Is there a difference between evidence that supports a theory and evidence that is merely consistent with a theory?

This post comes from a post by Stephen Law on the same subject in relation to Creationism and subsequent discussions on his blog. One of Stephen’s main criticisms of the Creationist position is that they subtly change their theory in order to save it in the face of evidence. The result is a theory that is compliant with evidence, but not supported by it. Much of the discussion (at least from me) centred around Stephen’s requirements for a theory to be capable of being supported by evidence:

“o First, the theory must allow us to deduce observational consequences:
consequences that can be clearly and precisely stated.
o Second, there should
be a sense in which the prediction is surprising and unexpected.
o Third, the
prediction should turn out to be true.”

I advanced the (not very well worded) thesis that this reduced to Popper’s concept of corroboration: evidence supports theory-x only if its absence would falsify theory-x. Creationism’s tendency to hedge its bets progressively removes any support it has from the evidence. This post is an attempt to bring more clarity to the thesis.

Starting Point
Let us, first of all, characterise "Creationism" and "Evolutionism":

Creationism: “God created all the species on the planet within six 24 hour days less than 10,000 years ago” we will call this ψ (“psi”)

Evolutionism: “The species evolved from a small number of common ancestors by means of evolution over a period in excess of 4 billion years”. We will call this φ (“phi”)

ψ entails certain things, it means certain things. If we were to give the full meaning of ψ we would list out an infinite number of things that result from the truth of ψ. ψ means ‘A and B and C and D….’. Using a full-stop for “and”, an arrow → for “means” gets us to ‘ψ→A.B.C.D.E.F….’. For the purposes of this illustration I shall truncate the meaning to just the first four propositions entailed:

1. ψ→A.B.C.D (psi means A and B and C and D)

If one of these (A, B , C or D) is incorrect then, in that respect ψ is incorrect. If one of these is correct then, in that respect ψ is correct. I shall characterise the meaning of φ as follows:

2. φ→G.H.I.J (phi means G and H and I and J )


Possible Changes
If the adherent to ψ subsequently finds out, say, ¬A (‘not A’) he has four options:

Drop the belief in ψ:

3. ¬ ψ (not ψ)

Drop the entailment of A by ψ:

4. ψ→B.C.D (ψ means, just, B and C and D)

Add additional possible outcomes (with “v” as “or”):

5. ψ→(AvL).B.C.D (ψ means “A or L” and B and C and D)

Replace the entailment of A with the entailment of something else:

6. ψ→L.B.C.D (ψ means L and B and C and D)

If more negative evidence is found the same choices are open.

At the limit (all entailments appear negative) we get to:

3*. ¬ ψ

4*. ψ→ (ψ means nothing)

5*. ψ→(AvL).(BvM).(CvN).(DvO)

6*. ψ→L.M.N.O

(Notice that in 4,5,6,4*,5* and 6* the meaning of ψ changes. None of “A.B.C.D”, “B.C.D”, “(AvL).B.C.D”, “L.B.C.D”, “ ”, “(AvL).(BvM).(CvN).(DvO)” or “L.M.N.O” are identical.)

Following the strategy in 3 and 3* the adherents of ψ stop being adherents of ψ. Following the strategy in 4 and 4* the adherents of ψ remove all meaning from ψ.
What of the situation in 5*: (AvL).(BvM).(CvN).(DvO)? Well “L” most decidedly is not “A”, we got to “L” from our evidence “¬A”. We got to “M” from our evidence “¬B” and so on. ¬A.¬B.¬C.¬D together with (AvL).(BvM).(CvN).(DvO) entail:

7. (Av¬A).(Bv¬B).(Cv¬C).(Dv¬D).

Each of those disjunctive pairs entails nothing, thus, means nothing and can be discounted. However ψ does say something, we also added entailments to the meaning of ψ. In full what we have is:

8. ψ→(Av¬AvL).(Bv¬BvM).(Cv¬CvN).(Dv¬DvO)

Again each of the meaningless disjunctive pairs can be discounted, with the result:

9. ψ→L.M.N.O

This is the same as the result of following the strategy in 6 and 6*.

Evidence

We may be able to check whether A or ¬A holds. If so then A or ¬A, whichever does hold, can count as evidence. If there are entailments of ψ that we are unable to check up on, say B and ¬B, these may still form part of the meaning of ψ but cannot be either evidence for or against ψ (or indeed evidence for or against any proposition).

Let now expand our conception of ψ a little. We shall say that ψ means:

a) Evidential statements L, M, N and O
b) Non-evidential statements P, Q, and R

10. ψ→L.M.N.O.P.Q.R


Bring Evolutionism Back Into Things

Of course whilst all this adaption of ψ has been going on the evolutionists have been up to the same tricks, φ has been revised as well. Let’s say:

11. φ →L.M.N.O.S.T.U

With S, T and U being non-evidential statements.

We can notice two things:

a) Neither proposition can be proven. Proving ψ requires ascertaining P, Q and R the truth of which we cannot ascertain. Proving φ requires ascertaining S, T and U the truth of which we cannot ascertain.
b) The totality of their evidential statements, (L, M, N and O), are exactly the same.

For the purposes of assessing the evidence we may drop the non-evidential statements from our definitions:

12. Evidentially, ψ→L.M.N.O and
13. Evidentially, φ→L.M.N.O

Evidentially speaking ψ and φ are the same proposition.
In this situation our choice between the two propositions cannot be based on evidence. Assuming:
- The above definitions are complete definitions of ψ and φ
- ψ and φ are the only propositions under consideration
- L, M, N and O hold true
Then based on the evidence we must adopt either ψ or φ. The evidence supports “either ψ or φ”.

L, M, N and O do not prove either proposition
L, M, N and O are evidence for the propositions because they force us to accept part of what they mean
L, M, N and O are no evidence at all for “P and Q and R” or “S and T and U”, there is no “force” to accept either of them.


A Difference

Right, let’s bring in a number of ways in which ψ and φ can be different:

14. ψ→L.M.N , φ→L.M.N.O
15. ψ→L.M.N.(Ov¬O) , φ→L.M.N.O
16. ψ→L.M.N.¬O , φ→L.M.N.O

In situation 14. “L.M.N” can be taken to be evidence for either ψ or φ it gives no indication as to which one to adopt. “O”, on the other hand is evidence for φ. It forces us to accept at least part of φ. It is not evidence for ψ, it does not force us to accept any of ψ.

In situation 15. we can rewrite “(Ov¬O)” as a blank “” bringing us back to the situation in 1. “O” is evidence for φ but not for ψ.

In situation 16. “O” is evidence for φ and evidence against ψ. Where O to hold ψ would have to be revised either to:

16.a ψ→L.M.N
16.b ψ→L.M.N.O

And we would be back to either 14. or the propositions would be evidentially similar.

Notice that O to be evidence for ψ we have to be able to write out ψ→O without adding ¬O. If we ascertain ¬O having stated ψ→O then we have to revise ψ. If we are in a situation where we do not have to revise ψ after ascertaining ¬O then we must either have started with

17. ψ→¬O or
18. ψ→(Ov¬O)

In neither situation is O evidence for ψ.

O is evidence for ψ if and only if ¬O forces a revision in ψ

The Fossil Record

Evolutionism makes some predictions about the fossil record. Not very precise ones and not very many, but some. For example evolutionism never says “There will be a fossil right there”, “FRT”. If we do not find an FRT we do not revise evolutionary theory. Thus evolutionary theory, φ, predicts “either there will be an FRT or there won’t be an FRT” or φ→(FRTv¬FRT).

Evolutionism does not say that there will be lots of fossils (LOF), that they will contain certain types of fossil (CTF) etc etc.

19. φ→(FRTv¬FRT).(LOFv¬LOF).(CTFv¬CTF), which cancels down to φ→

To have any meaning for evolutionary theory in the fossil record we have to find something that would force a revision in evolutionary theory. What could that be? One reply is “A rabbit fossil in the in the pre-cambrian”. Not much of a prediction but it is a predicition. Evolutionary theory means we do have a Rabbit Free Pre-Cambrian Type Fossil Record (RFPC).

21. φ→RFPC

RFPC is evidence for Evolutionism

And Creationism? Well RFPC is consistent with creationism. ¬RFPC is also consistent with Creationism. It does not say either way. So:

20. ψ→(RFPCv¬RFPC)

Which cancels down to ψ→.

RFPC is not evidence for Creationism

Surprise!

Let us say that both ψ→O and φ→O. O is evidence for ψ and φ. O is either evidence equally for ψ and φ or is not evidence at all for at least one of them. Very crudely, if ψ and φ were the only two propositions that →O, O would support them both 50%.

Let us add a third, χ (“chi”). χ→O. There are now three propositions ψ, φ and χ for which O is evidence. Very crudely again, O supports them 331/3% each. Add another, ω (“omega”) and the crude measure of evidential support becomes 25%.

Lets add a fifth, κ (“kappa”) but say that κ→(Ov¬O). We know that ¬O does not force a revision in κ, so O is not evidence for κ. There remain just the four theories that →O and so each remains supported 25% by O.

We could go on through the Greek alphabet and beyond but I think we have demonstrated the point.

The existence of support of ψ by O depends upon whether ¬O forces a revision in ψ. The amount of support of ψ by O depends upon the number of other theories where ¬O also forces a revision.

Now supposing we had ψ, φ, χ and κ all equally likely. How likely are we to have O? Well, ψ, φ and χ all firmly predict O if ψ, φ or χ are true then O is definite. If κ is true then we have no idea either way, 50%. The total likelihood of O is [(3x100%) + (50%)]/400% = 87.5%

Now suppose that φ, χ and κ do not predict O either way. The total likelihood of O is now [100% + (3x50%)]/400% = 62.5%

Suppose that φ, χ and κ predict ¬O. If they were true the likelihood of O would be zero. The total likelihood of O would be [100% + (3x0%)]/400% = 25%

As the number of rival theories not predicting O or firmly predicting ¬O increases the likelihood of O decreases. The term “surprising” springs to mind. As the number of rival theories not predicting O or firmly predicting ¬O increases the amount of support by O increases.

“Surprising” = exclusivity of firm prediction = amount of support


Now…

The situation right now is that there are bundles of evidence for evolution. Much of this is compatible with Creationism (Creationism has been changed to fit it) but is not a firm prediction of Creationism. Creationism does not say “if Creationism then Evidence”, it says “if Creationism then the Evidence or Not the Evidence”.

Much of the evidence for Evolutionism is surprising. We would not expect it if Evolutionism were not as it is. The apparent fusing of two ape chromosomes to produce one human chromosome is not just surprising it is, frankly, flabbergasting. (Have your flabber-gasted at this link)

Creationism on the other hand presents no evidence what-so-ever that is not also evidence of Evolution. There is nothing that forces a revision of Creationism that would not also force a revision of Evolutionism.

If Author thinks there is any evidence for Creationism he is welcome to put it forward. Just make sure that:

21. Creationism would have to be revised if it were to be false
22. Godless-communist-fag-Evolutionism would not also have to be revised were it to be false.

Bayesianism, the Brunson Objection

Bayesian’s use probability theory to explain the degrees of belief or acceptance that people assign to propositions.

I do not like the use of probability in epistemology. I have always (well, since someone told me about it) viewed probability as a matter of coping with lack of knowledge rather than supporting knowledge. More than that, we need knowledge in order to make probability “work”. The probability of a fair coin landing heads on any particular toss is 0.5 Why? Because we know that the coin has two sides and we know it will land on just one of them. Our ignorance is limited to which particular one it will land on. Assume that the coin has an unspecified number of sides and the assignment of a probability to heads is impossible.


Specifying Probabilities

Never-the-less Bayesian’s use complicated maths to, allegedly, give us some idea of probability of some hypotheses. And so what? Am I to accept a proposition because it is more likely than not? Because it is more likely than the relevant alternatives? Because it is 0.x probable? The likelihood of ‘there is free beer in the pub across the road’ need only be very, very, low for me to be induced to accept the proposition. The likelihood of ‘this plane will not crash’ has to be, slightly, higher.


Valuing Consequences

We can explain my varying attitude by considering the value I place on the end results:

• Having beer and
• Not dying

And we can consider the combination of probability and consequences mathematically by using expected values. If I have a defined probability and I can assign a numerical value then I can calculate a definite figure for the expected value.
Let us say I bet £1 on a number at (English) roulette.

• The probability of me being right is 1/37
• If I am right I get 35 times my original stake, plus my stake : up £35
• If I am wrong I lose my stake: down £1
• The expected value is (1/37 x £35) + (36/37 x -£1) = a little under 3p negative.
  

It's a bad bet and I should pass it up.

Without nice neat probabilities and nice neat valuations (how to value the benefit of discovering Neptune against the benefit of discovering where Newton got it wrong?) we can't get nice neat expected values. In lots of situations, naturally, we do not need nice neat expected values. The negative-benefit of dying so far outweighs the benefit of almost all plane journeys that the expected value of air travel given any appreciable risk is negative.

But that is still not enough to explain our decisions.

Now this guy knows probability:

He is Doyle Brunson, world famous Poker Player and author of the poker player's 'Bible': 'Super/System A Course in Power Poker'.

In Super/System Doyle looks at the following scenario:

Doyle's Scenario: You are offered odds of 10:1 on a single toss of a fair coin provided you bet all the money you have in the world.

Now we know exactly the probability of "this coin will be heads on the next toss": 0.5. We can put a monetary value on the likely outcome: 11 times all the money you have in the world if you win, 0 times all the money you have in the world if you lose.

So the expected value is very easy to calcualte: 0.5 x 11 x all the money you have in the world = 5.5 times all the money you have in the world.

What would your decision be in Doyle's Scenario? I wouldn't bet, I suspect that your wouldn't bet either. Doyle would:

But, I’d do it. I surely would. I’d just have to. I couldn’t
pass up the opportunity to take 10 to 1 on an even-money shot. (Brunson,
Doyle. Super/System A Course in Power Poker. Las Vegas, Nevada. B
& G Publishing. page 512)

Even if the Bayesians can establish their probabilities, even if they can assign numbers to the values of consequences it still does not explain our actions. We need to assign numbers and have a method of calculating the effect of the absolute value of the possible actions.

Our Empirical Basis has no Empirical Basis

I read Graham Priest's book "Logic: A Very Short Introduction" ( http://www.oup.com/uk/catalogue/?ci=9780192893208 ). Very enjoyable it was too (and quite droll that a man called "Priest" uses refutations of arguments for the existence of God as his main examples!).


If the Argument from Design is hogwash.....

One argument got me thinking. This was Graham's take on the "Argument from Design". The "Argument from Design" takes the observation of the order ("O") in the universe as confirmation (to varying extents) of the existence of God ("G"). Using "P(A¦B)" as "The propbability of A given B" the claim that O confirms G is:

(1) P(G¦O) > P(¬G¦O) (the probability of God given the existence of Order is greater than the probability of not-God given the existence of order).

(2) This, Graham asserts, is equivalent to P(G) > P(¬G). The probability of God before any evidence is greater than the probablity of not-God before any evidence. If God is, prior to any evidence, more likely to exist than not exist then any evidence adds confirmation to his existence. If God is, prior to any evidence, more likely to not-exist then any evidence adss confirmation to his non-existence. Graham then disputes that, prior to any evidence, God is more likely to exist than not.

(a more detailed working is in M. Colyvan, J. Garfield and G. Priest, 'Problems with the Argument for Fine Tuning', Synthese 145 (2005), 325-38)

It got me thinking as to whether Graham's argument applied to other hypotheses, such as "I see a pint of beer" or "I see X" and, where X is "this is my empirical basis", whether the whole empirical basis disappears.


Lets Stick Some Numbers In

The probability of hypothesis (H) after some evidence (E), P(H¦E), is given by Bayes' Theorem:

(3) P(H¦E) = [P(E¦H) x P(H)]/P(E)

(The probability of a hypothesis given the evidence is

• the probability of the evidence given the hypothesis times the probability of the hypothesis ignoring the evidence (ie the probability that you are seeing the evidence because of the hypothesis)
• divided by the probabiliy that we would see the evidence anyway)

(The best explanation Bayes' Theorem on the web is Eliezer Yudkowsky's interactive explanation on http://yudkowsky.net/bayes/bayes.html)


Substituting

• G and ¬G for H (our hypothese are "God" and "Not-God"), and
• O for E (our evidence is the order in the universe

We get

• Probability of God given Order: [P(O¦G)xP(G)]/P(O)
• Probability of not-God given Order: [P(O¦¬G)xP(¬G)]/P(O)

Now can we add numbers to these? We can, at least have a go.

(4) P(G) and P(¬G) = 0.5. Before any evidence, before any consideration we should be completely indifferent between these two hypotheses. Thus the prior probabilites are both 0.5

(5) P(O¦G) = 1. Our conception of God is such that He would create order. So the probability of order, given the existence of God, is 1.

(6) P(O¦¬G) = 0.5 If there is no God, what are the chances of there being order? We have no ****ing idea again: so it is 0.5.

(7) P(O) = 0.75. The probability of order (given no evidence) is the probability of order, given God plus the probability of order given no God:

P(O) = [P(O¦G) x P(G)] + [P(O¦¬G) x P(¬G)] = (1 x 0.5) + (0.5 x 0.5) = 0.75

Substituting these numbers into our formulae:

• Probability of God given Order: (1x0.5)/0.75 = 2/3
• Probability of not-God given Order: (0.5x0.5)/0.75=1/3

It's considerably more likely that God exists given order than that God does not exist.

Ah, but....

What about a self-ordering non-divine universe (S)? On the same basis as steps (4) to (7) the probability of a self-ordering non-divine universe given that we have order is 2/3.

Or add something to the concept of God: "God is an Englishman". This can be translated as "there exists a God who is English" (say "(G&E)" and call this "ѱ"). ѱ and ¬ѱ will have the same relationship as G and ¬G; and S and ¬S. The probability of ѱ given there is order in the universe is 2/3

Or add something completely ridiculous to the concept of God: "God ordered the universe and koppites are not gobshites" ("ϕ").........

The posterior probabilities, where we have absolutely no idea before we gather evidence, are dependent upon the way we formulate the hypothesis. Careful formulation of any possible hypothesis will always make it more likely than not given the evidence. So what is determining our hypotheses appears to be something prior to evidence.

Now there may well be objections to the way that I have formulated the prior pobabilities of God, order, self-order or Koppites being Gobshites. Many may well be able to come up with different prior probabilites that are equally acceptable. They will, however, be equally acceptable. They will not overrule mine, nor yours. They will not be fixed and not, in any way, be fixed by evidence.

And what about the evidence itself?

What we commonly call 'evidence' is, itself, a theory dependent on the really basic evidence of our experience. I may be mistaken about the pint of beer in front of me (the "experiential statment, "ES"), despite the fact that I have all the experiences compatible with a pint of beer being in front of me (the experiences, "E"). It remains possible that E is true whilst ES is not. Thus the relation between E and ES is not "E if and only if ES" (E ↔ ES) but "E only if ES" (ES → E). Experience is inconclusive evidence for experiential statements. From above, prior to any evidence statements, the likelihood of our conclusion ES depends upon non-evidential statements about the likelihood of and formulation of ES itself.

Is "Truth-tracking" an alternative term for "Certainty"?

The Gettier Problem
As every schoolkid who has taken 'Theory of Knowledge' knows Edmund Gettier ( http://www.ditext.com/gettier/gettier.html ) questioned the sufficiency of the 'traditional' definition of knowledge as:

Justified (J)
True (T)
Belief (B)

by giving counterexamples were 'Smith' or 'Jones' or whoever had Justified True Belief but not, intuitively, knowledge. One solution to the probem is to expand the definition of knowledge to include a degettierising "truth-tracking" condition (DG): if the proposition were not true then the subject would not believe it.

Fallibilism
Now, just about everyone (not just schoolkids taking TOE) has a pretty strong intuition that justification need not be certain. If we insisted that our justifications alone entailed truth we would have very little justification and very little knowledge indeed.

The Infallibilist position can be characterised as justification implying truth. If it is justified, then it is true; if it is not true then it is not justified. What we wish to avoid is the entailment of truth by Justification. The problem with truth-tracking is that it implies that a believed proposition can only be justified if it is certain.

Nasty Consequence (NC) : B→ ¬(J&¬T) (if it is believed then it cannot be justified and false).

Lets characterise the truth-tracker degettierising condition:

(DG) ¬T→¬B (if it is not true then it is not believed)

Does DG entail NC? We need a counterexample to:

1. (¬T→¬B) ├ B→ ¬(J&¬T) (if it is not true then it is not believed entails if it is believed then it cannot be justified and false)

A counter example would be a believed proposition that is both justified and false:
2. B & (J&¬T)

Without DG that is a piece of cake, with DG, however:

3. DG is true in two circumstances:
3.a ¬B, we do not believe the proposition
3.b T the proposition is true

Both of these conflict with 2.


And? Your Point Is?
A truth-tracking condition implies certainty in knowledge, you cannot add a truth-tracking condition to the Justified True Belief conception of knowledge and retain fallible knowledge. The DG condition requires anything that is justifed and believed to be certain. As any knowledge must be justified and believed any knowledge with DG must be certain.

Swap "reliably formed" for justification and, for the most part, we fail to solve the problem. Most conceptions of reliability include truth-tracking within their make up.

We have three choices:
1. Accept it that knowledge must be certain
2. Accept that the examples given by Gettier are examples of genuine knowledge.
3. Come up with some other solution.