Church's Solution to the Knowability Paradox

In Church's "First Anonymous Referee Report on Fitch's 'A Definition of Value'" from 1945 we find not only the first formulation of the proof today known as the knowability paradox, but we find the first proposed solution. Church rejects any principle stating that propositional attitudes necessitate (other) propositional attitudes. For instance, Church rejects plausible closure principles for belief. He explains,

To be sure, one who believes a proposition without believing its more obvious logical consequences is a fool; but it is an empirical fact that there are fools. It is even possible there might be so great a fool as to believe the conjunction of two propositions without believing either of the two propositions; at least, an empirical law to the contrary would seem to be open to doubt. On this ground it is empirically possible that a might believe k' at time t without believing k at time t (although k' is a conjunction one of whose terms is k.

Church here denies that belief is closed under conjunction elimination. The context reveals that he is also denying that knowledge is so closed. More generally the idea is that if a propositional attitude is the result of another, it is so contingently. Notice that we have today departed dramatically from Church's thought. In all of the literature on the knowability paradox, for instance, it is granted that it is at least metaphysically necessary that one knows the conjuncts of known conjunctions. An exeception is found in a position articulated in Kelp and Pritchard's very interesting forthcoming " Anti-realism, Factivity and Fitch".

To pick another example,

...there is no valid law of psychology according to which anything whatsoever about my desires may be inferred from the fact that I know so-and-so.

It should be noted that Church (in the second referee report) allows Russell's theory of types to avoid such obvious counterexamples as "Necessarily, if a knows that she desires that p then a desires that p". Accordingly, the approach blocks the instance of the factivity of knowledge that is employed in the knowability paradox---viz., K(p & ~Kp) => (p & ~Kp). In this regard Church forshadow's Bernie Linsky's independent thoughts on the matter, forthcoming as "Logical Types in Arguments about Knowability and Belief".

In sum, Church's position is that his knowability paradox is indeed an invalid proof. On his view knowledge is neither closed under conjunction elimination nor unrestrictedly factive.

Fitch, Church and Nagel Papers

The following is a chronological catalog of archival documentation pertaining to the early history of Fitch's knowability paradox. The items were identified for the first time by myself or those aiding my research. I discuss their content in "Knowability Noir: 1945-1963", which will appear in New Essays on the Knowability Paradox.

The documents can be found in one of three archives:

(FFP) Frederic B. Fitch Papers: Manuscripts and Archives. Yale University Library.

(ACP) Alonzo Church Papers: Department of Rare Books and Special Collections. Princeton University Library.

(ENP) Ernest Nagel Papers: Rare Book and Manuscript Library. Columbia University.

*****************************************

* Church, Alonzo. "First Anonymous Referee Report on Fitch's 'A Definition of Value'". January or February 1945. Hand written by Alonzo Church to Ernest Nagel, coeditor of JSL. Contains the first proof of the modal epistemic result, today known as the knowability paradox. (ENP: Box 1)

An edition of this and the second referee report (listed below) is being prepared by Julien Murzi and myself for publication in New Essays on the Knowability Paradox.

* Nagel, E. "Letter to Church: March 6, 1945". Explains that Fitch has returned the manuscipt and offered replies to the first report. (ACP: Box 20)

* "Second Anonymous Referee Report on Fitch's 'A Definition of Value'". Late March or early April 1945. Includes a more formal characterization of the knowability result in Lewis and Langford's proof theory. Includes replies to Fitch's discussion of the first report. (ENP: Box 1)

* Nagel, E. "Letter to Church: April 13, 1945". Announces that Fitch has withdrawn his paper owing to a defect in his definition of value. (ACP: Box **)

* Fitch, F. "A Logical Analysis of Some Value Concepts". Fitch's December 23, 1961 Presidential Address to the Association for Symbolic Logic. (FFP: Box 33)

* Fitch, F. "A Logical Analysis of Some Value Concepts". Penultimate draft. (FFP: Box 33)

* Postcard to Fitch: January 18, 1963. Regarding remaining typographical edits to be made prior to the printing of "A Logical Analysis of Some Value Concepts" in JSL. (FFP: Box 33)

Introduction to New Essays

I've attached here a pdf of a draft of the introduction to New Essays on the Knowability Paradox. Comments Sought.

Fibonacci Program

I’ve come up with a draft of a program for testing my Fibonacci betting strategy for statistical reliability. It applies to a basic game of craps. Informal discussion of the strategy appears in the previous post.

The program defined below has three possible outcomes:
1. it yields a bankroll value B that is 500 or greater, and halts. FORMAL WIN
2. it yields a bankroll value B that is too low to continue, and halts. FORMAL LOSS
3. it yields neither a formal WIN nor a formal LOSS after exceedingly long play and stops itself.



Variables:

B = bankroll value. (I use BB to calculate a change in the bankroll value (e.g., BB = (B - x), and then reassign B to that number.)

R = outcome of the roll of two fair six-sided dice = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.

P = point made = 4, 5, 6, 8, 9, or 10.

x = pass line bet = 1, 2, 3, 5, 8, 13, or 21 (defined by the Fibonacci series).

y = free odds bet = 10(x).

C = Counter #. (Shuts program down at 25. I use CC to calculate a change in the counter (e.g., CC = (C+1)), and then reassign C to the new number.)


Program:

S0: Let B = 400. C=0. Go to S1.
[Notes: Initial bankroll and counter values]

S1: Let CC=(C + 1). Let C=CC.
If C=26, then END.
If C<26, then Go to S2.
[Counter to stop play after 25 starts.]

S2: Let x = 1. Go to S3
[Notes: Begin Fibonacci series]

S3: Let y = 10(x). Go to S4.

S4: If (y + x) ≤ B, then Roll dice. Outcome=R. Go to S5.
If (y + x) > B, then END.
[Notes: on come out roll, game shuts down if bankroll is lower than the required bet]

S5: If R=7 or R=11, then Let BB = (x + B). Let B = BB. Go to S11.
If R=2, 3, or 12, then Let BB = (B - x). Let B = BB. Go to S4.
If 6, then let P = 6 and Go to S6.
If 8, then let P = 8 and Go to S6.
If 5, then let P = 5 and Go to S7.
If 9, then let P = 9 and Go to S7.
If 4, then let P = 4 and Go to S8.
If 10, then let P = 10 and Go to S8.

S6: Roll dice. Outcome = R.
If R=P, then Let BB = ((x + y + .2(y)) + B). Let B = BB. Go to S10.
If R=7, then Let BB = (B - (x + y)). Let B = BB. Go to S9.
If R≠P and R≠7, then Go to S6.

S7: Roll dice. Outcome = R.
If R=P, then Let BB = ((x + y + .5(y)) + B). Let B = BB. Go to S10.
If R=7, then Let BB = (B - (x + y)). Let B = BB. Go to S9.
If R≠P and R≠7, then Go to S7.

S8: Roll dice. Outcome = R.
If R=P, then Let BB = ((x + 2(y)) + B). Let B = BB. Go to S10.
If R=7, then Let BB = (B - (x + y)). Let B = BB. Go to S9.
If R≠P and R≠7, then Go to S8.

S9:
If x = 1, then let x = 2 and Go to S3.
If x = 2, then let x = 3 and Go to S3.
If x = 3, then let x = 5 and Go to S3.
If x = 5, then let x = 8 and Go to S3.
If x = 8, then let x = 13 and Go to S3.
If x = 13, then let x = 21 and Go to S3.
If x = 21, then END.

S10: If B ≥ 500, then END.
If B < 500, then Go to S1.

S11: If B ≥ 500, then END.
If B < 500, then Go to S4.

Fibonacci Betting Strategy

Here is a Fibonacci-based betting strategy that I have been developing. It is the simplest and the safest of my strategies for beating a standard house at craps. It seems effective but I wonder whether its reliability can be decided more precisely

The basic game of craps is to match your point on the dice before you "seven-out". The probability that you will seven-out (and fail to match your point) depends on the point, but on average is .66.

Let us define a loss as sevening-out six times in a row. The probability that you will lose is .08. The probability of a win is then .92. And the probability of winning 5 times in a row is .65 and of winning 8 times in a row is .51. The strategy outlined below yields a 25% profit with roughly 5 to 8 wins. So it is likely on this betting algorithm to come out ahead by 25%.

Here's the strategy.

1.

Begin with $400. Remember to quit when you have earned $100 (or go bankrupt, which is about 6 losses in a row).

2.

Bet only Fibonacci numbers (in order) on the "pass line"

1, 2, 3, 5, 8, 13 ...

So, always begin with $1 on the pass line. If you fail to match your point, then bet $2 on the passline. If you fail to match your next point, then bet $3, and so on, traversing the Fibonacci series. If at any stage you do match your point, then begin the sequence again.

Most importantly, always take 10x odds behind the line. Completely ignore wins and losses on the come out roll, since the gains and losses there will be negligible (i.e., the game play that matters begins only after the point is established and 10X odds are placed behind the line ). The game is over when you're up $100 (and walk with $500 total) or you go bankrupt (and walk with $0 total).

Give the method a try here. Click on "options" to set your virtual bankroll and to set the game to 10X odds.

What is the precise reliability of the strategy described above? If the problem is undecidable (as I expect it is), then might one nonetheless be able to design a computer program that would run the strategy a sufficient number of times for statistical probability?