Tuesday, December 31, 2019
Sunday, December 22, 2019
Saturday, November 30, 2019
Thursday, October 31, 2019
Monday, September 30, 2019
A Less Mysterious Solution to the Paradox of Analysis
Moore's paradox of analysis strikes me as having a simpler resolution
than what recent commenters, such as Richard Fumerton in "The Paradox of Analysis," have proposed. The major premise of the paradox, as
Fumerton states it,
"If 'X is good' just means 'X is F' then
the question 'Is what is F good?' has the same meaning as the
question 'Is what is F, F?' " (478)
is unsound because it assumes that an object being "good" is
an identity statement (or at least, more charitably, that what is sought is an
identity statement). One arrives at the paradox by assuming that an analysis
just involves a chain of identities. In fact, what one wants here is an
equivalence, conditions such that if and only if they obtain, something is
good-- not an identity with the same intensional value under a different name. This latter
notion just perpetuates the basic error of the Meno.
For example, an informative analysis of "good" could ask how its extension is formed by treating it as a binary relation and asking what sub-properties it restricts its members to, as restrictions on the equivalence class of x := the set of things that are good. Taking good as a relation G, we might propose, for instance, that when x is mapped to the set u of things with a positive utility value by the relation G, the statement "x is good" is true.
A logical analysis along the lines of the above
without a necessary commitment to any one semantic interpretation could avoid
being either merely lexicographical or mysteriously phenomenological.
I'm not sure if it would be satisfying for all conceptual
analyses to be only the product of equivalences, but I would claim that
there is no satisfying conceptual analysis that is only the product of an
identity. Indeed, since this is already an intuition most people
have, Moore's reduction of conceptual analysis to an identity statement is
what makes the paradox of analysis work. The only problem is
that this reduction is wrong. In most cases, we do want an
equivalence statement with a conceptual analysis to tell us how and
when a concept obtains. (Maybe one could evoke Aristotle's four causes
here and say that an informative analysis should not just give us a noun phrase
that the concept noun stands in for but should give us one or more of these four
constitutive explanations.)
For example, if the concept is "triangle,"
there might be an identity relation with "three-sided polygon," but
this would not be informative-- it would just be a definition, a substitution
of a noun phrase for the concept noun, not an analysis-- while adding, that
"a triangle is any three-sided polygon such that it has three internal
angles that sum to 180 degrees" would be informative (e.g., by this
analytic definition we could claim that a three-sided polygon in non-Euclidean space
might not be a triangle). So, one could say that a conceptual analysis can
include identity definitions but cannot be constituted only by identity
definitions.
Also, because a satisfying conceptual analysis, such as the analysis of "triangle," can be constituted only from a combination of equivalence statements and identity statements, we can at least say that there are some paradigm cases of analysis that do not require some phenomenal state in addition to succeed.
Thoughts on Modal Distance
Duncan Pritchard’s modal luck/risk model in Epistemic Luck (2005) addresses a real problem, which can be illustrated as
follows:
Say that Lottie has a lottery ticket that has a 1/50,000,000
chance of winning in an upcoming draw. And say that while walking home in
relatively safe conditions, I have a 1/10,000,000 chance of being struck by
lightning. I’m convinced that it would irrational for Lottie to decide to tear
up her ticket before the draw, but it would be rational for me to think I won’t
be struck by lightning and thus decide to walk home.
However, note that these are cases of prediction,
such that what we are concerned with justifying here is not knowledge at all
but confidence, i.e. justified likely prediction. Even if Lottie’s
prediction that she won’t win turns out to be correct, the sense in which she
“knew” the outcome is different from the sense of concurrent knowledge
of an already true belief*. So, Pritchard’s model addresses the problem with
relying solely on probability to justify confidence, not knowledge, as safe.
I’m not sure if we can always rely solely on probability with the safety
principle in justifying concurrent true belief— I haven’t seen any instances
where this would be a problem, but there may be some.
Note that there could be a problem
for Pritchard’s Modal Distance model, as a measure of physical difference
between possible worlds, if there were exceptions to apparent closeness in
particular instances, such as a particular instance where the energy required
to make a given lottery draw when the needed ball is at the bottom of the pile
is greater than the energy required in a particular instance for a loose piece
of paper to jam the basement door in a well-maintained garbage chute. I thought that this problem
could be solved by putting the Modal Distance together with probability, but
the way I initially tried to do this was wrong**. The correct solution is to instead use
the Average Modal Distance, as the sum of the Modal Distances of
all the outcomes in close possible worlds over the number of outcomes: (MD1 +
MD2 + … + MDn)/n.
In any case, we can calculate the Modal Distance between two
possible outcomes in an event as follows:
Where: P := conditions needing to obtain
for any given outcome A to occur in an event E
Q := conditions needing to obtain for a target
outcome B to occur in E
O := the total set of outcomes of E in
close possible worlds
p := {x|(x ∈ P) &
(P ⊆ A)}
q := {y|(y ∈ Q) &
(Q ⊆ B)}
Modal Distance of A from B: MD[A,
B] := |(p - q∩p)| i.e.
the quantity of p minus the quantity of the intersection
of q with p
In the lottery case, if the quantity of p, i.e.
the relevant conditions needing to obtain for a non-winning draw A to
occur, is 1.0, and the quantity of the intersection of q, i.e. the
relevant conditions needing to obtain for the winning draw B to
occur, with p is .95, then the Modal Distance of a non-winning
draw from a winning draw is .05. In the lightning case, if the quantity of p,
i.e. the relevant conditions needing to obtain for a safe walk home A to
occur, is 1.0, and the quantity of the intersection of q, i.e. the
relevant conditions needing to obtain for a deadly lightning strike B to
occur, with p is .05, then the Modal Distance of a safe walk
home from a deadly lightning strike is .95.
Thus, while the probability of being struck by lightning is
actually greater than the probability of winning the lottery, the Modal
Distance between winning and not winning the lottery is much closer than the
Modal Distance between walking home safely and being struck by lightning. This
latter disparity seems to explain why it would be irrational to tear up the
ticket but rational to walk home.
But how does Modal Distance explain this?
When should Modal Distance be appealed to? After all, we would still want to
say that buying a lottery ticket with 1/50,000,000 odds is not a safe bet and
thus is not a rational purchase, no matter how modally close any ticket is to
winning.
I have a
couple thoughts on how to answer these questions:
1. One solution that I don't particularly like is to just
say that while modal distance can be measured quantitatively and does have an
impact on our intuitions about rational decision making, e.g. in the lottery
versus the lightning cases, these intuitions are actually emotion-based rather
than strictly rational. If that were so, a purely rational subject (e.g. an
AI), once fully apprised of all the external circumstances, would not
have these intuitions and would base its decisions solely on simple probability
over possible worlds.
Take Pritchard's bullet example: say the bullet that
misses a soldier named Duncan by several meters is fired by a very accurate
sniper who is sitting on a boat that just happens to be rocked by a wave at the
moment he pulls the trigger; this boat had been steady all day and the wave at
that moment is a fluke with a very low probability. Then, say the bullet that
misses Duncan by only a few centimeters is fired by a sniper with a bent scope
such that he routinely misses by a few centimeters, and thus his missing by a
few centimeters has a very high probability. In this case, even if Duncan were
to become aware of these facts, he might still want to say that the bullet that
missed him by a few centimeters put him at greater risk than the one that
missed him by several meters.
To explain this, Duncan might appeal to a concept of
physical danger that we could only translate through something like the modal
distance model, not through strict probability. But this physical danger
concept might only be employed to capture a subjective emotional response to a
situation as one experiences it in the moment, and not a strictly rational
assessment. Perhaps we could avoid this conclusion by appealing to the idea
of access to possible worlds within a given knowledge frame.
Within the knowledge frame of Duncan's experience, he has access to all possible
worlds resulting in the two bullets missing him, and it is rational for him to
compare the Average Modal Distances of the shots over all those worlds,
unrestricted by the facts of the sniper being rocked by a chance wave and the
other sniper having a bent scope. This knowledge frame consideration might land
us back in internalist territory, though, which I was trying to avoid
because I prefer the strictly externalist Robust Anti-luck Epistemology
account.
2. Another solution is to take a closer look at the lottery
case in particular and point out some considerations that might make this case
exceptional. With a
lottery draw where Lottie has already purchased a ticket, one can do a cost-benefit
analysis: while there is a very small chance that she will have a huge
payoff, now that she already has the ticket, it costs her nothing to keep it
and check the result-- or at any rate, it actually costs her less energy to
keep the ticket and check the result (in whatever fashion is most convenient to
her) than it does to tear the ticket up and throw it away. So tearing up the
ticket is irrational because it a loss to her. The value of the ticket before
the draw may be much less than the $2 or whatever she paid for it, but because
the payoff is so large, it's probably worth more than a penny-- it might be
worth something between a nickel and dime. Most people don't throw away nickels
and dimes. On the other hand, my being able to walk home might be of
considerable benefit to me, much more, quantitatively, than the cost of risking
the extremely low probability of being struck by lightning.
I like this solution better than previous one because it
remains an externalist account, but I don't like that it deflates the
interesting distinction between modal probability and modal (physical)
potential through basically a game theory analysis***.
_____
*How to interpret “knowledge” of future events is of course an ancient problem going back to Aristotle’s “sea battle” example, but more recent developments in modal logic look to have solved this problem.
**Initially I had thought that modal distance could just be
put together with probability over possible worlds in a simple way, as a ratio,
but I found that this yielded bad results, e.g. it would have said
that a given outcome in a lottery ball draw with fifty possible outcomes was
modally closer than a given outcome in a coin flip with two
possible outcomes, even though the inertia displacement from outcome to outcome
is roughly the same for both scenarios.
***Actually, I'm surprised Pritchard doesn't bring up game theory in his account of luck and risk. I wonder how he sees game theory fitting in with his account? I'm not sure that he really wants to rule out probability for modal distance altogether so much as say that the safety principle can't be reduced to a simple probability, i.e. 1/(# of possible outcomes). A game theoretical account could of course fully accommodate Bayesian induction.
The Gettier Getaway: a Gettier Example
Here is a Gettier-type (per Gettier's "Is Justified True Belief Knowledge?") case I came up with:
A car speeds past you. Directly behind this car follows a
speeding police cruiser with its siren on. The speeding cars, one after the
other, round a corner and disappear from sight.
You believe on the basis of this incident that the driver of
the speeding car is being pursued by the police, perhaps for a criminal act.
You are justified in doing so. And in fact, the driver is being pursued by the
police for a criminal act.
However, when you saw the driver, he was not trying to
escape the police cruiser behind him but was in fact speeding back to his
apartment because, after he had robbed a bank earlier that day, he remembered
he had left his oven on. The police cruiser behind him was not chasing him but
had actually been dispatched to go to the bank the driver had just robbed, in
order to pursue the robber (the officer had no idea that that robber was right
in front of her). This officer noticed that the car in front of her was
speeding, but since she had more important crimes to worry about than traffic
violations, she was not pursuing it at that moment.
(i) You believe that the driver is being
pursued by the police.
(ii) It is true that the driver is being
pursued by the police.
(iii) You are justified in believing the
driver is being pursued by the police.
But because what you saw does not actually constitute
evidence of his being pursued, you do not know that the driver is being pursued
by the police.
This case is different in an important respect from both the
first Gettier case and cases like the sheep case (qv. Chisholm) and the clock case (qv. Russell).
In those cases, the major premise of the valid inference is a sound
implication, but the minor premise is false.
First Gettier Case
(i) If Jones (who has ten coins in pocket) will get the job
(p), the person who will get the job has ten coins in pocket (q). [p ⊃ q]
(ii) Jones will get the job. [p]
∴ The
person who will get the job has ten coins in pocket. [q]
In fact, Jones will not get the job [~p], though
the person who will get the job has ten coins in pocket. [q]
Thus, Smith has made a valid but unsound inference due to
the minor premise being false.
Sheep Case
(i) If Roddy sees a sheep in the field (p), there is a sheep
in field (q). [p ⊃
q]
(ii) Roddy sees a sheep in the field. [p]
∴
There is a sheep in the field. [q]
In fact, Roddy does not see a sheep in the field (he sees a
sheep-like object) [~p], though there is a sheep in the field (that
he does not see). [q]
Thus, Roddy has made a valid but unsound inference due to
the minor premise being false.
Clock Case
(i) If the clock is working (p), the time it reads is the
actual time (q). [p ⊃
q]
(ii) The clock is working. [p]
∴ The
time it reads is the actual time. [q]
In fact, the clock is not working [~p], though
the time it reads is the actual time (it just so happens). [q]
Thus, a valid but unsound inference has been made due to the
minor premise being false.
Gettier Getaway Case
(i) If a speeding car is closely followed by a police car
with its siren on (p), the driver of the speeding car is being pursued by the
police (q). [p ⊃
q]
(ii) A speeding car is closely followed by a police car with
its siren on. [p]
∴ The
driver of the speeding car is being pursued by the police. [q]
In fact the speeding car is closely followed by a police car
with its siren on [p], and the driver of the speeding car is being
pursued by the police. [q]
Since, unlike the other cases, the minor premise in the
Gettier getaway case is true, and you have made a valid inference, something
else must have gone wrong. It is the major premise that is
unsound. We find that [p ⊃
q] in this case is not necessarily true. It is false
in some cases*.
Major premise cases, I think, take us closer to the problem
with the “no false lemmas” objection than minor premise cases because with them
we see that to be Gettier-proof, the major premise must be necessarily true in
all possible worlds, not just in almost all instances. This is a problem
because this would seem to set too high a threshold for justification. How
could one ever obtain certainty that one’s major premise is necessarily true in
all possible worlds? And how many natural inferential bases would be ruled out
due to not being necessarily true?
_____
*The garbage chute case (Sosa) is the most similar in this respect to
the Gettier getaway, except that (i) the garbage chute case is
inference-based instead of directly perceptual, as the would-be knowledge
holder does not see the basement but only infers its state from partial
evidence, and (ii) its consequent has to be made false [~q] in
order to show that it can go wrong.
Saturday, August 24, 2019
Wednesday, July 31, 2019
POLYSEMOUS POLYGRAPHY
Consider the sentence:
As Violet sat by the bank, after her trip over the spring, she saw the jam she had made for herself from the dates with her contemplative pupil.
This sentence contains at least 64 different narratives, depending on one of two meanings for the six words: bank, trip, spring, jam, dates, and pupil. (2^6 = 64) Every reading is coherent.
As Violet sat by the bank (building or slope), after her trip (voyage or tumble) over the spring (season or aquifer), she saw the jam (problem or condiment) she had made for herself from the dates (trysts or fruit) with her contemplative pupil (student or inner-eye).
Call this Oulipian method of composition Polysemous Polygraphy.
As Violet sat by the bank, after her trip over the spring, she saw the jam she had made for herself from the dates with her contemplative pupil.
This sentence contains at least 64 different narratives, depending on one of two meanings for the six words: bank, trip, spring, jam, dates, and pupil. (2^6 = 64) Every reading is coherent.
As Violet sat by the bank (building or slope), after her trip (voyage or tumble) over the spring (season or aquifer), she saw the jam (problem or condiment) she had made for herself from the dates (trysts or fruit) with her contemplative pupil (student or inner-eye).
Call this Oulipian method of composition Polysemous Polygraphy.
Friday, July 19, 2019
Sunday, June 30, 2019
Saturday, June 1, 2019
Monday, May 27, 2019
Tuesday, April 30, 2019
Friday, April 5, 2019
Sunday, March 31, 2019
Thursday, February 28, 2019
Reality Is Impossible
Things that are true are not possibly true. They are just true!
Likewise, things that are false are
not possibly false. They are just
false!
If something is necessarily true,
then it is true, which means that it is not possibly true. And if something is
necessarily false, then it is false, which means that it is not possibly false.
Yet Alethic Modal Logic tells us
that something being necessarily true just means that it is not possibly not true.
Does this definition, or pair of
definitions for Necessity N and Possibility P, [(Na = -P-a) & (Pa = -N-a)], make sense, given our two Reality Principles, (1) [(a > -Pa) & (-a > -P-a)] and (2) [(Na > a) & (N-a > -a)]?
Let's see:
Wednesday, February 27, 2019
False by Definition
With the Liar
Statement—i.e. “This statement is false,” which we can translate as “p = this
statement (p) is false = (p = ~p)”— the definition
of p is false, not p itself. What does it mean for a definition to be false? It
means that it is formally invalid. This in turn
means that it is not permitted within our logic to use a definition of this
form. Note that the statement of a definition implies the statement that it defines, but it is not identical to the statement it defines.
I.e. Def. p > (p = -p).
Def. q > ( q = -q) S
Def. q S
q = -q MP
q S
-q ES
q & -q CONJ
-q IP
q ES
q & -q CONJ
-(Def. q) IP
(Def. q > (q = -q)) > -(Def. q) CP
∀p [(Def. p > (p = -p)) > -(Def. p)] UG
(1), QED
In any case, the above proves that the Liar Statement is falsidical and not a real antinomic paradox.
Thursday, February 7, 2019
Thursday, January 31, 2019
Legislation: a Card Game
Legislation
E.B. Nelson, 1.31.19 (c)
A Card Game for 3 to 8 Players
Components:
16 Representative cards; 81 Bill
cards; Pro and Con Pledge Token sets numbered for players 1-8; Agenda Point
Tokens
Rules:
You play as an elected Representative
of Congress. Your goal is to get as many Bills that promote your Agenda passed as
you can over the course of the legislative Session, while blocking all Bills
that are detrimental to your Agenda.
You first draw a Representative
to play as from the Representative Deck. The Representative Deck must be
constructed according to the number of players so that there is an even
distribution of Representatives per Agendas (see the Deck list below).
Next, shuffle the deck of 81 Bills
and deal 9 face down to each player. You will look at these cards but conceal
them from the other players.
On the first turn, players decide
which of their cards they wish to keep, if any, and which they wish to discard.
Discard into your own Discard Pile in front of you.
On the second turn, players draw
the number of cards needed to make a full hand of 9 from any of the other
players’ Discard Piles and/or the remaining Deck. On each subsequent turn, at the
beginning of your turn, you can discard into your Discard Pile one card and
pick up one card from any of the other piles, or you can choose not to discard.
On the third turn, place 3 Bills
in front of you face up that you wish to place On Deck for future votes. Once
you place a card On Deck, you cannot discard and replace it. It must eventually
be voted on.
On the fourth turn, you seek Pledges
to vote on your Bills from the other players. You do this by bargaining with
each of them by offering to trade cards you have in your hand, to make Pledges
on their Bills, or to make anti-Pledges on their opponent’s Bills. Pledges made
by you and the other players are final only when your turn ends.
On the fifth turn, choose one of
your three Bills On Deck and put it to a vote. Everyone votes, and the vote is “Yay,”
“Nay,” or “Abstain.” Representatives can choose to betray their Pledges by voting
against their Pledged or anti-Pledged votes or by abstaining, but if they do
so, whether once or multiple times on a Vote round, they must skip the next
Pledge turn entirely, neither seeking Pledges nor making any (although players
can still Pledge on their On Deck Bills per bargains with other players). A Bill
pass if it has a positive vote total; otherwise it fails. A “Yay” counts as 1,
a “Nay” counts as -1, and an “Abstain” counts as 0. If your Bill passes, you
gain the Agenda Points listed for your Agenda on the Bill, as do the other
players with that Agenda, and others lose points as listed. Place passed Bills
in the Passed Bills Pile. If the Bill does not pass, place it in the Failed
Bills Pile. Finally, replace the empty On Deck spot with a new face up Bill.
Next is a Pledge round again,
followed by another Vote round, and so on. The game ends when all players run
out of Bills in their hands and On Deck, i.e. after 9 votes for each player
(optional: players can have hands of 6 instead for a shorter game).
Some cards also have Special
Effects if they pass, and Representatives each have a Special Effect they
cannot enact once per game. Follow the rules for these Effects as described.
The winner is the player with the highest Agenda Point total at the end of the
Session.
Representatives (two of each type):
Single Agenda:
- Social Progressive
- Social Conservative
- Fiscal Progressive
- Fiscal Conservative
- Social Progressive, Fiscal Progressive
- Social Progressive, Fiscal Conservative
- Social Conservative, Fiscal Conservative
- Social Conservative, Fiscal Progressive
Representative Deck
Builds per Number of Players:
Where X= Social Progressive, Y= Social Conservative
Where X= Social Conservative, Y= Social Progressive
Where A= Fiscal Progressive, B= Fiscal Conservative
Where A= Fiscal Conservative, B= Fiscal Progressive
3 Players: X, A, YB
4 Players: All single or all dual
5 Players: XA, YB, XB, Y, A
6 Players: X, Y, A, B, XA, YB
7 Players: XA, XB, YA, YB, X, Y, A
8 Players: All
Bill Deck Chart:
|
Bill
|
Agenda Points Scored if Passed per Representative Agenda
|
|||
|
# & Name
|
Social Progressive
|
Social
Conservative
|
Fiscal Progressive
|
Fiscal
Conservative
|
|
1
|
+1
|
+1
|
+1
|
+1
|
|
2
|
+1
|
+1
|
+1
|
-1
|
|
3
|
+1
|
+1
|
+1
|
0
|
|
4
|
+1
|
+1
|
-1
|
+1
|
|
5
|
+1
|
+1
|
-1
|
-1
|
|
6
|
+1
|
+1
|
-1
|
0
|
|
7
|
+1
|
+1
|
0
|
+1
|
|
8
|
+1
|
+1
|
0
|
-1
|
|
9
|
+1
|
+1
|
0
|
0
|
|
10
|
+1
|
-1
|
+1
|
+1
|
|
11
|
+1
|
-1
|
+1
|
-1
|
|
12
|
+1
|
-1
|
+1
|
0
|
|
13
|
+1
|
-1
|
-1
|
+1
|
|
14
|
+1
|
-1
|
-1
|
-1
|
|
15
|
+1
|
-1
|
-1
|
0
|
|
16
|
+1
|
-1
|
0
|
+1
|
|
17
|
+1
|
-1
|
0
|
-1
|
|
18
|
+1
|
-1
|
0
|
0
|
|
19
|
+1
|
0
|
+1
|
+1
|
|
20
|
+1
|
0
|
+1
|
-1
|
|
21
|
+1
|
0
|
+1
|
0
|
|
22
|
+1
|
0
|
-1
|
+1
|
|
23
|
+1
|
0
|
-1
|
-1
|
|
24
|
+1
|
0
|
-1
|
0
|
|
25
|
+1
|
0
|
0
|
+1
|
|
26
|
+1
|
0
|
0
|
-1
|
|
27
|
+1
|
0
|
0
|
0
|
|
28
|
-1
|
+1
|
+1
|
+1
|
|
29
|
-1
|
+1
|
+1
|
-1
|
|
30
|
-1
|
+1
|
+1
|
0
|
|
31
|
-1
|
+1
|
-1
|
+1
|
|
32
|
-1
|
+1
|
-1
|
-1
|
|
33
|
-1
|
+1
|
-1
|
0
|
|
34
|
-1
|
+1
|
0
|
+1
|
|
35
|
-1
|
+1
|
0
|
-1
|
|
36
|
-1
|
+1
|
0
|
0
|
|
37
|
-1
|
-1
|
+1
|
+1
|
|
38
|
-1
|
-1
|
+1
|
-1
|
|
39
|
-1
|
-1
|
+1
|
0
|
|
40
|
-1
|
-1
|
-1
|
+1
|
|
41
|
-1
|
-1
|
-1
|
-1
|
|
42
|
-1
|
-1
|
-1
|
0
|
|
43
|
-1
|
-1
|
0
|
+1
|
|
44
|
-1
|
-1
|
0
|
-1
|
|
45
|
-1
|
-1
|
0
|
0
|
|
46
|
-1
|
0
|
+1
|
+1
|
|
47
|
-1
|
0
|
+1
|
-1
|
|
48
|
-1
|
0
|
+1
|
0
|
|
49
|
-1
|
0
|
-1
|
+1
|
|
50
|
-1
|
0
|
-1
|
-1
|
|
51
|
-1
|
0
|
-1
|
0
|
|
52
|
-1
|
0
|
0
|
+1
|
|
53
|
-1
|
0
|
0
|
-1
|
|
54
|
-1
|
0
|
0
|
0
|
|
55
|
0
|
+1
|
+1
|
+1
|
|
56
|
0
|
+1
|
+1
|
-1
|
|
57
|
0
|
+1
|
+1
|
0
|
|
58
|
0
|
+1
|
-1
|
+1
|
|
59
|
0
|
+1
|
-1
|
-1
|
|
60
|
0
|
+1
|
-1
|
0
|
|
61
|
0
|
+1
|
0
|
+1
|
|
62
|
0
|
+1
|
0
|
-1
|
|
63
|
0
|
+1
|
0
|
0
|
|
64
|
0
|
-1
|
+1
|
+1
|
|
65
|
0
|
-1
|
+1
|
-1
|
|
66
|
0
|
-1
|
+1
|
0
|
|
67
|
0
|
-1
|
-1
|
+1
|
|
68
|
0
|
-1
|
-1
|
-1
|
|
69
|
0
|
-1
|
-1
|
0
|
|
70
|
0
|
-1
|
0
|
+1
|
|
71
|
0
|
-1
|
0
|
-1
|
|
72
|
0
|
-1
|
0
|
0
|
|
73
|
0
|
0
|
+1
|
+1
|
|
74
|
0
|
0
|
+1
|
-1
|
|
75
|
0
|
0
|
+1
|
0
|
|
76
|
0
|
0
|
-1
|
+1
|
|
77
|
0
|
0
|
-1
|
-1
|
|
78
|
0
|
0
|
-1
|
0
|
|
79
|
0
|
0
|
0
|
+1
|
|
80
|
0
|
0
|
0
|
-1
|
|
81
|
0
|
0
|
0
|
0
|
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