Posts filed under Logic

On the Island of Logical Robots

Suppose you are a maintenance tech who works on an island with many different robots, in three different models. (1) Truthinator 9000, (2) D-Seevr 4.0, and (3) Benders. All three kinds of robots look exactly alike, so you can’t tell them apart based on what they look like. They all know everything there is to know. But Truthinators are programmed always to tell the truth; D-Seevrs are programmed always to tell lies. Benders mostly lie, of course, but they can choose to say false things and they also can choose to say true things. All of them only output intelligible declarative sentences with definite consistent truth values.[1]

Now suppose that a robot needs a repair and wants to identify its model to you. Truthinator 9000 would say “I am Truthinator 9000.” But then, D-Seevr might also say that, and so might Bender, if he chooses to lie, which he often does. Bender could truthfully say “I’m a Bender, bub” but a D-Seevr might say that too.

  1. Is there one sentence that a Bender could say that would allow you, as a sufficiently logical maintenance tech, to identify the robot as a Bender rather than a Truthinator or a D-Seevr? If so, what?

  2. Is there one sentence that a Truthinator could say that would allow you to definitely identify the robot as a Truthinator and not as a Bender or a D-Seevr? If so, what?

  3. Is there anything that a D-Seevr could say in one sentence that would allow you to definitely identify the robot as a D-Seevr, and not as a Truthinator or a Bender?

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  1. [1]This is of course a modified Knights and Knaves problem, of the sort popularized by Raymond Smullyan. But the presence of Benders on the island should significantly change the sort of strategies available.

The Self-Confidence Argument for Anarchism Re-visited: Premise 5 and Marco Polo

Back in December, I posted about an original argument against the legitimacy of the state, which I called The Self-Confidence Argument for Philosophical Anarchism. Here’s the argument, again:

  1. This argument is a valid deductive argument. (Premise.)
  2. If this argument is a valid deductive argument and all of its premises are true, then its conclusion is true. (Premise.)
  3. Its conclusion is No state could possibly have legitimate political authority. (Premise.)
  4. If No state could possibly have legitimate political authority is true, then no state could possibly have legitimate political authority. (Premise.)
  5. All of this argument’s premises are true. (Premise.)
  6. This is a valid deductive argument and all of its premises are true. (Conj. 1, 5)
  7. Its conclusion is true. (MP 2, 6)
  8. No state could possibly have legitimate political authority is true. (Subst. 3, 7)
  9. ∴ No state could possibly have legitimate political authority. (MP 5, 8)

Q.E.D., and smash the state.

The problem, of course, is that if this argument is sound, then it seems like you could construct another argument that must also sound, simply by substituting Some states have legitimate political authority everywhere in lines 3, 4, 8 and 9 that No state could possibly have legitimate political authority. And then you’d get an apparently perfectly sound Self-Confidence Argument for the State. It’s easy enough to figure out that there has to be something wrong with at least one of those arguments. Their conclusions directly contradict each other, and so couldn’t both be true. But they are formally completely identical; so presumably whatever is wrong with one argument would also be wrong with the other one. But if so, what’s wrong with them? Are they invalid? If so, how? Whichever argument you choose to look at, the argument has only four inferential steps, and all of them use elementary valid rules of inference or rules of replacement. Since each inferential step in the argument is valid, the argument as a whole must be valid. This also, incidentally, provides us with a reason to conclude that premise 1 is true in both. Premise 2 seems true by definition, under any standard definition of deductive validity. Premise 3 is a simple empirical observation. If you’re not sure it’s true, you can just look down the page to line 9 and find out. Premise 4 is a completely uncontroversial application of standard disquotation rules for true sentences. That seems to leave Premise (5). And premise (5) may seem over-confident, perhaps even boastful. But what it says is that just all the premises of the argument are true; so if it’s false, then which premise of the argument are you willing to deny? Whichever one you pick, what is it that makes that premise false? On what (non-question-begging) grounds would you say that it is false?

On my first post, a commenter named Lexi made the following observation, in order to suggest that you might nevertheless be able to reject Premise 5 — they noted that Premise 5 makes a statement about the truth of all the premises in the argument. But one of the premises it makes the claim about is Premise 5 itself. And perhaps that allows you to cut the knot:

Premise 5 is, at least, unsupportable. In order for all the premises to be true, premise 5 must also be true. The only way to justify premise 5 is by circular reasoning. Given that, maybe it’s not so surprising that you can support any conclusion X with the argument, since circular reasoning can establish any proposition as true.

–Lexi, comment (23 December 2015)

They’re certainly right to observe since premise 5 itself is among the statements premise 5 is quantifying over, its truth conditions would have to be something like:

(T5) Premise 5 is true ≡ Premise 1 is true & Premise 2 is true & Premise 3 is true & Premise 4 is true & Premise 5 is true

That might seem curious, and it involves a certain sort of circularity, but I can’t say I see how it makes the premise insupportable, if that is supposed to mean that you couldn’t give non-circular reasons to believe that Premise 5 is true.

After all, statements like this really are a part of ordinary language in non-philosophical cases. For example, Marco Polo begins his Description of the World by making the following statement in the Prologue:

. . . We will set down things seen as seen, things heard as heard, so that our book may be an accurate record, free from any sort of fabrication. And all who read this book or hear it may do so with full confidence, because it contains nothing but the truth.

This is a pretty common conceit in traveler’s tales: the author frequently assures the reader that everything they say — incredible as it might seem — is true.

But that statement is among the statements in Polo’s book; if he asserts that it contains nothing but the truth, then that sentence, inter alia, asserts that it is itself true:

(M) Marco Polo and his brothers traveled the Silk Road to China, and there he befriended the Emperor Kublai Khan, and along the way they observed the decadent customs of Lesser Armenia, and along the way they traveled among the Turkomans, and . . ., and (M) is true.

Which makes its truth-conditions something like:

(TM) (M) is true ≡ Marco Polo and his brothers did travel the Silk Road to China, and there he did befriend the Emperor Kublai Khan, and along the way they did observe the decadent customs of Lesser Armenia, and along the way they did travel among the Turkomans, and . . ., and (M) is true.

But here’s the thing. It doesn’t seem to me like (M) is insupportable or viciously circular. In ordinary cases, wouldn’t we determine whether it’s true or not by going through the book and checking out the other statements? I.e., some people reading the book might take everything else Marco Polo says there as true; and if so, then they’d take (M) as true as well. Call someone with this attitude towards (M) and its truth-conditions the True Believer. On the other hand, some people doubt parts of his tale — some people for example doubt that he even went to China at all. If so, they typically think not only that the first conjunct is false, but also the last one — if one of his statements is an assurance that all the statements are true, and any of the other statements are false, then that makes at least two falsehoods in total. Call someone with this attitude towards (M) and its truth-conditions the Normal Skeptic.

But now imagine a reader who insisted that they were a skeptic about Polo’s claims — but then, when asked one-by-one, signed off on every one of his other statements, except that they denied the statement that the book contains nothing but the truth. Call someone who takes this attitude towards (M) and its truth-conditions the Degenerate-Case Skeptic. Would Degenerate-Case Skepticism even make sense, as a position you might take with respect to the truth value of the claims in the book? Would it be a supportable claim? If so, how? If anything, it seems like the fault of circular here is mots easily attributed to someone who denies (M), or who mutatis mutandis denies Premise (5), based solely on Degenerate-Case skepticism. If (M) or (5) is false for no other reason that you even in principle could give other than its sui generis falsity, then that seems like a particularly radical form of question-begging.

Of course, you might say that it is insupportable, but so is the alternative, the True Believer’s claim that all the statements are true. So there’s no non-question-begging reason you could give to say that (M) or (5) is true, and there’s no non-question-begging reason you could give to say that (M) or (5) is false. Since the function of an argument is to give reasons to believe that its conclusion is true, if one of the premises cannot have any non-question-begging reason given either for its truth or falsity, then it seems like the argument can’t provide reasons for any conclusions that depend logically on that premise. (As the conclusion of any Self-Confidence Argument does; the MP in the conclusion cites Premise 5.) So you could say that. But now the question is, why say that? Isn’t it normally possible to give reasons for being a True Believer, and reasons for being a Normal Skeptic, even if there are no reasons you can give for being a Degenerate-Case skeptic? Is this kind of claim of radical insupportability the way we normally read texts that make assurances about themselves, like Marco Polo? Should it be?

If it’s not, and it shouldn’t, then should it be the way that we read Premise 5 here, even though it’s not the way we read Marco Polo? If there’s some difference between the two, that suggests reading Marco Polo in this way but not reading Premise 5 in this way, then what if any reason (preferably a principled reason that’s not question begging, and not simply ad hoc) could we give for the difference in semantic treatment?[1]

  1. [1]Or is it a difference in their semantics? Or a difference in something else, e.g. the pragmatics of their use?

The Self-Confidence Argument

Some of you know that I am a philosophical anarchist. This conclusion is controversial: most people think that states can in principle have legitimate political authority over the people in them, and that some states really do. So no state can have legitimate political authority is a conclusion in need of some argument to justify it. I’ve tried looking at the issue a couple of ways in a couple of different places. But those are both arguments that start from within a pretty specific, narrow dialectical context. They’re intended to address a couple of fairly specific claims for state legitimacy (specifically, individualist defenses of minimal state authority, and defenses of state authority based on a claim of explicit or tacit consent from the governed). Maybe a more general argument would be desirable. So here is a new one. It is a general deductive argument with only five premises. All of its inferences are self-evidently valid, and most of the premises are either extremely uncontroversial logical principles, or else simple empirical observations that are easily verified by any competent reader. I call it The Self-Confidence Argument for Philosophical Anarchism.[1] Here is how it goes:

  1. This argument is a valid deductive argument. (Premise.)
  2. If this argument is a valid deductive argument and all of its premises are true, then its conclusion is true. (Premise.)
  3. Its conclusion is No state could possibly have legitimate political authority. (Premise.)
  4. If No state could possibly have legitimate political authority is true, then no state could possibly have legitimate political authority. (Premise.)
  5. All of this argument’s premises are true. (Premise.)
  6. This is a valid deductive argument and all of its premises are true. (Conj. 1, 5)
  7. Its conclusion is true. (MP 2, 6)
  8. No state could possibly have legitimate political authority is true. (Subst. 3, 7)
  9. ∴ No state could possibly have legitimate political authority. (MP 5, 8)

Q.E.D., and smash the state.

Now, of course, just about every interesting philosophical argument comes along with some bullets that you have to bite. The awkward thing about the Self-Confidence Argument is that if it is sound, then it also seems that you can go through the same steps to show that this argument, The Self-Confidence Argument For The State, is also sound:

  1. This argument is a valid deductive argument. (Premise.)
  2. If this argument is a valid deductive argument and all of its premises are true, then its conclusion is true. (Premise.)
  3. Its conclusion is Some states have legitimate political authority. (Premise.)
  4. If Some states have legitimate political authority is true, then some states have legitimate political authority. (Premise.)
  5. All of this argument’s premises are true. (Premise.)
  6. This is a valid deductive argument and all of its premises are true. (Conj. 1, 5)
  7. Its conclusion is true. (MP 2, 6)
  8. Some states have legitimate political authority is true. (Subst. 3, 7)
  9. ∴ Some states have legitimate political authority. (MP 5, 8)

. . . which admittedly seems a bit awkward.

It’s easy enough to figure out that there has to be something wrong with at least one of these arguments. Their conclusions directly contradict each other, and so couldn’t both be true. But they are formally completely identical; so presumably whatever is wrong with one argument would also be wrong with the other one. But if so, what’s wrong with them? Are they invalid? If so, how? Whichever argument you choose to look at, the argument has only four inferential steps, and all of them use elementary valid rules of inference or rules of replacement. Since each inferential step in the argument is valid, the argument as a whole must be valid. This also, incidentally, provides us with a reason to conclude that premise 1 is true. Premise 2 is a concrete application of a basic logical principle, justified by the concept of deductive validity itself. Sound arguments must have true conclusions; validity just means that, if all the premises of an argument are true, the conclusion cannot possibly be false. Premise 3 is a simple empirical observation; if you’re not sure whether or not it’s true, just check down on line 9 and see. Premise 4 is a completely uncontroversial application of disquotation rules for true sentences. And premise 5 may seem over-confident, perhaps even boastful. But if it’s false, then which premise of the argument are you willing to deny? Whichever one you pick, what is it that makes that premise false? On what (non-question-begging) grounds would you say that it is false?

See also.

  1. [1]I owe the idea behind the form of this argument to a puzzle that Roderick Long gave me a couple years ago.

How to Save Money On Books

Just buy these two. The Law of Excluded Middle proves, apriori, you won’t need any others:

Here's a phot of two books.

1. Philip Delves Broughton, What They Teach You at Harvard Business School. 2. Mark H. McCormack, What They Don’t Teach You at Harvard Business School (subtitled: Notes from a Street-Smart Executive).

Tertium non datur.

(Via Anna O. Morgenstern.)

Overtaking Zeno

Shared Article from Slate Magazine

Zeno’s Paradox Is a Trick—But a Very Interesting Trick

The Greek philosopher Zeno wrote a book of paradoxes nearly 2,500 years ago. “Achilles and the Tortoise” is the easiest to understand, but it’s …

David Plotz @ slate.com


O.K., so, briefly: If you think that the point of Zeno’s Paradoxes of motion is to prove that the arrow never will reach its target, or that Achilles never does pass the tortoise, &c. — then I think that you are mistaken about the point of raising the paradox in the first place. Of course, it’s hard to be confident about the motives of dead philosophers who have no surviving books. But what we do know is that Zeno was a student of Parmenides; and Plato tells us that his books were written to defend Parmenides’s doctrines, by negative means,[1] showing that the views of his opponents led to contradictions.

So the most charitable understanding of Zeno’s aims is not that he’s trying to show you that Achilles can never catch the tortoise. Of course he does; just watch them race and you’ll see it happen. His point is to ask, given that Achilles passes the tortoise, well, how is that possible? And, for good or for ill, to argue from the paradox that you can only make sense of Achilles passing the tortoise if you reject presentism, and accept eternalist and Parmenidean conclusions about the nature of time and being.

Maybe he’s right about that, and maybe he’s wrong. (I’m inclined to think he’s wrong.) But note that if your solution is to try and settle the issue by introducing a lot of mathematical notation and conceptual apparatus from modern calculus — for example infinitesimal limit processes, convergent and divergent series, etc. — as is done in the Slate article here, and as is probably the overwhelmingly most common first response to Zeno’s paradoxes by mathematically-trained writers — then probably you are doing a better job than any pre-classical Greek philosopher could do in elaborating the precise nature of the problem.[2] But you’re not obviously refuting Zeno’s claims in any way, at least not yet. At the most you’re kicking the can down the road, and really you’re sort of strengthening Zeno’s own position. After all, naive formulations of mathematical notation are more or less always going to involve you in all kinds of specifically eternalist language, for example about moments in past and future time actually existing, instantiating the value of functions, etc. You cannot normally take the limit of ΔS(t) over values of t that don’t exist (no longer exist, do not yet exist).[3]

Or perhaps you can. But if you can, then doing so, and explaining what you’re doing when you do it, will take some very non-naive reinterpretation of ordinary mathematical language — and some nice metaphysics, too, to justify your reinterpretation. In any case the solution is going to have to be deeply philosophical, not just a matter of applying a technical innovation in maths.

  1. [1]In the Parmenides: I see, Parmenides, said Socrates, that Zeno would like to be not only one with you in friendship but your second self in his writings too; he puts what you say in another way . . . You affirm unity, he denies plurality. . . . Yes, Socrates, said Zeno. . . . The truth is, that these writings of mine were meant to protect the arguments of Parmenides against those who make fun of him and seek to show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is addressed to the partisans of the many, whose attack I return with interest by retorting upon them that their hypothesis of the being of many, if carried out, appears to be still more ridiculous than the hypothesis of the being of one.
  2. [2]Since the 19th century, we’ve done a lot to really nicely rigorize the mathematics of infinites and infinitesimals, in ways that sometimes anticipated by but never fully available to ancient mathematicians.
  3. [3]If anything, this is even more true of late-modern mathematics than it was of classical mathematics. Contemporary mathematics constantly helps itself to a lot of the language of existence, actuality, etc., for mathematical objects, in areas where Euclid and other classical mathematicians were typically much more circumspect about making existence claims for mathematical objects that hadn’t yet been constructed.