B.C. Askins

The Man With the Golden Gun

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.


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3 thoughts on “The Liar Paradox and Presuppositional Apologetics 2: A Critique

  1. Pingback: Pat Mefford on Liar Paradox and Titus 1.12-13a (Again)

  2. Pingback: The Liar Paradox and Presuppositional Apologetics 3: A “Final” Reply to Patrick Mefford « B.C. Askins

  3. Pingback: The Liar Paradox and Presuppositional Apologetics 4: Defending Classical Logic « B.C. Askins

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