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Category: Philosophy

The philosophy of science, empiricism, a priori reasoning, epistemology, and so on.

December 4, 2017 | 2 Comments

There Is No “Problem” Of Old Evidence In Bayesian Theory

Update I often do a poor job setting the scene. Today we have the solution to an age-old problem (get it? get it?), a “problem” thought to be a reason not to adopt (certain aspects of) Bayesian theory or logical probability. I sometimes think solutions are easier to accept if they are at least as difficult as the supposed problems.

I was asked to comment by Bill Raynor on Deborah Mayo’s article “The Conversion of Subjective Bayesian, Colin Howson, & the problem of old evidence“.

Howson is Howson of Howson & Urbach, an influential book that showed the errors of frequentism, but then introduced a few new ones due to subjectivity. We’ve talked time and again on the impossibility that probability is subjective (where probability depends on how many scoops of ice cream the scientist had before taking measurements), but we’ve never yet tackled the so-called problem of old evidence. There isn’t one.

Though there is no problem of evidence, old or new, there are plenty of problems with misleading notation. All of this is in Uncertainty.

The biggest error, found everywhere is probability, is to only partially write down the evidence one has for a proposition, and then that information “float”, so that the one falls prey to equivocation.

Mayo:

Consider Jay Kadane, a well-known subjective Bayesian statistician. According to Kadane, the probability statement: Pr(d(X) >= 1.96) = .025

“is a statement about d(X) before it is observed. After it is observed, the event {d(X) >= 1.96} either happened or did not happen and hence has probability either one or zero” (2011, p. 439).

Knowing d0= 1.96, (the specific value of the test statistic d(X)), Kadane is saying, there’s no more uncertainty about it.* But would he really give it probability 1? If the probability of the data x is 1, Glymour argues, then Pr(x|H) also is 1, but then Pr(H|x) = Pr(H)Pr(x|H)/Pr(x) = Pr(H), so there is no boost in probability for a hypothesis or model arrived at after x. So does that mean known data doesn’t supply evidence for H? (Known data are sometimes said to violate temporal novelty: data are temporally novel only if the hypothesis or claim of interest came first.) If it’s got probability 1, this seems to be blocked. That’s the old evidence problem. Subjective Bayesianism is faced with the old evidence problem if known evidence has probability 1, or so the argument goes.

Regular readers (or those who have understood Uncertainty) will see the problem. For those who have not yet read that fine, award-eligible book, here is the explanation.

To write “Pr(d(X) > 1.96)” is to make a mistake. The proposition “d(X) > 1.96” has no probability. Nothing has a probability. Just like all logical argument require premises, so do all probabilities. They are here missing, and they are later supplied in different ways and equivocation occurs. In this case deadly equivocation.

We need a right hand side. We might write

     (1) Pr(d(X) > 1.96 | H),

where H is some compound, complex proposition that supplies information about the observable d(X), and what the (here anyway) ad hoc probability model for d(X) is. If this model allows quantification, we can calculate a value for (1). Unless that model insists “d(X) > 1.96” is impossible or certain, the probability will be non-extreme (i.e. not 0 or 1).

Suppose we actually observe some d(X_o) (o-for-observed). We can calculate

     (2) Pr(d(X) > d(X_o) | H)

and unless d(X_o) is impossible or certain, then again we’ll calculate some non-extreme number. (2) is almost identical with (1) but with a possibly different number than 1.96 for d(X_o). The following equation is not the same:

     (3) Pr( 1.96 >= 1.96 | H),

which indeed has a probability of 1.

Of course! “I observed what I observed” is a tautology where knowledge of H is irrelevant. The problem comes in there to put the actual observation, of the right or left hand side.

Take the standard evidence of a coin flip C = “Two-sided object which when flipped by show one of h or t”, then Pr(h | C) = 1/2. One would not say because one just observed a tail on an actual flip that, suddenly, Pr(h | C) = 0. Pr(h | C) = 1/2 because that 1/2 is deduced from C about h. (h is the proposition “An h will be observed”).

Pr(I saw an h | I saw an h & C) = 1, and Pr(A new h | I saw an h & C) = 1/2. It is not different from 1/2 because C says nothing about how to add evidence of new flips.

Suppose for ease d() is “multiply by 1” and H says X follows a standard normal (ad hoc is ad hoc, so why not?). Then

     (4) Pr(X > 1.96 | H) = 0.025.

If an X of (say) 0.37 is observed, then what does (4) equal? The same. But this is not (4):

     (5) Pr(0.37 > 1.96 | H) = 0,

but because of the assumption H includes, as it always does, tacit and implicit knowledge of math and grammar.

Or we might try this:

     (6) Pr(X > 1.96 | I saw an old X = 0.37 & H) = 0.025.

The answer is also the same because H like C says nothing about how to take old X and modify the model of X.

Now there are problems in this equation, too:

     (7) Pr(H|x) = Pr(H)Pr(x|H)/Pr(x) = Pr(H).

There is no such thing as “Pr(x)” nor does “Pr(H)” exist and we already seen it is false that “Pr(x|H) = 1”.

Remember: Nothing has a probability! Probability does not exist. Probability, like logic, is a measure of a proposition of interest with respect to premises. If there are no premises, there is no logic and no probability.

Better notation is:

     (8) Pr(H|xME) = Pr(x|HME)Pr(H|ME)/Pr(x|ME),

where M is a proposition specifying information about the ad hoc parameterized probability model, H is usually a proposition saying something about one or more of the parameters of M, but it could also be a statement about the observable itself, and x is a proposition about some observable number. And E is a compound proposition that includes assumptions about all the obvious things.

There is no sense that Pr(x|HME) nor Pr(x|ME) equals 1 (unless we can deduce that via H or ME) before or after any observation. To say so if to swap in an incorrect probability formulation, like in (5) above.

There is therefore no old evidence problem. There are many self-created problems, though, due to incorrect bookkeeping and faulty notation.

December 3, 2017 | 1 Comment

Summary Against Modern Thought: Every Agent Acts for a Good

Previous post.

Every agent acts for what it thinks is a good, even if it misses the mark.

That every agent acts for a good

1 Next after this we must show that every agent acts for a good.

2 That every agent acts for an end has been made clear from the fact that every agent tends toward something definite. Now, that toward which an agent tends in a definite way must be appropriate to it, because the agent would not be inclined to it except by virtue of some agreement with it. But, what is appropriate to something is good for it. So, every agent acts for a good.

Notes Be careful: a good is not necessarily the Good.

3 Again, the end is that in which the appetitive inclination of an agent or mover, and of the thing moved, finds its rest. Now, the essential meaning of the good is that it provides a terminus for appetite, since “the good is that which all desire.” Therefore, every action and motion are for the sake of a good.

4 Besides, every action and movement are seen to be ordered in some way toward being, either that it may be preserved in the species or in the individual, or that it may be newly acquired. Now, the very fact of being is a good, and so all things desire to be. Therefore, every action and movement are for the sake of a good.

5 Moreover, every action and movement are for the sake of some perfection. Even if the action itself be the end, it is clear that it is a secondary perfection of the agent. But, if the action be a changing of external matter, it is obvious that the mover intends to bring about some perfection in the thing that is moved. Even the thing that is moved also tends toward this, if it be a case of natural movement. Now, we call what is perfect a good. So, every action and movement are for the sake of a good.

6 Furthermore, every agent acts in so far as it is in act, and in acting it tends to produce something like itself. So, it tends toward some act. But every act has something of good in its essential character, for there is no evil thing that is not in a condition of potency falling short of its act. Therefore, every action is for the sake of a good.

7 Again, an intelligent agent acts for the sake of an end, in the sense that it determines the end for itself. On the other hand, an agent that acts from a natural impulse, though acting for an end, as we showed in the preceding chapter, does not determine the end for itself, since it does not know the meaning of an end, but, rather, is moved toward an end determined for it by another being.

Now, the intelligent agent does not determine the end for itself, unless it do so by considering the rational character of the good, for an object of the intellect is only motivating by virtue of the rational meaning of the good, which is the object of the will. Therefore, even the natural agent is neither moved, nor does it move, for the sake of an end, except in so far as the end is a good; for the end is determined for the natural agent by some appetite. Therefore, every agent acts for the sake of a good.

Notes In that first paragraph is the (correct) notion of design. Even suicide is seen as a good at the moment, for instance to remove or relieve mental torment. But suicide is not Good.

8 Besides, there is the same general reason for avoiding evil that there is for seeking the good, just as there is the same general reason for moving downward and for moving upward. But all things are known to flee from evil; in fact, intelligent agents avoid a thing for this reason: they recognize it as an evil thing. Now, all natural agents resist corruption, which is an evil for each individual, to the full extent of their power. Therefore, all things act for the sake of a good.

9 Moreover, that which results from the action of an agent, but apart from the intention of the agent, is said to happen by chance or by luck. But we observe that what happens in the workings of nature is either always, or mostly, for the better.

Thus, in the plant world leaves are arranged so as to protect the fruit, and among animals the bodily organs are disposed in such a way that the animal can be protected. So, if this came about apart from the intention of the natural agent, it would be by chance or by luck. But this is impossible, for things which occur always, or for the most part, are neither chance nor fortuitous events, but only those which occur in few instances. Therefore, the natural agent tends toward what is better, and it is much more evident that the intelligent agent does so. Hence, every agent intends the good when it acts.

Notes Recall the luck and chance are not real, but pertain to your mental state, your understanding. A fruit tree will produce fruit unless it is prevented in doing so by some cause or mechanism.

10 Furthermore, everything that is moved is brought to the terminus of the movement by the mover and agent. So, the mover and the object moved must tend toward the same thing. Now, the object moved, since it is in potency, tends toward act, and so toward the perfect and the good, for it goes from potency to act through movement. Therefore, both the mover and the agent always intend the good in their movement and action.

11 This is the reason why the philosophers, in defining the good, have said: “the good is what all desire.” And Dionysius states that “all crave the good and the best [De div. nom. IV, 4].”

November 26, 2017 | 15 Comments

Summary Against Modern Thought: Every Agent Acts for an End

Previous post.

Every agents does indeed act for an end, and never for nothing. For nothing is the absence of anything. There is no such thing as “random”.

How every agent acts for an end

1 The first thing that we must show, then, is that in acting every agent intends an end.

2 In the case of things which obviously act for an end we call that toward which the inclination of the agent tend the end. For, if it attain this, it is said to attain its end; but if it fail in regard to this, it fails in regard to the end intended, as is evident in the case of the physician working for the sake of health, and of the man who is running toward a set objective.

As far as this point is concerned, it makes no difference whether the being tending to an end is a knowing being or not. For, just as the target is the end for the archer, so is it the end for the motion of the arrow.

Now every inclination of an agent tends toward something definite. A given action does not stem from merely any power but heating comes from heat, cooling from cold. Thus it is that actions are specifically distinguished by virtue of diversity of active powers. In fact, an action may sometime terminate in something which is made, as building does in a house, and as healing does in health. Sometimes, however, it does not, as in the cases of understanding and sensing. Now, if an action does in fact terminate in some thing that is made, the inclination of the agent tend through the action toward the thing that is produced. But if it does not terminate in a product, then the inclination of the agent tends toward the action itself. So, it must be that every agent in acting intends an end, sometimes the action itself, sometimes a thing produced by the action.

Notes Insert What’s My Line about product joke here.

3 Again, with reference to all things that act for an end, we say that the ultimate end is that beyond which the agent seeks nothing else; thus, the action of a physician goes as far as health, but when it is attained there is no desire for anything further. Now, in the action of all agents, one may find something beyond which the agent seeks nothing further. Otherwise, actions would tend to infinity, which is impossible. Since “it is impossible to proceed to infinity,” the agent could not begin to act, because nothing is moved toward what cannot be reached. Therefore, every agent acts for an end.

Notes You can’t go to where you can’t get to, and don’t know where it is. Infinity is tricky, as the next argument shows. This even holds after the resurrection. Infinity is infinitely far away, always. Heaven, and the other place, can’t become boring.

4 Besides, if the actions of an agent are supposed to proceed to infinity, then there must be as a consequence to these actions either something that is produced, or nothing. Supposing that there is something that results, then the existence of this thing would come about after an infinite number of actions. But that which presupposes an infinite number of things cannot come into existence, since it is impossible to proceed to infinity. Now, that which is impossible in regard to being is impossible in regard to coming into being. And it is impossible to produce that which cannot come into being. Therefore, it is impossible for an agent to begin to produce something that presupposes an infinite number of actions.

Supposing, on the other hand, that nothing follows as a product of these actions, then the order of such actions must either depend on the ordering of the active powers (as in the case of a man who senses so that he may imagine, imagines so that he may understand, and then understands so that he may will); or it depends on the ordering of objects (thus, I think of body so that I may be able to think of soul, which latter I think so that I may be able to think of immaterial substance, which in turn I think so that I may be able to think about God).

Indeed, it is impossible to proceed to infinity, either through a series of active powers (for instance, through the forms of things, as is proved in Metaphysics [Ia, 2: 994a 1–b6], for the form is the principle of action) or through a series of objects (for there is not an infinite number of beings, because there is one First Being, as we demonstrated earlier [I:42]). So, it is not possible for actions to proceed to infinity. There must, then, be something which satisfies the agent’s desire when it is attained. Therefore, every agent acts for an end.

5 Moreover, for things which act for an end, all things intermediate between the first agent and the ultimate end are as ends in regard to things prior, and as active principles with regard to things consequent. So, if the agent’s desire is not directed to some definite thing, but, rather, the actions are multiplied to infinity, as was said, then the active principles must be multiplied to infinity. This is impossible, as we showed above. Therefore, the agent’s desire must be directed to some definite thing.

6 Furthermore, for every agent the principle of its action is either its nature or its intellect. Now, there is no question that intellectual agents act for the sake of an end, because they think ahead of time in their intellects of the things which they achieve through action; and their action stems from such preconception. This is what it means for intellect to be the principle of action.

Just as the entire likeness of the result achieved by the actions of an intelligent agent exists in the intellect that preconceives it, so, too, does the likeness of a natural resultant pre-exist in the natural agent; and as a consequence of this, the action is determined to a definite result. For fire gives rise to fire, and an olive to an olive. Therefore, the agent that acts with nature as its principle is just as much directed to a definite end, in its action, as is the agent that acts through intellect as its principle. Therefore, every agent acts for an end.

Notes Hello, free will!

7 Again, there is no fault to be found, except in the case of things that are for the sake of an end. A fault is never attributed to an agent, if the failure is related to something that is not the agent’s end. Thus, the fault of failing to heal is imputed to the physician, but not to the builder or the grammarian. We do find fault with things done according to art, for instance, when the grammarian does not speak correctly, and also in things done according to nature, as is evident in the case of the birth of monsters. Therefore, it is just as true of the agent that acts in accord with nature as of the agent who acts in accord with art and as a result of previous planning that action is for the sake of an end.

8 Besides, if an agent did not incline toward some definite effect, all results would be a matter of indifference for him. Now, he who looks upon a manifold number of things with indifference no more succeeds in doing one of them than another. Hence, from an agent contingently indifferent to alternatives no effect follows, unless he be determined to one effect by something. So, it would be impossible for him to act. Therefore, every agent tends toward some determinate effect, and this is called his end.

Notes Another way to put this is to think that the number of things you are not now doing is infinite. But you’ve picked one thing.

9 Of course, there are some actions that do not seem to be for an end. Examples are playful and contemplative actions, and those that are done without attention, like rubbing one’s beard and the like.

These examples could make a person think that there are some cases of acting without an end. However, we must understand that contemplative actions are not for another end, but are themselves ends. On the other hand, acts of play are sometimes ends, as in the case of a man who plays solely for the pleasure attaching to play; at other times they are for an end, for instance, when we play so that we can study better afterward. Actions that are done without attention do not stem from the intellect but from some sudden act of imagination or from a natural source. Thus, a disorder of the humors produces an itch and is the cause of rubbing the beard, and this is done without intellectual attention. So, these actions do tend to some end, though quite apart from the order of the intellect.

10 Through this consideration the error of the ancient natural philosophers is refuted; they claimed that all things come about as a result of material necessity, for they completely excluded final cause from things.

November 20, 2017 | 9 Comments

The Static Theory of Time and Free Will — Guest Post by The Cranky Professor

It is often argued that if the block universe theory of time is correct then people lack free will. The reason for this contention is because on a B-theory of time, future events already exist and so a person has already made future choices and that these future choices can never be changed. In other words, since the future of a person is already laid out and accomplished, a person cannot do anything other than what she does in the future.

The problem with this argument is that it commits a common fallacy in modal logic: it wrongly identifies any kind of “unchangeable” truth or fact with necessary, non-contingent truth. To identify the fallacy in the argument it is important to know the proper meanings and distinctions between “contingent” and “necessary” truths.

The modal distinction between contingent and necessary truths is not, contrary to Aristotle, the difference between “changeable” and “unchangeable” truths. Contingent truths are just as unchangeable as necessary truths. Sound far-fetched? Think about something that has happened in the past. Let’s say the person Jill has bought ice cream yesterday. Now Jill cannot change the fact that she went to the store and bought some ice cream but it is also true that she could have decided not to buy any ice cream yesterday. So long as Jill could have refrained from buying the ice cream at that event, then she was free to act in anyway at that period of time.

So an unchangeable truth like Jill buying ice cream yesterday cannot be a necessary truth like “2+2=4” or “a triangle has three sides”. And if Aristotle were correct in saying that all unchangeable truths are necessary truths then any kind of reconciliation of divine omniscience of the future and human free will would be doomed to fail whether one advocates a presentist theory of time or a block time theory.

It’s also worth noting that one of the reasons why Aristotle held the idea that propositions about future human actions are neither true nor false in his answer to the logical fatalists of his time, is because in his system, “necessary truth” meant “unchangeable truth”. So if he were to say, for instance, that there will in fact be a sea battle tomorrow it would indicate to him that people absolutely had to engage in a sea battle because once you have an unchangeable truth about the future established then you also have a necessary truth about the future according to his system. But thanks to developments in logic rooted in later medieval philosophers like Duns Scotus and William of Ockham, we know that the distinction between contingent and necessary truths cannot be the difference between changeable and unchangeable truths.

The real difference between contingent and necessary truths lies in this: contingent truths are true propositions that happen to be true but could have been false, whereas necessary truths are true propositions that absolutely have to be true and cannot be conceivably false. For instance, the proposition that “Jill buys ice cream on October 31st, 2018” is clearly a contingent truth because while it is unchangeably true, it could have been false provided that Jill decided to do something else on that day. The mathematical proposition “2+2=4” is a necessary or non-contingent truth meaning that this particular content has to be true and cannot ever be false because it is impossible to add 2 onto 2 and have it equal anything other than 4.

All that the B-theorist has to do to show that we have free will is to point out that facts and truths about the future are essentially contingent facts particularly when we consider human behavior. And by “free will” I am assuming, of course, the libertarian concept of human freedom where a person has several choices she can make as such and that the individual alone determines for herself what choice is made as such.

So if it is true, for instance, that Jill will rob a bank thirteen years into the future from our relative present, then while that future event may already be happening in its respective time and place, it is nonetheless a contingent state of affairs. Now it is true, that Jill cannot change her future event of robbing the bank because once Jill makes that decision within the relative future, it cannot be changed or erased just like once something is done within the past, it cannot be erased. However, Jill could have decided not to rob the bank in that particular event in time; and the fact that she could have refrained from committing theft is sufficient for her to be able to act with free will. In other words, the block time theory is compatible with contingent facts and truths in the world and the time theory does not imply the Spinozian view of necessitarianism (the idea that only necessary facts and truths exist). For that matter, an unchangeable future whether on a static theory or dynamic theory of time, does not imply necessitarianism.

As a matter of fact, the B-theory of time is not the only time theory that entails that the future is unchangeable. Presentists, growing block theorists, shrinking block theorists and moving spotlight theorists can all agree that the future is unchangeable. For it is impossible, as many philosophers point out, to bring about that a future event will not occur because if a future event ended up not occurring then it would not be a future event at all. Hence, whatever will happen, will happen and this is impossible to change or erase. It is literally nonsense to change the future.

Even on an Aristotelian view of time, it would be impossible to change the future. This is because on Aristotle’s view where the Law of Excluded Middle does not apply to the future, there is not even an established future about human behavior to change in the first place. So it is simply a non-sequitur fallacy to argue that from our inability to change the future to the conclusion that people lack free will.

It also equally a non-sequitur to argue that since on a B-theory of time, future moments exist, that there is only one conceivable future that people have to follow as such. What is only implied by an eternalist theory of time is that only one possible future among many possible futures is actualized on the block universe. To argue that an existent, actualized future entails only one possible, non-contingent future is like arguing that since the past consists in a series of actualized events that there was only one possible, non-contingent past or a line of events that necessarily had to take place. Such reasoning is clearly specious and wrong.

Therefore, the block universe theory of time in no way implies fatalism or some sort of nonsensical predestinarianism like Calvinism. The B-theory of time is perfectly compatible with libertarian free will. The contention that the B-theory of time implies fatalism or a denial of free will is nothing but nonsensical hype that is groundless.

Now many religious believers would be, of course, interested in seeing how the B-theory of time would influence one’s views of divine foreknowledge and human freedom. A lot can said about this issue of foreknowledge and free will and unfortunately not every aspect of this issue can be covered within a brief article. However, I will simply say this: I find no logical incoherency in the idea of someone knowing the outcome of future events and persons acting with free will whether one is assuming a block time theory, or a presentist theory or any other time theory. Modal logic can sufficiently show that there is no logical incompatibility between a mind knowing the outcome of future events and individuals having free will.

However, the B-theory of time does offer at least one advantage over any dynamic theory of time that says that the future does not currently exist. The block time theory, because it affirms the existence of the future, it better secures God’s knowledge of the future. The philosophical problem with advocating a presentist theory or a growing block theory for resolving the omniscience and free will puzzle, is how God can know what will happen in the future if the future events do not yet exist and at the same time, persons can make several possible choices with their free will?

Moreover, on a presentist model and a growing block model, God would clearly be temporal which may make it more difficult to account for God’s knowledge of future free choices. If it is true, for instance, that thirteen years from now, Jill will rob a bank, and a presentist theory is true, then God would literally have to wait for thirteen years, causing new present moments to exist one after another before the future event comes to his view. And how is God going to know for certain that Jill will freely choose to rob the bank if it’s possible that she could refrain from performing the action and the future event has not yet occurred?

With a block time theory, if it’s true that thirteen years from now, Jill will steal money from a bank, then God would not have to wait for thirteen years for the event to occur. With a B-theory of time, God would already have the event before his view and there would be no question at all whether Jill freely chose to rob a bank thirteen years ahead of our earlier moment in the block timeline.

So the B-theory of time gets rid of this problem of how God can know future free choices by holding that the future already exists before God. Of course, I suppose I cannot absolutely prove that a temporal God on a presentist theory of time cannot know future free decisions. Nonetheless, being a B-theorist, I conveniently don’t have to worry about how God can know future free choices because of how the static theory of time presents all temporal events before God. At any rate, like with other theories of time, there is no logical incompatibility with the static theory of time and persons acting out of free will.