Jonah over at Bishop Berkeley, Bacon, & Bird put up a nice little taxonomy of all the main paradoxes today. It got me thinking about the Liar's Paradox which is the grand daddy of all the semantic paradoxes. But what if, instead of considering truth as a purely semantic issue (i.e. a property of sentences or at least propositions) we made it more complex. Initially I was curious if anyone had done anything with Heidegger's sense of truth. I didn't find much on Google on that. But one of the links mentioned a paper about Peirce and the Liar's Paradox. I hadn't realized he'd actually written on the subject. To make things even nicer the paper on the subject was actually available online. (Not always a sure thing with philosophy papers - especially older ones)
The basic idea on the paradox comes from Peirce's discussion of truth.
. . . a proposition is true only if whatever is said in it is true, but is false if anything said in it is false. (5.340)
With regards to variations on the Liar's Paradox we therefore have the problem that
the proposition in question, therefore, is true in all other aspects but in its implications of its own truth. (ibid)
Peirce's solution is
'This proposition is false,' far from being meaningless, is self-contradictory. That is, it means two irreconcilable things. That it involves contradiction (that is, leads to contradiction if supposed true), is easily proved. For if it be true, it is true; while if it be true, it is false. Every proposition besides what it explicitly asserts, tacitly implies its own truth. The proposition is not true unless both what it explicitly asserts and what it tacitly implies, are true. This proposition, being self-contradictory, is false; and hence;, what it explicity asserts is true. This proposition, being self-contradictory, is false; and hence, what it explicitly asserts is true. But what it tacitly implies (its own truth) is false.
Am I missing something or is he basically just saying self-referential statements aren't allowed? Which appears to be similar to Russell's theory of types which tries to solve the same problem.
I don't see him making any claims about self-referential sentences. Rather he is claiming that all sentences asserted claim their truth and that a contradictory sentence can't assert its truth, therefore it is false. In other words he's just invoking a hidden aspect to assertion. Were I to draw a parallel it would be to intuitionalist logic. Although clearly there are some big differences.
I'm not sure I find Peirce's comments particularly persuasive. But I need to think through them some more. I came upon them as I was packing on vacation. So I hadn't thought through them too deeply.
If you find this statement persuasive: "Every proposition besides what it explicitly asserts, tacitly implies its own truth", it is a short hop, skip and jump to the ethics of Levinas and Derrida.
Yes, I agree, which was partially why I found the article interesting and relevant to Heidegger on the Liar's Paradox. (Since I tend to read Levinas and Derrida as largely being in basic harmony with Heidegger - other than some quibbles) I did a quick google search, as I said, prior to my trip to see if anyone had written on Heidegger and the Liar's Paradox. Not as good as a library search. But it's kind of a hassle now days to make it up to campus. (Perils of not being a student or single anymore)
Thank you for the link on your sidebar, Clark. For the record I need to state that I'm not really a Peirce scholar; my scholarship is too weak for that. One might say that I'm a Peirce fan.
I found Peirce's account of the Liar Paradox momentarily convincing till I tried the same method with "This statement is true," "This statement is true or false," and "This statement is true and false." Peirce's result with "This statement is false" seemed to break a surface pattern which seemed traced out by my results with the others. As if a spell had been broken, Peirce's account then appeared under a different aspect. His distinction into explicit and tacit had seemed to leave the two levels free from logical dependence on each other; but I think that it doesn't do so and that it doesn't escape the Liar's Paradox. (Still, there's that moment when one thinks that it does....)
Not that I'm so confident about the pattern which I thought I discerned. I'd say I'm 65% confident.
(I added "evidently" hereunder to avoid calling a statement "neither true nor false" but the statement in question may in fact be in a sense evidently neither-true-nor-false.)
"This statement is true" ~ ~ ~ ~ ~ "This statement is false"
Explicit assertion: ~ ~ ~ ~ ~ ~ ~ ~ ~ Explicit assertion:
Neither evidently true ~ ~ ~ ~ ~ [Peirce:] True.
nor evidently false.
Tacit assertion: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Tacit assertion:
Neither evidently true ~ ~ ~ ~ ~ ~ [Peirce:] False.
nor evidently false.
"This statement is true or false" ~ "This statement is true & false"
Explicit assertion: ~ ~ ~ ~ ~ ~ ~ ~ ~ explicit assertion:
Evidently true. ~ ~ ~ ~ ~ ~ ~ ~ ~ Evidently false.
Tacit assertion: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~tacit assertion:
Evidently true. ~ ~ ~ ~ ~ ~ ~ ~ ~ Evidently false.
(It would seem that "This statement is false," if it follows the surface pattern, would come out both evidently true and evidently false both in its explicit assertion and in its tacit assertion. That's what made me take a fresh look at Peirce's account.) My problem is with "This statement is true." It's not logically or otherwise evidently true and it's not logically or otherwise evidently false, but it's not contingent on anything, either; the answer just flatly doesn't exist, it's not imaginably veiled behind infinities like some mathematical undecidable's answer. So it can't be true or false at all. But then it must be false in calling itself true.... But it has already violated the given that every statement is true or false, which gives the Liar Paradox its sting -- and in violating it, paradoxically it takes on a truth value and thus conforms to it. But "This statement is true" doesn't dispute that every statement is true or false, it's merely seeming to counter-instantiate it. Now, "This statement is false" seems to both dispute tacitly and counter-instantiate it. So I figure it's okay to say that "This statement is true" comes out neither (evidently) true nor (evidently) false both in its explicit assertion and in its tacit assertion, though in some sort of Liar's Paradox problematic way. But there's no tying this stuff up in a neat package.
Ben, thanks for the comments. I'm rather busy at the moment catching up with work. So I may not be able to respond for a few days. But this is definitely something I considered when I first read the article but rejected. However since it all took place while I was packing for my vacation, I confess not to having given it the musing it deserved. So I may be completely wrong. But I want to return to this in the context of Derrida and Heidegger. (I'm behind on many posts, including a few over on Peirce-L regarding realism)
Sorry for the delay answering your excellent critique Ben. I've been rather busy at work and have a backlog of things to write.
I take Peirce's claim that every proposition tacitly implies its own truth to be tied up with Peirce's theory of propositions. I find propositions are often taken for granted in philosophy without being clear about what we mean by them. I frequently find that this causes no end of problems. Anyway, as you know but perhaps others don't, Peirce wisely considers propositions to be potential assertions. Thus what I take Peirce to be discussing with "tacit truth" is whether someone can logically assert the statement in question. That's why I brought up intuitionalist logic.
Of course I may be missing some of what you are arguing. For instance you take, "this statement is true" to have an explicit assertion that is neither evidently true nor evidently false. I can see that. I'd simply say that the explicit assertion doesn't make a truth claim. Now can one assert the truth of something that you admit doesn't have a truth value? No. Thus I think Peirce's point holds.
Now this line of reasoning has been brought up about the Liar's paradox many time before. So the more sophisticated version is something like this. "This proposition is not true, not false, and not indeterminate." With presumably "indeterminate" being neither true nor false.
But can one assert such a thing? I think Peirce would say not. So I think Peirce's solution works even for the more complicated forms.
I should add that Peirce is able to take this route because of his claim, "every proposition actually asserted must refer to some non-general subject; for the doctrine that a proposition has but a single subject has to be given up in the light of the Logic of Relations." (CP 5:506) Now if a proposition is a possible assertion but not an actual assertion then can a proposition refer in this fashion? That is can we referentially deal with self referring propositions in this fashion?
Let us change it to "John asserts an assertion that is both true and false." Now can that be true? I don't see how. How about "John asserts and assertion that is neither true nor false." Can that be true? No, because an assertion that is neither true nor false can't be an assertion, as I understand Peirce. We can talk about indeterminacy in the Pericean sense. But we have to be careful because in the context of the Liar's paradox I believe indeterminacy means something quite different from what Peirce means by it. Peirce by indeterminate simply means some predicate that has not yet been predicated of a subject. In the general case this is left to the listener to predicate. (Thus "some man is a bachelor" allows us to pick which man this is) In the vague case this is not left to the listener but the listener must discover by investigation the predicate. (Thus "John saw a rabbit" doesn't allow us to pick which rabbit John saw) Now this is relevant because Peirce is explicit about limiting generals with regard to subject when we make an actual assertion. We may say things. But the assertion has a certain character that I believe would rule out the Liar's paradox to be asserted.
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