What the Tortoise Said to Achilles
So I've come across a story (What the Tortoise Said to Achilles) that may pose some problems for deductive logic. I'm actually tempted to call it the 'problem of deduction'. I'm curious to see what some other people think about this. https://wmpeople.wm.edu/asset/index/cvance/Carroll
Comments (43)
Me too: https://thephilosophyforum.com/discussion/comment/377693
Quoting 83nt0n
Different to this, though?
https://www.semanticscholar.org/paper/The-Enduring-Scandal-of-Deduction-Is-Propositional-D'Agostino-Floridi/6ff51e3f704044fac00b2c7430cf1ac775283820
Yes, I think so.
It is not a problem for deductive logic so much as an observation about it: There is a difference between a premiss (such as A or B) and a logical leading principle (such as C and D and so on). The latter is also called an inference rule, and all other deductive inference rules can be derived from modus ponens once certain axioms are established. One must recognize that such rules are intrinsically truth-preserving in order to understand and employ deductive logic. The turtle is only compelled to accept Z if his goal is to adopt true beliefs.
Again, that is why it is a leading principle or inference rule for deductive arguments in general, not an additional premiss (or infinite series of premisses) in each individual argumentation that employs it.
Are we trying to use deductive logic to prove that we shouldn't use deductive logic in this story?
Absolute truth exists, but it is very often hard to obtain.
How do you justify that definition?
Quoting 83nt0n
How do you justify that assertion?
Quoting 83nt0n
How do you justify that response?
Quoting 83nt0n
How do you justify that self-characterization?
Quoting 83nt0n
How do you justify that judgment?
If that is your definition of justified, then modus ponens is entirely justified, since it always takes us from true beliefs to true beliefs. For suppose toward a contradiction that P and if P, then Q are both true, but Q false. Since P is true and Q is false, it follows that if P then Q is false, contrary to what we have assumed. Therefore, if we believe in P and we believe in if P then Q, we are rationally justified in believing that Q.
Expanding a bit, it is always possible to justify our rules of deduction by soundness proofs, which are typical in mathematical logic.
This is how I am choosing to use the word.
Quoting aletheist
Again, this is how I am using the word.
Quoting aletheist
Because I don't know. And no I don't know that I don't know. Maybe I do know, I just don't know it.
Quoting aletheist
I am not saying this as an assertion; skeptics compose sentences like this not to assert something but to describe the way it appears.
Quoting aletheist
That's why I included the word seem. I could be wrong, but others have not met the burden of proof.
If you have questions about skepticism, check out the YouTube channel carneades.org and his series in defense of skepticism. However, this is not really the topic of this discussion. I am specifically interested in discussing deduction.
You seem to be using modus ponens to support modus ponens. 1) If that is your definition of justified, then modus ponens is entirely justified. 2) That is your (read:my) definition of justified. 3) Therefore modus ponens is entirely justified (from 1 and 2 modus ponens).
Also you seem to assume that if a rule goes from true beliefs to true beliefs, it is justified. Once again, this is using modus ponens to prove modus ponens. This is the point behind Carroll's story. For Achilles to justify modus ponens, he has to assert modus ponens.
No offense intended when I say this: you seem to be driven more by hubris in contrast to the pursuit of knowledge. If you've articulated your perspective to the best of your ability I can't follow your reasoning. It's almost as if you hold the stance that nothing has meaning since everything is built on presuppositions that are built on further abstractions.
I think Tim layed it out nicely: no matter what game we play the rules must be established first before any player can proceed.
If your beef is exclusively with modus ponens, then rest assured that it is dispensable (well, sort of, for some systems).
But that does not seem to be your problem with modus ponens; rather, you seem to be wary of using any rule of inference at all (incidentally, note that this is not my reading of Carroll's story at all---I think he is pressing the need for distinguishing axioms from rules of inference). Behind this wariness there seems to be some kind of linear propositional support requirement, namely that "a proposition or theory must be supported by inference from accepted premises to a conclusion, and that the conclusion not appear among the premises, premises of the premises, etc." (Paul Gregory, Quine's Naturalism, Chapter 1) The name derives from the fact that this requirement imposes a linear structure on our knowledge, i.e. P < Q iff P supports Q. It is well known that this requirement leads to skeptical conclusions, in the form of the Agrippan trilemma. Ironically, you seem to apply modus ponens to this argument (If there is a requirement for linear propositional support, then we must embrace skepticism; there is such a requirement; therefore, we must embrace skepticism), whereas it seems to me that it would be better to apply modus tollens and reject the linear propositional support requirement. (For what is worth, that is precisely Quine's strategy in "Epistemology Naturalized".)
Quoting ISeeIDoIAm
Yes, I tend to agree that rules need to be established, but the question is how do we go about establishing the rules. How do we establish a system of logic that allows us to know?
Quoting Nagase
I don't necessarily think that propositions need linear support. In fact, it seems that this kind of linear analysis is flawed (because of Agrippa's Trilemma and other problems), however, nonlinear analysis seems to me to be unable to avoid circular reasoning. If I am mistaken, please enlighten me. I am relatively new to philosophy; I have been an autodidact for about a year now.
How do you know that you do not know anything? How do you know that you want to have knowledge?
Quoting 83nt0n
We already have, but you claim not to accept it, even though you cannot avoid employing it.
I do not know this. I could be wrong. However, skeptics like me do not assert as true what we're saying. We just explain how it appears to us to be able to hold a conversation.
Quoting aletheist
Have we? Deductive logic (at least classical) seems unable to guarantee the conclusions it validates. I do admit that I probably cannot avoid employing it, but this has no bearing on whether it allows us to know.
A wise way to live so long as you have stable footing.
"I am the wisest man alive, for I know one thing, and that is that I know nothing."
I have had a similar thought pattern in regards to what you describe. How do we know what we see is true? How can we ever know what is absolute in this world of imagination and illusions? The short answer: we can't.
Trial and error is the only tried and true method of determining value in a reality that technically may not even exist. For all we know life could be a simulation. But that thought is pointless imo. I can't do anything with that. So I, like many others, chart my waters by experience rather than where the wind might take me. To do anything you need a basis (a reference point), no matter how wide you set your goalposts you need parameters to work from.
You can never truly know if you walk "the yellow brick road". If the game doesn't work well enough, the only course of action is to scrap the old rules and make new ones until you find a set that works. And if you can't find a better version than what exists then that's the state of things until they aren't. There's an infinite amount of ways a plan can fail, and only a handful where it works out as intended.
How do you know that you do not know this?
Quoting 83nt0n
How do you know that you could be wrong?
Quoting 83nt0n
This is your third assertion in a row that something is true.
Quoting 83nt0n
Why should I believe you?
Quoting 83nt0n
The inference rules of deductive logic, including modus ponens, are intrinsically truth-preserving; if the premisses are true, then the conclusion is necessarily true. What deduction cannot guarantee is that the premisses are true.
Quoting 83nt0n
Another assertion that something is true. Do you not realize that thoroughgoing skepticism is self-defeating? In order to be consistent you would have to be just as skeptical about skepticism as you claim to be about everything else.
They are intrinsically true so long as the original presupposition rings true. I think that's the point he's trying to lay out. Like many layers stacked on each other: the whole thing comes tumbling down if the foundation is faulty.
What do you mean by "the original presupposition"?
Again, I'm not asserting my statements/positions as definitely true. This is how it appears to me. INCLUDING THIS.
Quoting aletheist
Maybe you shouldn't.
Quoting aletheist
But the axioms that classical deductive logic employs are unsupported.
Quoting ISeeIDoIAm
Exactly.
Quoting aletheist
Again, this is not an assertion. This is how it seems to me. I have to utilize appearances, as I have nothing else to allow me to hold a conversation. Skepticism the position doesn't seem to be self defeating. However, if skepticism as a way of action is incoherent, this is probably due to human fallibility. And I am skeptical of my skepticism (or at least it appears that way). It could be that someone does have knowledge, but as of yet I haven't found any.
This is why I consider medical depression a "existential question". Half baked thought but I consider the two tied in some manner.
If you do not believe that anything you say is definitely true, including your assertions about your own beliefs, then how on earth is anyone supposed to have a meaningful conversation with you?
Quoting 83nt0n
There you go again, making an assertion. You need to stop doing that if you want to convince people that you are a genuine skeptic, but I guess you have no way of knowing whether you want to do that. Anyway, I suggest looking up the definition of "axiom."
Quoting 83nt0n
Now you are asserting that an assertion is not an assertion--self-defeating, just like I said.
Quoting 83nt0n
Asserting how it seems is still an assertion.
Quoting 83nt0n
See, you just used modus ponens. "If I have nothing else to allow me to hold a conversation, then I have to utilize appearances. I have nothing else to allow me to hold a conversation. Therefore, I have to utilize appearances."
Quoting 83nt0n
We have exchanged several posts now, all utilizing the English language. Unless you wish to claim that we have been throwing gibberish at each other, clearly you and I both have knowledge of the English language.
Sorry, I still have no idea what you are talking about or how it relates to my exchange with @83nt0n.
As I explained previously, I only say how things appear to me. This allows for a conversation.
Quoting aletheist
axiom: a statement or proposition which is regarded as being established, accepted, or self-evidently true
I could assert that "my big toe is purple" is an axiom, but I need to demonstrate that it is true. Same with other axioms. You should look into Agrippa's Trilemma a bit more.
Quoting aletheist
I did not say that as an assertion. And I AM NOT asserting this either.
Quoting aletheist
Again, I may not be able to help but use deductive logic, but how does my inability to be a complete skeptic have any bearing on skepticism as a position?
Quoting aletheist
This might not be the same as propositional knowledge, which is the knowledge that I am talking about.
If you're saying that because I use the English language, I know how it works, this seems to be flawed reasoning. Just because someone uses their brain doesn't mean they know how it works.
Overall, no offense, but I don't think you understand my position at all. Check out the Youtube channel carneades.org (he's better at explaining things than I am).
Another assertion.
Quoting 83nt0n
No, you do not. Read the definition of "axiom" that you quoted again.
Quoting 83nt0n
Two more assertions.
Quoting 83nt0n
It is not so much your individual inability as the fact that no one can be a complete skeptic--again, such a position is self-defeating--so it then becomes a matter of which beliefs you adopt, just like anyone else.
Quoting 83nt0n
You and I are competent users of the English language. This is a true proposition that we both justifiably believe. Therefore, we both have propositional knowledge.
Quoting 83nt0n
Right back at you.
When you started posting.
You are right; my example is not a good one. But the way I see it, no beliefs/truths are self-evident.
Quoting aletheist
This is probably true, but adopting beliefs would not consist of knowledge, which is what I want.
Quoting aletheist
All of your objections to my position are very common. I understand your objections, but they do not hold up. I think I am going to stop arguing with you, so I advise that you watch carneades.org playlist "In defense of skepticism".
Hypothetically if deductive logic was flawed, what would you replace it with? The most you could say against deductive logic is that out of the trillions variables that reality deals with, not all those variables are known to apply to the overall equation. In other words deductive logic isn't flawed, however people only have so much information at their disposal.
Its a clever story, but if that story was instead written as a mathematical proof we could all get to the bottom of it real quick. Like i said before deductive reasoning only fails when not all variables or not all important variables are known to answer the question.
Perhaps next time you should show the story like you did but then place the mathematical proof next to it. Otherwise you are only going to fool uninformed people, which you may have succeeded in doing that.
Agreed.
Quoting 83nt0n
I disagree. It might just be recognising soundness as a self-evident virtue. Another modus ponens (aside from the one being justified) needn't be involved. You were just on a roll with that objection, no? It is the tortoise's expected refrain, true, but the tortoise doesn't talk about this combination, wherein the student accepts A, B and Z (from true to true) but not C (the rule). The tortoise invites us to justify Z on the basis of A and B, and then of course he claims to need C (and then D etc).
I only mention this in case it connects the tortoise's problem to the alleged 'scandal' of deduction: of its telling us no more than we already knew; of soundness being an empty (as well as self-evident) virtue. If Z does indeed follow from A and B as C claims, then C goes without saying. So much then for,
Nonlinear reasoning does not avoid circular reasoning; it argues that, in some cases, it is both unavoidable and not vicious (or, perhaps, more positively, indeed virtuous). That is Quine's stance: we always start in the middle of things, so to speak, and there is no problem in using our background knowledge to understand how we came to have that knowledge in the first place. The point is that, once we realize that our knowledge is not linearly arranged, but rather forms an intricate web, we give up the search for foundations (so that the aim of epistemology is not to secure knowledge---i.e. it is precisely not to argue against the skeptic), and rather try simply to further the knowledge we already have.