A good link-history of the recent discussion surrounding the Liar Paradox and presuppositional apologetics can be found at the beginning of Chris Bolt’s latest post. So I won’t re-tread any of that here.
In his final post to Chris Bolt, he did respond to me as well – so I’ll give some final thoughts in return. ‘Tis the season for giving and all…
There was a bit of exegetical back-and-forth between Chris and Patrick over Titus 1:12-13a, specifically regarding whether or not Paul presents a case of the Liar Paradox when he quotes a Cretan prophet stating, “Cretans are always liars.”
Given that the broader context of Titus 1 has to do with the careful selection of church leaders according to certain character qualities, it makes the best sense to understand Paul’s quotation of Epimenides as proving his point – that close evaluation of men from Crete would be necessary in choosing church leaders. Cretans were notorious scoundrels, and even their own prophets (whom one might expect to evaluate their own heritage and culture patriotically) can be called as witnesses to the fact – “we’re all liars, gluttons and monsters.” This seems more in line with the whole letter to Titus – rather than the introduction of dialetheism without further commentary of any sort from the author.
However, I don’t think Patrick intended to get into an extended exegetical debate over the issue. His point seems to me to have been an attempt at providing a stronger reason for a Christian apologist to want to resolve the Liar Paradox – “look, it’s even in the Bible.” I’ll give Patrick credit for trying to bring some further rhetorical force to his objection, but his conclusion (unfortunately) requires a rather forced interpretation of Titus – which Chris, to his credit, resisted.
Regardless of the exegesis of Titus, the Liar Paradox remains an issue for both parties, since Chris desires to maintain a classical binary logic and there’s a few problems with Tarski’s position (which Mefford has proposed).
Patrick quotes a section of this post, where I’m attempting to summarize Tarski’s semantic hierarchy as relevant to the issue of the Liar Paradox. He quotes me summarizing Tarski, then points out that “this does not accurately describe what I laid out.” Nor was it meant to. It’s a summary of Tarski, not Mefford.
Admittedly, my summary borders on an oversimplification of Tarski, but it was an attempt to break things down to laymen’s terms (as much as I might be able); but to criticize it for not being something it was never intended to be is hardly appropriate. So he misread me.
Misreading me… or Tarski… or himself?
Regarding his own formulation, Mefford states: “There is no bottom level object language that does not contains (sic) words like ‘true’ or ‘false’. This is made explicit.”
However, quoting Mefford’s original post, he formulates the Liar Sentence within the hierarchy as:
(A1b) “P2 is false” [ln] is true [Tn+1] if and only if P2 is false [ln]
He explains, “The liar paradox cannot be stated in this hierarchy, because any language it is written in will not have the proper truth predicate.”
So, on the one hand, he says: (X) the bottom-level object language (ln-1) does not contain the proper truth predicate (i.e. words like ‘true’ or ‘false’), which is how the hierarchy eludes the paradox.
Or, on the other hand, he says: (Y) “there is no bottom level object language that does not contains (sic) words like ‘true’ or ‘false’.” So which is it, (X) or (Y)?
These propositions exist. They are a contradiction. How do they stand in relation to Patrick Mefford?
Well, most likely, he has either misread me or Tarski (or himself?) or some combination thereof. I don’t know which.
(Let me make the point more explicitly: if Mefford’s object language contains a truth predicate then it is susceptible to the Liar Paradox; if it does not contain the relevant truth predicate then he has no basis for his objection to my summary.)
Misreading me again?
Mefford goes on to quote my point regarding the infinite nature of Tarski’s hierarchy. But for some reason he thinks I’m raising this as a “mistake or problem.” He says, “…asking me where the hierarchy stops is to assume to (sic) the collection is finite instead of infinite.” But I never asked where the hierarchy stops. I didn’t ask, because that’s a dumb question. I was merely pointing out that Tarski’s hierarchy is infinite. I agree with Mefford when he says, “An infinite regress isn’t a mistake or a problem. Infinite regression is fine n dandy in mathematics…” So, I’m not sure why he disagrees with me, unless it’s due to the jocular reference to “turtles all the way down.”
It’s my understanding that the phrase “turtles all the way down” has specific historical reference to the infinite regress problem within the domain of cosmology – but when used outside of that domain it can refer to an infinite regress of whatever kind (in this case a non-problematic infinite regress). Since we weren’t at all discussing cosmology, I assumed Mefford would know that I was merely referencing a colorful illustration of the infinite recourse within Tarski’s hierarchy.
However, if Mefford objects that I shouldn’t have borrowed a term from cosmology for illustrative purposes, then I’ll simply point out (in good fun) that his illustration from Tristram Shandy is a pure cock and bull story.
Unfortunately, Mefford chose not to interact with any of my actual criticisms of Tarski’s hierarchy (such as that the semantic hierarchy is incapable of useful self-reference, it relies on natural language in a problematic way, and that it doesn’t resolve the Liar Paradox for natural languages). Rather, it would appear that his engagement with presuppositional apologetics in general could be characterized by the sorts of misreadings outlined above.
Since C.L. Bolt was kind enough to “leave the rest” to me, I’ll be following this post with a (brief) defense of classical logic, Lord willing.