B.C. Askins

The Man With the Golden Gun

Archive for the month “December, 2012”

The Liar Paradox and Presuppositional Apologetics 3: A “Final” Reply to Patrick Mefford

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.

It appears that Patrick Mefford has decided to take his ball and go home. Entirely his prerogative. We salute your service, Mr. Mefford, and wish you a happy new year.

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…

Misreading Paul?

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).

Misreading me?

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.

Bottom line

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.


The Liar Paradox and Presuppositional Apologetics 2: A Critique

English: The Pinocchio paradox. When Pinocchio...

The Pinocchio paradox. When Pinocchio says “My nose grows now” it creates a Liar sentence and makes Pinocchio’s nose to grow if and only if it does not grow. (Photo credit: Wikipedia)

Pat Mefford has said that “conventional notions of truth and falsity in our natural language and in our everyday discourse are what I consider to be a useful fiction.” He proposes the adoption of Alfred Tarski’s semantic conception of truth, an artificial hierarchy of languages whereby the truth predicate for an initial “object-language” is only found in a “meta-language” (neatly escaping the Liar Paradox, i.e. “This sentence is false.”). However, as we’ll see momentarily, if the intuitively appealing use of “true” and “false” in our natural language is a “useful fiction,” as Mefford put it, then Tarski’s hierarchy could be considered a “useless fiction” in contrast.

Tarski’s proposal is that we can save consistency in the face of the Liar Paradox, not for natural languages, but for restricted and regimented artificial languages, wherein no language contains its own truth predicate. At the bottom level, we have the “object language,” which does not contain words like “true” or “false” at all. Above that we have the “meta-language,” which contains such words, but where they can only be applied to object language sentences. It really is a very clever way of avoiding the Liar, since no self-referential, truth-value-ascribing sentences are possible within any given language.

However, if we want to contend with the truth-value of an assertion made in the meta-language, we would need further recourse to a meta-meta-language to consider whether or not the truth predication of the meta-language (regarding the object-language) is correct. What if there’s a question regarding the truth ascribed in the meta-meta-language? Well, we need a meta-meta-meta-language. See the pattern? It’s turtles all the way down.

Consider the following example:

(1) All dogs go to heaven.

(2) It is true that “all dogs go to heaven.”

(3) It is false that “all dogs go to heaven.”

Sentence (1) is written in the object-language while (2) and (3) are in the meta-language. How do we express the disagreement between (2) and (3)? If we were allowed to do that in the meta-language, then the meta-language would contain its own truth predicate, and we could construct a meta-language sentence like (1), but the problem of the Liar wouldn’t have been avoided. So we have to step up into a meta-meta-language to make a statement like:

(4) It is false that “it is true that ‘all dogs go to heaven.’”

Note that most explanations of Tarski’s linguistic hierarchy are commonly made in our natural language, not within the hierarchy itself. This isn’t clearly self-refuting, since such a project appears possible, but it would seem awkward and impractical to parse out each sentence’s place within the hierarchy while explaining the hierarchy itself; consistency would require such a task though. So, while this hierarchy of languages presents a resolution for the special case of the semantics associated with the Liar Paradox it makes the semantics associated with the rest of discourse cumbersome and unnatural.

Further, there are statements which seem intuitively true and which would be useful in discussing Tarski’s hierarchy but which simply can’t be said in any of his artificial languages. For instance:

(5) No sentence anywhere in the hierarchy says of itself that it is false.

(6) No sentence anywhere in the hierarchy is both true and false.

Sentences like (5) and (6) are commonly used in persuading someone to employ Tarski’s hierarchy, but have no place within the hierarchy itself. There is no “ultimate-meta-language” or “trans-meta-language” which can use a truth predicate in reference to itself (much less to the entire hierarchy). This is a severe problem for the theory, since the principle which helps it to escape the Liar Paradox also prevents it from being useful in broader discourse, especially when attempting to demonstrate the value of its proposed solution.

To add insult to injury, the Liar Paradox may not be predicable in Tarski’s hierarchy, but it is still present in English – and Tarski’s solution says nothing about that.

In fact, some* interpret Tarski to be a “proto-dialetheist” of sorts, in that he believed semantic paradoxes of self-reference within natural languages were inescapable (he famously, and controversially, stated that natural languages were “inconsistent”). This may be why he was willing to abandon natural languages to preserve logical consistency (albeit of an artificial sort). (*See The Law of Non-Contradiction: New Philosophical Essays by Graham Priest, et al, p. 118f.)

So, with regard to Mefford’s proposed solution to the Liar Paradox (which he raised as an objection to Chris Bolt’s presuppositional/covenantal apologetic) there are some severe problems which leave him in the regrettable position of having no answer for his own objection.

As I said in my last post, (if I get the time and people seem interested) I’ll propose a solution to the Liar Paradox, as well as some ethical considerations on the whole issue.

The Liar Paradox and Presuppositional Apologetics

So Pat Mefford has raised an objection (here, here, and here) to Chris Bolt’s presentation of VanTilian Presuppositional (sometimes voguely referred to as Covenantal) Apologetics. Chris asked Pat for clarification, then responded here and here. Pat raises the “Liar Paradox” as an objection to Chris’ account of logic, specifically the Law of Non-Contradiction (LNC), as grounded analogically in the divine thoughts.

(His original objection was that Chris was “smuggling” classical binary logic into his account of logical laws – something which Chris’ opponent may not be willing to grant. Instead, Mefford proposed the adoption of Kleene’s ternary logic, which is paracomplete (it denies the Law of the Excluded Middle (LEM), not the LNC). Then he presented the Simple Liar Paradox (i.e. “This sentence is false.”) as the basis for adopting a (presumably paraconsistent) multi-valued logic (MVL). Initially, it was unclear which objection Mefford was proposing: the adoption of Kleene’s MVL (denying the LEM) or the acceptance of the Liar Paradox (denying the LNC). His follow-up posts have focused exclusively on the Liar Paradox, possibly because I pointed out that Mefford’s original argument was self-refuting, since he argued for denying the LEM via disjunction – a philosophical solvent which dissolves itself.)

Laying all of that to the side, I must admit that (despite Mefford’s clarifications) I’m still not sure what the substance of the objection is. To quote Mefford:

In what way does Chris’ simulacrum of God’s divine system answer this paradox? In what way are we thinking God’s thoughts after him when think (sic) of this scriptural passage that was at the top of my original post?… This proposition exists. It is a contradiction. How does it stand in relation to the Triune God? How is this proposition grounded in Almighty God? How does Chris account for it?

To my reading, it seems that Mefford is merely asking for an account of the Liar Paradox from the perspective of the Presuppositionalist (specifically one who holds to a classical, binary logic as Chris does). I don’t see how this is a very significant objection, but that may be a reflection of my poor reading skills rather than Mefford’s argumentation.

For example, Mefford believes Tarski’s semantic conception of truth solves the Liar Paradox. So Chris might say (arguendo, at the very least) that Tarski’s hierarchy more closely reflects God’s thinking with respect to the Liar. (I might suggest to both of them that Saul Kripke’s partial predicate T-schema seems to be a less problematic resolution. Kripke’s construction seeks to maintain classical binary logic with the Liar falling into a “truth-gap,” rather than outside of Tarski’s overly-restrictive semantic hierarchy.) Does this result in some wholesale denial of classical logic? Hardly. It simply provides a semantic “extension” of classical logic which “subsumes the binary apparatus into the new system” (to use Mefford’s own terms). Whether Tarski’s or Kripke’s or some other account is best is irrelevant to answering Mefford’s objection, since the Presuppositional Apologist is only committed to viewing human logics as analogs of divine “Logic” anyway.

Further points to consider:

Mefford’s treatment of Titus 1:12-13 is superficial. To prove this is a biblical example of affirming the Liar Paradox would require Mefford to establish that Paul intends to say that “absolutely every Cretan lies every time he speaks.” It seems much more natural to interpret Paul as hyperbolically generalizing about Cretans, citing one of their own poets as an authoritative source. Chris made this point quite clearly.

For the sake of argument, however, let’s assume Mefford’s interpretation. So Paul says the Liar Paradox is true. This would simply be a proto-affirmation of Graham Priest’s contention that there are true dialethias. This interpretation would put Christians in the minority in the history of Western philosophy, but dialetheism is not as easily refuted as some might like to think. So the Christian might just say that a paraconsistent, non-explosive logic is the most accurate way of thinking God’s thoughts after him. Mefford’s objection is again refuted.

Mefford seems to think that the adoption of a non-classical logic would dull the sharpness of the Classical Presuppositionalist’s double-edged Thesis-Antithesis approach to apologetics. However, he’s never demonstrated why this must be the case, and I’m not interested in doing the heavy lifting for him. To the contrary, the adoption of a classical-logic-with-truth-value-gaps may serve to strengthen the Presuppositionalist’s case in certain ways, a la Peter Strawson‘s conception of presupposition.

However, my intent is not to elaborate on that point right now, but rather to show that Mefford’s dilemma is hornless. Chris could defend classical, binary logic (hardly a losing proposition in my mind) or adopt a non-classical, multi-valued logic without losing any ground to his interlocutor. In my estimation, a VanTilian apologetic could comport with the more classical monaletheism or with a robust form of dialetheism, having a well-formulated paraconsistent logic.

If I get the time, I may attempt to present a reasoned defense of classical logic in this context – or Chris may beat me to the punch; I’d be willing to give it a shot, if others would find such an endeavor useful.

Finally, I find it interesting that Mefford sees the Liar Paradox as “rooted in epistemology and metaphysics,” but does not mention ethics. Being a good tri-perspectivalist, I think there must be some ethical aspects to any explanation for the Liar, which I may also consider at a later date. Remember, lying is a sin…

Tullian Tchividjian, Rick Phillips and Total Depravity

So the evangelical blogospheric dust-up last week between Tullian Tchividjian (TGC) and Rick Phillips (Ref21) over the relationship between justification and sanctification in the Christian life appears to have settled somewhat. Tullian wrote an article responding to the question “Are Christians Totally Depraved?” His answer, in short: Yes – in a sense.

Rick Phillips criticized Tullian’s article in his own blog post, “Thank God that Christians Are Not Totally Depraved.” Tullian responded with a post which was largely an exercise in missing the point, quoting extensively from various Reformed confessions while turning the discussion away from the question of depravity and to an argument about the relationship between justification and sanctification. Phillips responded with a comparatively irenic post, attempting to clarify his own and Tchividjian’s positions. Meanwhile, the TGC combox brought the usual heat rather than light, with Tullian’s fanboys and girls rallying around their hero.

Several have commented that the issue seems like two sides of the same coin, arguing about differences of emphasis (rather than substance) in relating justification and sanctification. Phillips rounded out this discussion by posting on common misconceptions in that regard.

While it appears accurate to see both Phillips and Tchividjian as orthodox in their views of justification, sanctification, and the relationship between the two, the heart of the disagreement is regarding total depravity, not justification and sanctification.

Bottom line: Tullian misused the term “total depravity” when he applied it to Christians. He says Christians are totally depraved “in a sense,” that sense being half of the standard theological definition of total depravity. And “half-totally depraved” is a confusing concept.

A definition of total depravity incorporates (at least): (1) the radical corruption of every human faculty by sin and (2) a complete inability to not sin (non posse non peccare; cf. Berkhof p. 246-7, Reymond p. 450f., Grudem p. 497f., Shedd p. 601-2). Tullian stipulates that he is using the term only to refer to the first half of the definition – but this is a confusing misuse of the term. While it may carry rhetorical force in illustrating his point regarding the pervasive nature of sin even in the life of a Christian, it is ultimately a confusing rather than clarifying way of teaching the doctrine of indwelling sin in the life of a believer.

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