been blocked. It seems then, that the inappropriate modification of the belief operator in (1) is to blame for the unwanted results. Since this argument and proposition (1) on which it turns were designed to mirror Mr. Magoo’s argument in its (WP-dis) formulation, the failure of the former argument can be instructive regarding the latter. Although this is not a direct argument against the coherence of (WP-dis), the present argument does suffice to cast doubt on the plausibility of disjunctive conditionals like (WP-dis). The argument shows that a disjunctive proposition similar to (WP-dis) in both form and intent leads to a contradiction independent of (KK). It is therefore reasonable to suspect that it is also the disjunctive modification in (WP-dis) that leads into trouble and not (KK).²⁷

Let us take another disjunctive principle of the (WP-dis)-type, this time, one pertaining not to knowledge but rather to the norms of belief-formation. Once again, we show, the disjunctive character of the principles leads to inconsistency regardless of (KK) or any analogous principle.

Say Mr. Magoo is deliberating about his own perceptual beliefs. Which of the following belief forming norms, structurally analogous to (WP-KK) and (WP-dis), should Mr. Magoo adopt?

(BFN-dis) If Mr. Magoo believes by looking or reflecting (on what he believes by looking) that a tree is not i inches tall, then he may also form a belief that a tree is not i+1 inches tall.

Or, alternatively, is he justified in adopting: ²⁸

(BFN-KK) If Mr. Magoo believes by looking alone that a tree is not i inches tall, then he may also form the belief that it is not i + 1?

Notice that (BFN-dis) is more permissive than (BFN-KK). Like (WP-dis), it applies to other sources of belief, besides the purely visual. As we will now demonstrate, this permissiveness makes (BFN-dis) unacceptable since it can lead one to an incoherent belief system. In other words, the permissiveness of this norm leads to incoherence just as its analogue, (WP-dis), leads to a contradiction.

Imagine that, looking at a tree, Mr. Magoo forms the true belief that it’s between 5 and 10,000 inches tall. We therefore have the following:

$$\begin{eqalign} B_M^{looking} t_{5-10,000}\quad\quad(\hbox{Mr. Magoo believes by looking that the tree is between 5 and 10,000\,inches tall}). \end{eqalign}$$

(12)

From (12) we can infer:

$$ B_M^{looking\,\&\,reflecting} \sim t_4\quad\quad(\hbox{Mr. Magoo believes by reflecting and looking that the tree is not 4\,inches tall}). $$

(13)

Now recall our candidate for belief-forming norm:

(BFN-dis)²⁹

$$ B_M^{looking\vee reflecting} \sim t_i \rightarrow B\sim t_{i+1} $$

Following the direction of (BFN-dis), Mr. Magoo forms on the basis of (13) the belief:

$$ B_M^{looking\,\&\,reflecting} \sim t_5\quad\quad(\hbox{Mr. Magoo believes by reflecting and looking that the tree is not 5\,inches tall}). $$

(14)

He then iterates this move, advancing 1 inch at a time. Imagine that he is recording these steps by writing them in succession on paper. Therefore, at least in this case, if he has a belief (Bp) he also knows that he has it (KBp). He repeats these steps again and again, forming his beliefs according to the direction of his accepted norm (BFN-dis). Finally, after surveying his records, he concludes that:

$$ B_M^{looking\,\&\,reflecting} \sim t_{5--10,000}\quad\quad(\hbox{Mr. Magoo believes by reflecting on what he believes by looking and reflecting that the tree is not of any height between 5 and 10,000}) $$

(15)

But (15) is in direct conflict with (12). Although a person may (arguably) hold incompatible beliefs, the fact that (BFN-dis) inevitably leads to contradiction is good reason not to adopt it as a norm of belief-formation. This conclusion is strengthened by the fact that there is another candidate available that has the virtues of (BFN-dis) (it is justified by the same considerations) without its vices, namely (BFN-KK):

$$ (\hbox{BFN}\hbox{-KK})\;B_M^{looking} \sim t_i \rightarrow B_M^{looking\,\&\,reflecting} \sim t_{i+1} $$

To see the advantage of this norm, imagine that we start with the visually formed belief:

$$ B_M^{looking} \sim t_4\quad\quad(\hbox{Mr. Magoo believes by looking that the tree is not 4\,inches tall.}) $$

(13′)

Under the guidance of (BFN-KK) he can form the belief that the tree is not 5 inches tall.

$$ B_M^{looking\,\&\,reflecting} \sim t_5\quad\quad(\hbox{Mr. Magoo believes by looking and reflecting}, \hbox{that the tree is not 5\,inches tall}). $$

(14′)

But this belief-forming norm can carry Mr. Magoo no further than this, since (14′) cannot be posited as the antecedent in (BFN-KK). The more moderate character of (BFN-KK) prevents one from getting into contradictions, by adopting it. Unlike its more radical disjunctive counterpart: (BFN-dis).

Notice that the problematic result of (BFN-dis) does not involve (KK). It seems then that it is just the disjunctive nature of (BFN-dis) that is to blame for the inconsistency. And so, by analogy it is plausible to claim that the disjunctive nature of (WP-dis), rather than (KK), should be identified as responsible for Magoo’s contradiction. It is this principle’s insensitivity to the source of knowledge it operates on (which, ironically, was the reason for accepting this principle in the first place) that entitles us to reject it. ³⁰