Physical impossibility and ought implies can

I was watching The Unbearable Lightness of Being when a scene suddenly made me realize a possible problem with regularity theory of physical laws and the ought implies can principle.

Regularity theory’s view of physical impossibility

Regularity theory implies that what it means to say that something is physically impossible is just to say that it will never happen (in this world). N. Swartz writes this in his book about about the theory:

“Indeed, the Regularity Theory – in withholding physical possibility from rivers of Coca-Cola – asserts only what Molnar himself hypothesized: namely, that there never are any such rivers.”1

Here the example offered was a river of Cola-Cola. Supposing that there never is such a river and that regularity theory is true, then it follows that rivers of Cola-Cola are physically impossible.

The ought implies can principle

Ought implies can is an ethical principle that seems to fit many people’s intuitions. Suppose someone killed another person. We are initially disposed to think lowly of that person. But if it comes to our attention that the person doing the killing could not have reasonably avoided doing it, then we don’t think lowly of the killer. One example of this would be if someone jumped outside your car when you were driving at normals speed.

I’m further supposing that this “can” expresses physical possibility. It does not seem like logical possibility or epistemic possibility. Thus, ought implies physically possibility. Contra-position: Physical impossibility implies not-ought.

Deontic logic and formalizing

We have to be careful with “ought” and “not”. There is a difference between “He ought not to fuck other women.” and “It is not the case that he ought to fuck other women.”. Formalizing them requires deontic logic. I’m using “O” to mean “It is obligatory that “ or equivalently “It ought to be the case that “, “P” to mean “It is permissible that “, and “F” to mean “It is forbidden that “. “A” is taken to mean whatever action it is we are talking about. I follow standard notation in deontic logic according to SEP2.

“He ought not to fuck other women.” is thus simplistically formalized as “O¬A” which is equivalent with “FA” which means “It is forbidden that he fucks other women.”. “It is not the case that he ought to fuck other women.” is thus simplistically formalized as “¬OA”. For simplicity I will refer to this latter use of “ought” as “not-ought” and the former use as “ought not”. Keep this distinction in mind.

I write “simplistically formalized” because that I omitted the agent in the formalizing.

The problem

Now that we’ve understood the concepts in use, let’s look at an example slightly inspired by the aforementioned film. Suppose that Tereza and Tomas are married, and that Tomas cheats on her every time we has the chance. In other words: It is never the case that when (a description that uniquely identifies Tomas) has chance to cheat on (a description that uniquely identifies Tereza), then (description of Tomas) cheats on (description of Tereza). This together with a regularity theory of physical laws implies that it is physically impossible for Tomas not to cheat on her. [¬◊PhyA] using “◊phy” to mean physical possibility. Suppose further that he knows that she will find out when he cheats on her, and that this hurts her, but he still continues to do it. The question now is: Is it immoral of him to cheat on her? I think we’d have to say yes. But if he physically cannot not-cheat on her, then it is not the case that he ought not to cheat on her. [¬◊PhyA→¬O¬A] Thus, it is not the case that he ought not to cheat on her. [¬O¬A] Further: if it is not the case that he ought not to cheat on her, then he is not acting immorally by cheating on her.[¬O¬A→¬Ia] using “Ix” to mean “x is immoral” and “a” for the action. Thus, he is not acting immorally by cheating on her. [¬Ia] In other words: The action of cheating on her is not immoral. This conflicts with our judgment from before. [Ia] So someone who accepts regularity theory and the ought implies can principle would have to find someway out of this situation and regain consistency among his beliefs.

1Norman Swartz, “The Concept of physical law”, 1985, Cambridge University Press, p. 62

Quote: Bertrand Russell

“Philosophy, throughout its history, has consisted of two parts inharmoniously
blended; on the one hand a theory as to the nature of the world, on the other an
ethical or political doctrine as to the best way of living. The failure to separate
these two with sufficient clarity has been a source of much confused thinking.
Philosophers, from Pla to to William James, have allowed their opinions as to the
constitution of the universe to be influenced by the desire for edification: knowing,
as they supposed, what beliefs would make men virtuous, they have invented
arguments, often very sophistical, to prove that these beliefs are true. For my part I
reprobate this kind of bias, both on moral and on intellectual grounds. Morally, a
philosopher who uses his professional competency for anything except a
disinterested search for truth is guilty of a kind of treachery. And when he assumes,
in advance of inquiry, that certain beliefs, whether true or false, are such as to
promote good behavior, he is so limiting the scope of philosophical speculation as
to make philosophy trivial; the true philosopher is prepared to examine  all
preconceptions.
When any limits are placed, consciously or unconsciously, upon
the pursuit of truth, philosophy becomes paralyzed by fear, and the ground is
prepared for a government censorship punishing those who utter “dangerous
thoughts”  – in fact, the philosopher has already places such a censorship over his
own investigations.”

My emphasis. From The Philosophy of Logical Analysis
(Chapter XXXI of “A History of Western Philosophy”)

“But you don't know that you know that!”

Fast:

“We need to keep in mind that “I know that I know” is ambiguous between 1) I am certain and 2) I have knowledge that I know. The latter is very often true, but the former is very rarely true.”

Emil:

“Why do you think it is ambiguous between them? It seems to me that it is not ambiguous at all and only means (2).”

Fast:
“Well, it may mean the second, but when someone rebuts, “But, you don’t know that you know!,” the implication of what they mean is almost always the first. Um, the opposite (“you are not certain”) since I used the word, “don’t”.

The point is the utterance of “you don’t know that you know” is oftentimes based on the confusion between knowledge and certainty. If you don’t confuse them, then you probably don’t mean the first, and even if the phrase isn’t technically ambiguous, there’s certainly a difference between what the person means by what he says and what the person says means.”

Source.

Induction, deduction and the lack of justification

It has been thought for many years (especially since Hume’s first Enquiry1) that induction lacks a justification. This justification for induction, it has been thought for long, is necessarily if it is rational to use induction. The argument for why there is no justification, indeed there cannot be a justification, can be presented like this:

(A1)

n Proposition Explanation
1 If induction cannot be justified inductively or deductively, then it cannot be justified. Premise
2 Induction cannot be justified inductively. Premise
3 Induction cannot be justified deductively. Premise
4 Induction cannot be justified. From 1, 2, 3, Conj., MP.
5 Induction is not justified. From 4, M2

It has then been thought that since we only ought to use justified reasoning systems and that induction is not a justified reasoning system, then we ought not to use induction, and thus that using induction was in some sense irrational.

But something is terribly wrong with this approach. First, is it really irrational to use induction? Is induction and deduction not what defines human rationality? What other reasoning systems are available to us? Second, an argument analogous to (A1) above can be constructed against deduction:

(A2)

n Proposition Explanation
1 If deduction cannot be justified inductively or deductively, then it cannot be justified. Premise
2 Deduction cannot be justified inductively. Premise
3 Deduction cannot be justified deductively. Premise
4 Deduction cannot be justified. From 1, 2, 3, Conj., MP.
5 Deduction is not justified. From 4, M

Then, should we really accept that since deduction is not justified, then we ought not to use it? That’s a paradoxical conclusion since we just used deduction to reach it! Something is terribly wrong.

It seems a good idea to dispose with the premise used above that:

(P1). For all reasoning systems, if it is not justified, then we ought not to use it.

That seems to do the job. Now we can no longer infer that we ought not to use deduction or induction. But now the special problem for induction seems to have “vanished”, there is no problem for induction that there is not for deduction too. What is curious is that deduction has tended to be assumed to be a good reasoning system (and not in need of justification) while induction had to have a justification before we could rationally use it. Call adherents to this view deductivists.

But an inductivist, that is, someone who thought of it the other way could use (A2) above against the deductivist.3

1An Enquiry concerning Human Understanding published 1748.

2M is an alethic logic axiom. It follows from the definition of ¬◊, substitution and M.

3Note that the argument will not have any force for the inductivist since he does not trust deduction but the deductivist does.

Human rationality

Induction and deduction are foundational to human rationality because it is impossible for a human to stop using them. Indeed what it means to be humanly rational is that one uses these reasoning systems.

There is no sense in trying to discover any justification for these two reasoning systems because even if we discovered that one of them ought not to be used, then we could not stop using them. If we cannot stop using them even if we ought to, then it is futile, indeed a waste of time, to examine if we ought not to use them.

Those who think that ought implies can might also reason like this: If we cannot stop using them, then it is not the case that we ought not to use them. This follows from the contra-position of the proposition ‘ought implies can’, that is, ‘cannot implies ought not’. If it is not the case that we ought not to use them, then they are justified. This follows from the contra-position of the proposition if we are not justified in using a reasoning system, then we ought not to use it, that is, ‘if it is not the case that we ought not to use a reasoning system, then we are justified in using that reasoning system’. Therefore, we are justified in using induction and deduction. Reasoning like this eliminates the arbitrariness of “choosing” these two reasoning systems. There are no other reasoning systems that we cannot stop using if we wanted to.

Note that the foregoing reasoning uses deduction. I’m not sure if that is a problem. It may be.

Valid arguments, contingent truths & forms

n Proposition Symbol Explanation
1 Peter is gay. P1 Premise
2 Peter is male. Q From 1

Valid? Perhaps. A truth table with show:

P Q ¬Q P∧¬Q
T T F F
T F T T
F T F F
F F T F

If the superconjunction of all the premises and the negation of the conclusion comes out as necessarily false (F on all rows), then the argument is valid.

Or equivalently: An argument is valid iff the superconjunction of all the premises and the conclusion is a necessary truth. (I.e. all rows come out as T.)2

In the above case the argument is apparently invalid. But still it is “valid” in a sense. Why is this? It is because there is a necessary connection between the two propositions: Necessarily if p, then q [□(P→Q)]. It is impossible that (Peter is gay and Peter is not male) [¬◊(P∧¬Q)].

Now consider another argument:

n Proposition Symbol Explanation
1 Peter is male. P Premise
2 Peter is gay. Q From 1

Suppose further that it just happens to be the case (contingent truth) that all males are gay in a possible world. Formally this means that P implies Q in that world. So, is the second argument valid or not? Is it valid only in a specific world?

If we go by the first proposed definition of validity, then both arguments are invalid. But still there seems to be a crucial difference between their “validity” is some sense. It seems to me that it may be the necessity of the connection between P and Q that makes the difference. In the first case there is a necessary connection [□(P→Q)] because it is true per definition that a gay person is a man.3 But it is not necessarily true that all males are gay, that is only a contingent truth. There is no necessary connection between P and Q in the second argument [¬□(P→Q)]. Thus, the first argument is “valid” and the second is not even though it happens “to work” in some possible world. I think validity should not be relative to what possible world one considers the argument in. No matter what world one considers the first argument in, it comes out as valid in this sense. The second argument does not.

I suspect there is some deep confusion in this way of thinking about validity and I have tried to locate it (if it is there) by being clear, but I don’t think I’ve succeeded. I’ve also failed to properly state the meaning of ‘validity’ used in the second sense. I will continue to think about this.

Validity as a function of argument form

I’m not convinced that it is not a good idea to define validity as a sole function of the argument form. If that is done, then both the arguments above are invalid because the form they share is invalid.

From a pragmatic point of view it would be smart if people always reasoned in valid forms as it would make it much easier to check for validity instead of having to think about hidden necessary connections between the propositions in the argument.

Recap

There are three proposed ways to define validity:

1. An argument is valid iff the superconjunction of (all the premises and the conclusion) is a necessary truth. (Standard definition.)

2. An argument is valid iff it comes out as valid in all possible worlds. (This is not very clear.)

3. An argument is valid iff it has a valid form.

1I’m aware that this argument could be expressed in predicate logic but I’ve tried to keep it simple by using only propositional logic.

2This one is often mentioned in textbooks though they use the word ‘tautology’ instead of ‘necessary truth’. Probably to avoid talk of modalities and stick to truth table talk. A tautology is defined in that context as a proposition that comes out with only Ts, and a contradiction is defined as a proposition that comes out with all Fs.

3‘Gay’ is defined here as male that is homosexual.

Complete arguments

A complete argument is as argument that is stated or presented such that:

1. One can see all the propositions in the argument.

2. All the propositions in the argument are numbered.

3. It is explained what kind of purpose all the propositions have. (Premise, inference or assumption.)

4. All inferences are stated, it is stated form where they follow and what kind of inference it is.

Here is an example argument:

n Proposition Explanation
1 The earth exists. Premise.
2 If the earth exists, then the moon exists. Premise.
3 If the moon exists, then the moon is round. Premise.
4 If the earth exists, then the moon is round. From 2, 3, HS
5 The moon is round. From 1, 4, MP.

I find that it is a very good idea to use tables to present arguments with.

Complete formal argument

A complete formal argument is an argument that is a complete argument where the symbols representing the propositions or predicates are stated and the definitions of these symbols or predicates is stated (if needed).

Here is an example argument created by extending the above argument:

n Proposition Symbol Explanation
1 The earth exists. P Premise.
2 If the earth exists, then the moon exists. P→Q Premise.
3 If the moon exists, then the moon is round. Q→R Premise.
4 If the earth exists, then the moon is round. P→R From 2, 3, HS
5 The moon is round. R From 1, 4, MP.

Note that it is not necessary to state the definitions of the propositions or predicates. If the argument is complicated and uses predicate logic, then it is wise to define the terms before presenting the propositions.