Hypothesis Confirmation: The Paradox of the Ravens

[drug_mentor]

I hope this thread will be of some interest, it seems that some of the topics I post here are a little too abstract and/or removed from daily concerns to generate much interest. How we respond to this paradox has implications for the way science should be conducted, and I would hope this fact might bait some more people into the discussion. Immediately following this post, I am going to make a post where I prove the equivalence of a universally generalised conditional and the universal generalisation of its contrapositive. I don't expect this will be of much interest to anyone here, but if anyone doubts my claim that 'All F's are G's' is equivalent to 'All not-G's are not-F's', see post #2 for a formal proof of their logical equivalence.

Let me begin by stating two prima facie plausible principles of hypothesis confirmation:

Nicod's Principle: A universal generalisation is supported by any and all observations of its positive instances, and falsified by any and all observations of its negative instances. What this condition amounts to is this - a universal generalisation of the form 'All F's are G's' is supported by any observation of an F which is also a G, and is falsified by any observation of an F which is not also a G.

Equivalence Principle: Whatever confirms a universal generalisation also confirms any hypothesis logically equivalent to that generalisation, and vice versa. (If A and B are logically equivalent then one can infer B from A and vice versa.)

For now I am taking it for granted that these two principles are fairly intuitive and uncontroversial, but I am happy to provide a defense of them if anyone expresses some scepticism about them.

Before presenting the paradox, I want to emphasise that this is a logical problem about confirming scientific hypotheses which take the form of a universal generalisation. I am going to use the initial presentation of the paradox; I am aware that ravens apparently can be white, I hope it will be obvious that this counterexample does not in any way undermine the issue here.

Let us consider the hypothesis 'All ravens are black'. (In case it is not obvious, in this case 'being a raven' amounts to 'being an F, and 'being coloured black' amounts to 'being a G'.) By contraposition, this hypothesis is logically equivalent to 'All not-black things are not ravens'; by Nicod's principle, any observation of something which is not black and is not a raven supports the latter hypothesis. But, according to the equivalence principle, any observation which supports the latter hypothesis also supports the former hypothesis. We seem to have been led to the prima facie absurd conclusion that any observation of something which is not black and is not a raven serves as support for the hypothesis that 'All ravens are black'. It just seems wrongheaded to suppose that observing my white shoes and my brown coffee table constitute evidence in favour of the hypothesis that 'All ravens are black'; surely this kind of indoor ornithology is not respectable.

What do people think? Is there an obvious solution to this problem? Is it not really a problem at all?

 

[drug_mentor]

Let F be 'is a raven' and let G be 'is black', then the hypothesis of 'All ravens are black' is (x)(Fx-->Gx). (Strictly speaking, this says, 'take anything at all, if it is a raven then it is black', this is the only way to express the hypothesis in question in first-order logic, but I should hope it will be obvious that it amounts to saying the same thing as 'All ravens are black'.)

(x)(Fx-->Gx) -||- (x)(~Gx-->~Fx)

|- direction
{1} (1) (x)(Fx-->Gx) - Assumption
{1} (2) Fa-->Ga - 1, Universal Elimination
{3} (3) ~Ga - Assumption
{1, 3} (4) ~Fa - 2, 3 Modus Tollens
{1} (5) ~Ga-->~Fa - 3-4 Conditional Proof
{1} (6) (x)(~Gx-->~Fx) - 5, Universal Introduction
QED

-| Direction
{1} (1) (x)(~Gx-->~Fx) - Assumption
{1} (2) ~Ga-->~Fa - 1, Universal Elimination
{3} (3) Fa - Assumption
{3} (4) ~~Fa - 3, Double Negation Introduction
{1, 3} (5) ~~Ga - 2, 4 Modus Tollens
{1, 3} (6) Ga - 5, Double Negation Elimination
{1} (7) Fa-->Ga - 3-6 Conditional Proof
{1} (8 ) (x)(Fx-->Gx) - 7, Universal Introduction
QED

I have used classical logic here. It is true that the two hypotheses under consideration are not equivalent in some non-classical logics, for example, in intuitionistic logic (in IL the |- direction holds while the -| direction does not).

 

[GodandLove]

By following this algorithm, let us consider this:

All Humans are White, therefore all not-white things are not humans.

Here's a real paradox for you:

If a gender Non Binary person walks into a bar, what bathroom do they use?

The Men's Room or The Ladies Room?

The answer,

You guessed it! They sue the shit out of the bar for not having an "All-Gender" bathroom!

Man You're Good!

 

[drug_mentor]

By following this algorithm, let us consider this:

All Humans are White, therefore all not-white things are not humans.

This is a valid argument, it isn't sound because the premise 'All humans are white' is false.

I am not sure if this was supposed to be an attempt to demonstrate that contraposition isn't valid. Contraposition is a valid form of inference in (classical) mathematical and philosophical logic, and (at least) most other logics with a rich enough language to express conditionals (which is the overwhelming majority). Off the top of my head, the only logic I can think of which doesn't validate contraposition is a counterfactual logic using Lewis-Stalnaker semantics, and there is good reason for this. Contraposition is fallacious when one is using subjunctive conditionals; 'All ravens are black' is an indicative conditional, so one can validly infer the contrapositive 'All not-black things are not ravens' from it. This is pretty basic logic, and I did give a formal proof of this in post #2 of this thread.

As to the rest of your post, I have no idea what you are talking about. I really would prefer to confine the discussion to logic, epistemology and philosophy of science, rather than veering off into identity politics.

 

[GodandLove]

This is a valid argument, it isn't sound because the premise 'All humans are white' is false.

I am not sure if this was supposed to be an attempt to demonstrate that contraposition isn't valid. Contraposition is a valid form of inference in (classical) mathematical and philosophical logic, and (at least) most other logics with a rich enough language to express conditionals (which is the overwhelming majority). Off the top of my head, the only logic I can think of which doesn't validate contraposition is a counterfactual logic using Lewis-Stalnaker semantics, and there is good reason for this. Contraposition is fallacious when one is using subjunctive conditionals; 'All ravens are black' is an indicative conditional, so one can validly infer the contrapositive 'All not-black things are not ravens' from it. This is pretty basic logic, and I did give a formal proof of this in post #2 of this thread.

As to the rest of your post, I have no idea what you are talking about. I really would prefer to confine the discussion to logic, epistemology and philosophy of science, rather than veering off into identity politics.

So let me get his straight, your logic is as follows: My logic is more sound than your logic because some Ravens are perceived as being Black. If some Raves are perceived as being not-Black therefore I will deny they are Ravens.

In other words, if I don't agree with you that all Ravens are black, you will completely ignore the existence of Albino ravens. You will then convolute the issue by replying to my post with an
arbitrary term "indicative conditional" which has no formal meaning.

That about sound right?

 

[drug_mentor]

Before presenting the paradox, I want to emphasise that this is a logical problem about confirming scientific hypotheses which take the form of a universal generalisation. I am going to use the initial presentation of the paradox; I am aware that ravens apparently can be white, I hope it will be obvious that this counterexample does not in any way undermine the issue here.

So let me get his straight, your logic is as follows: My logic is more sound than your logic because some Ravens are perceived as being Black. If some Raves are perceived as being not-Black therefore I will deny they are Ravens.

In other words, if I don't agree with you that all Ravens are black, you will completely ignore the existence of Albino ravens.

This is a paradox about confirming hypotheses which take the form of a universally generalised conditional, the idea being that some reasonable seeming premises about hypothesis confirmation lead us to a very counter-intuitive conclusion about the nature of evidence. The ravens are just a toy example, I already noted in the OP that some ravens are white. If you need me to explain why a counterexample to some particular hypothesis doesn't undermine the paradox then I am happy to elaborate.

You will then convolute the issue by replying to my post with an
arbitrary term "indicative conditional" which has no formal meaning.

That about sound right?

It's true to say that philosophers and logicians don't agree on the truth conditions for indicative conditionals, but they do generally agree on both their syntactic structure and the fact that they contrapose. It is also true that in most logics, including classical logic, there is no way in the object language to distinguish between indicative and subjunctive conditionals. However, since it is universally agreed that the truth-conditions for subjunctive and indicative conditionals differ, a logic which only has one conditional connective will be given an interpretation and in practice will only be used to deal with one or the other. One can certainly criticise the truth conditions for the material conditional in classical logic as being inadequate for representing the indicative conditional in natural language, in fact, I am extremely sympathetic to said criticisms - but, these criticisms have nothing to do with contrapostion.

It is also true that the hypothesis 'All ravens are black' is not being expressed in natural language as a conditional at all, let alone as an indicative conditional. However, I think it is fairly uncontroversial to say that the truth conditions for 'All ravens are black' and 'Take anything at all, if it is a raven then it is black' are exactly the same, and the latter is an indicative conditional. The controversy over indicatives is mostly over issues like whether conditionals which express a 'will' clause in the consequent (e.g. 'if I go outside, then my allergies will flare up') should be regarded as indicative or subjunctive conditionals. There is no controversy over the fact that conditionals of the simple form 'if A then B' (or 'for any x, if it is F then it is G') are indicative.

I would prefer not to get bogged down further in semantics that are of minimal relevance to the thread topic. Do you deny that the following informal argument is deductively valid:
1. If A then B
2. Not-B
Therefore,
C. Not-A

This form of argument is called Modus Tollens and is universally regarded as deductively valid. If you think you have a counterexample I would be interested to hear it.

 

[GodandLove]

Can you please simplify your argument? Speak plainly. Limit your argument to two sentences.

Use this format: A = Presmise
I believe A
This is my reasoning for believing A

You seem like you're an intelligent person and all, but it is hard to comprehend someone who is trying to express themselves with words and concepts they don't fully understand. Perhaps it would be best if you didn't try posting past your depth.

Or maybe you are intentionally trying to be as incoherent as possible, who knows.

 

[drug_mentor]

Can you please simplify your argument? Speak plainly. Limit your argument to two sentences.

Use this format: A = Presmise
I believe A
This is my reasoning for believing A

I believe that the paradox of the ravens raises an interesting and important issue for the epistemology of science, and that the statement 'All ravens are black' is logically equivalent to the statement 'All not-black things are not ravens'. I provided a formal proof of their equivalence in post #2 of this thread; the fact that their equivalence can be proved using classical logic is my primary (but not my only) reason for believing that they are in fact logically equivalent.

You seem like you're an intelligent person and all, but it is hard to comprehend someone who is trying to express themselves with words and concepts they don't fully understand. Perhaps it would be best if you didn't try posting past your depth.

You are touching on issues that relate to the semantics of different types of conditionals, different ways that conditionals can be expressed in formal logic(s), and some philosophy of logic about the interpretation of formal systems as they relate to arguments in ordinary language. One could write a book on these topics and not come close to saying everything there is to say. I have tried to make a thread about an interesting paradox without getting bogged down in these details, which are really quite peripheral to the issue I want to discuss, and are likely to accomplish little more than to deter people from participating.

I am actually quite well-versed in formal logic, I have studied it fairly extensively at a well-regarded institution over the past couple of years. Just because you don't understand the words and concepts I am using, that doesn't mean I don't. You can't expect me to give you a crash course in philosophy of logic in order for you to participate in this thread. I tried to give a concise explanation as to why it is not infelicitous to take the conditional hypothesis under consideration as an indicative conditional; I appreciate that my explanation might be slightly hard to follow if you have little to no familiarity with semantics or formal logic. But, you have to understand that my bringing you up to speed in these areas is outside the scope of this thread. In any case, I have tried to emphasise that, while there are genuine controversies over exactly what an indicative conditional amounts to and whether classical logic can give a wholly adequate treatment of these conditionals, whatever tenable position one takes on these issues will leave the validity of the argument I presented for the paradox of the ravens untouched.

I am still waiting for you to provide a counterexample to modus tollens and/or contraposition.