B.C. Askins

The Man With the Golden Gun

The Liar Paradox and Presuppositional Apologetics 4: Defending Classical Logic

This post will be an overly-brief thumbnail sketch of a response to a broad and complex philosophical topic: dialetheism. From SEP: “A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true… dialetheism opposes the so-called Law of Non-Contradiction (LNC),” (i.e. for any A, it is impossible for both A and not-A to be true).

The Liar Sentence (“This sentence is false.”), considered as a semantic paradox, is the most common example of a proposed dialetheia, and has been the subject of my most recent blog series. A notional objection to presuppositional apologetics was proposed by atheist philosopher Patrick Mefford, roughly stating that the Liar Paradox presents a problem for the presuppositional apologist’s use of the LNC in arguing for the existence of God. Mefford proposed that the adoption of a multi-valued logic (rather than the classical binary logic) might blunt the force of the apologist’s reliance on the LNC in his argumentation. (Or possibly the objection was that if there are true dialetheias then God must believe falsehoods or create contradictions or some other such untrustworthy or nefarious thing… as I said, the objection wasn’t clearly stated).

In response to the objection, I proposed that the adoption of a multi-valued logic wouldn’t be as problematic as Mefford supposed (and I criticized his proposed solution as well).

However, there seems to have been some confusion surrounding what a multi-valued logic actually is. (This seems to have been due to Mefford’s recent familiarity with the subject, as evidenced by his acknowledged unfamiliarity with dialetheism and paraconsistency.) So, to be clear:

A classical binary logic has two truth-values: true and false.

A multi-valued logic (MVL) contains multiple truth-values: true, false, and at least one other value – such as “both,” “neither,” “undefined,” “unknown,” etc.

There are also infinite-valued logics, such as fuzzy logic, with truth-degrees between 0 and 1.

There are many different multi-valued logics and I have neither the time, desire, nor the expertise to discuss them all at length here. I would simply note that even Asenjo’s Logic of Paradox (promulgated by the foremost dialetheist Graham Priest) doesn’t deny the LNC outright, but seeks to outline a logic which incorporates the LNC with sentences that are inconsistent with it (i.e. dialetheias). To attempt to put it more simply, a classical, binary logic seeks to maintain logical consistency in light of the LNC, while certain multi-valued logics seek to maintain a kind of logical consistency (paraconsistency) which takes into consideration the LNC and certain, specific dialetheias – while not succumbing to the problem of trivialism (the undesirable view that all contradictions are true) through logical explosion (when the truth blows up and gets everywhere).

Dialetheism is an extreme minority position in the history of Western philosophy, but in its more robust forms it is a difficult theory to overturn. There are many complex and thorny philosophical issues in this regard which, again, go beyond the scope of a blog post. While there are many motives proposed for adopting dialetheism, it would not be inaccurate to say that the Liar Paradox is the central reason proffered for the position.

The most common (and misbegotten) objection to dialetheism is that it entails trivialism via logical explosion – that any sentence can be materially implied from a contradiction via disjunctive syllogism.

An example:

Assume that (A) “All cats go to hell” and (¬A) “All cats do not go to hell” are both true. From this we can validly infer anything, such as (B) “David Hume is David Bowie.”

(P1) Either all cats go to hell or David Hume is David Bowie (A v B)

(P2) All cats do not go to hell (¬A)

(C) Therefore, David Hume is David Bowie. (B by DS)

If dialetheism produces these sorts of logical conclusions then it would appear to be deeply flawed. However, paraconsistent logics are constructed purposefully to avoid triviality. So the argument that dialetheism entails triviality fails because paraconsistent logics are non-explosive (though the details in this regard can be quite technical and are not entirely uncontested).

A stronger response to paradoxes of self-reference is the proposal of MVLs which can account for sentences which appear to be both true and false (or neither true nor false, by intersubstitutivity). So a sentence like the Liar is accounted for by giving it a third truth-value (as described above). However, these MVLs all ostensibly fall prey to various “Revenge Paradoxes,” such as the “Strengthened Liar.”

The Strengthened Liar accepts the truth-values of whatever multi-valued logic may be proposed, but then reproduces the paradox of self-reference within the truth-values of that logic (i.e. “This sentence is not true” or “This sentence is neither true nor false nor both,” etc.). So even the adoption of MVLs with truth-value gaps (neither true nor false) or gluts (both true and false) falls prey to various Strengthened Liars. Whatever truth-values a given logic may contain, a Liar Sentence can be produced for those values. These sentences have been called “Revenge Paradoxes,” in that they respond to proposed solutions to the semantic paradoxes with a reformulation of the original paradox, seeking semantic vengeance on their objectors. (“Semantic Vengeance” would be a pretty good band name for a progressive metal group, don’t you think?)

To summarize, semantic paradoxes (such as the Liar) provide evidence for the dialetheistic cornerstone position that there are true contradictions. The paradoxical characteristics of sentences like the Strengthened Liar(s) are due to the ordinary features of natural language, such as self-reference and the presence of truth predicates (i.e. “is true”). Various proposed solutions fail, such as Tarskian metalinguistic hierarchies, since they only produce languages that are expressively weaker than English. MVLs are non-explosive but still fall prey to Strengthened Liars. Several other solutions have been proposed, but most simply beg the question in favor of classical logic. As I said, dialetheism is a difficult theory to overturn.

So what recourse is there for the defender of classical binary logic in the face of the Liar Paradox?

Recently, a defense of monaletheism has been advocated by Benjamin Burgis, in his doctoral dissertation (HT: Paul Manata). The essence of Burgis’ argument, as I understand it, is that sentences with truth-value ascriptions are meaningless unless they are “grounded-out” in sentences which contain no truth predicate (p. 112f., esp. n. 101).

The problem for the Liar is that this semantic paradox doesn’t ground its truth attributions in extra-semantic reality. Burgis alternatingly (and somewhat confusingly) calls this the “meaningfulness solution” or the “meaninglessness solution.” He states it more explicitly as the “Kripke/Tarski Thesis: We are making some sort of mistake when we attribute truth or falsity to a sentence that isn’t (directly or indirectly) about something other than truth” (p. 116). He argues that sentences like the Liar are actually meaningless (and thus not true dialetheias), though they give an initially plausible appearance of meaningfulness because they contain many of the characteristics of meaningful sentences, such as being grammatically well-formed, self-referential, truth-ascribing, etc.

The argumentation he presents is extensive and I would commend it to any with the time and interest in reading it. He seems to have a good case for a non-question-begging response to dialetheism, which is easier stated than demonstrated. Given our discussion above, it seems best then to briefly consider whether or not Burgis’ defense of monaletheism falls prey to any sort of Revenge Paradoxes.

A Revenge Paradox to Burgis’ meaningfulness solution could be formulated as: “It would be a mistake of some sort to call this sentence true.” If we say the sentence is true, we are mistaken – since it’s meaningless (per Burgis’ solution). If we say it is false, then we commit no mistake when we say it is true – but that’s exactly what the sentence says is a mistake. We make one kind of mistake in ascribing truth to a meaningless sentence, and another kind of mistake in ascribing falsehood to a true sentence. If it’s true, then we’re mistaken, if it’s false, then it’s true (and we’re mistaken), and if it’s meaningless then it’s true (and we’re still mistaken). There doesn’t appear to be a non-mistaken way to refer to the truth-value associated with this Revenge Paradox sentence.

So, given this analysis, a way of reformulating this sentence would be “This sentence is either false or meaningless.” It’s this disjunction which allows Burgis’ meaningfulness solution to escape the Revenge Paradox, since the first disjunct (“This sentence is false”) is meaningless and a disjunction must have two meaningful disjuncts in order to ascribe truth-value to it (per the meaningfulness solution). So if the disjunct is meaningless and it is saying the same thing as the Revenge Paradox above, then this Revenge Paradox is also meaningless (or begs the question against the meaningfulness solution).

So if the strongest candidate for a proposed dialetheia, the Liar Paradox, is meaningless, then one (the?) major objection to classical logic has been de-fanged.

In my limited and humble estimation, Burgis’ proposals give the strongest non-question-begging, non-ad hoc, intuitively plausible defense of monaletheism (and concomitant critique of dialetheism) available for pursuing a defense of classical binary logic in the face of semantic paradoxes such as the Liar.

So, to conclude this series, Mefford’s original objection can be answered by the presuppositional apologist through (1) demonstrating his dilemma is hornless by adopting a multi-valued logic (maintaining the same thesis-antithesis approach which incorporates the LNC but adopting an MVL as concerning the semantic paradoxes where necessary) – this is not problematic since the presuppositionalist in particular understands the relationship between divine and human logic as analogical; (2) criticizing his proposed solution in the Tarskian hierarchy; and (3) by defending classical logic, arguing that the semantic paradoxes like the Liar are meaningless.

In any case, it would hardly seem that the presuppositional apologist (or any apologist in general, I think) need fear anything from a consideration of the Liar Paradox.


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7 thoughts on “The Liar Paradox and Presuppositional Apologetics 4: Defending Classical Logic

  1. Thanks for a very useful article. My primary concern is with the first summary statement in your conclusion:

    “Mefford’s original objection can be answered by the presuppositional apologist through (1) demonstrating his dilemma is hornless by adopting a multi-valued logic (maintaining the same thesis-antithesis approach which incorporates the LNC but adopting an MVL as concerning the semantic paradoxes where necessary) – this is not problematic since the presuppositionalist in particular understands the relationship between divine and human logic as analogical.”

    I see at least 2 problems in using this defense when arguing with an unbeliever. First, if we adopt MVL for the “semantic paradoxes where necessary” don’t we admit that the LNC is not an absolute law of logic? And if it is not an absolute law of logic, then there is no need to ground it in an absolute being. Second, the unbeliever could likely point out that we are engaging in the same arbitrariness of which we accuse him. That is, when the going gets tough, we just redefine certain things and qualify our statements to make sense of what we need to.

    • JohnD,

      Thanks for reading and for the thoughtful comment.

      In the section which you quoted I was summarizing a few different apologetic tactics available to the presuppositionalist (of one kind or another) which I would consider valid. In my original discussion of (1) I suggested that the apologist could adopt a MVL for the sake of argument (at least), much the way VanTilians commonly adopt a skeptic’s position for the sake of refutation (often via reductio) – demonstrating there is no dilemma.

      Given your apparent belief that an “absolute law of logic” needs to be “grounded” in an “absolute being,” you may prefer tactic (3), defending classical logic. However, I don’t want to assume that I know what you mean by the terms in quotations in the last sentence, so you maybe you’d care to elaborate on that a bit?
      (More specifically, what is an “absolute law of logic”? What is an “absolute being”? And in what way is the former “grounded” upon the latter?)

      I don’t recall accusing anyone of arbitrariness anywhere in this series, so I don’t think your second criticism sticks to my prior presentation. Further, I’m not proposing an ad hoc adoption of MVL in the face of a particular objection to the LNC (that would be a problematic tactic) – rather, we should have good reasons for going that route, either for the purpose of reductio (as mentioned above) or because we believe a MVL more closely aligns with divine revelation.

      Thanks again for the comment. Cheers!

      • B.C. thanks for the response. Admittedly, I did not read the prior posts in the series so let me do that before I reply in depth again. Sorry for misunderstanding part of your post, because what you just said you weren’t doing is what I thought you were suggesting. Namely, “Further, I’m not proposing an ad hoc adoption of MVL in the face of a particular objection to the LNC (that would be a problematic tactic).” So, that clears up a lot.

  2. Hmm I did read the posts and it does leave me a little bit uncertain of your position B.C. Askins. Are you saying a robust MVL or dialetheism view of logic is possible as a Van tillian or is this simply an internal critique of that position?

    Are you a staunch defender of Law of Non Contradiction which cannot be assailed? Because, I too had the same question as JohnD. If the LnC doesn’t always obtain then what do we make of that? Would you say that we simply have a human view of LNC but God knows there are no contradictions? If yes, then what is the unbeliever accounting for. If he is accounting for the instances where the LNC seems to hold, how do we know which instances those are and that it is absolute since we just said it doesn’t always obtain. Perhaps some clarity on these issues would be benefit me and some of your readers. Thanks God bless.

    • Hi Peter,

      Thanks for reading the posts and taking the time to comment. I appreciate your questions and I’ll respond to each below, in sequence.

      You may be “a little bit uncertain of (my) position” because I never explicitly said which line of argumentation I would personally pursue in discussing the Liar Paradox with a logically astute skeptic like Mr. Mefford. Rather, I presented several arguments which I think would be consistent with most flavors of VanTilian apologetics. If one particular argument I presented doesn’t seem to match your particular view of apologetics, I think one of the others might (I suspect most apologists, particularly traditional VanTilians [the Dr. Greg Bahnsen, strong modal TAG “impossibility-of-the-contrary” types] would probably prefer option 3, defending classical logic).

      So, on to your questions:

      1a.) “Are you saying a robust MVL or dialetheism view of logic is possible as a Van tillian…”

      1b.) “…or is this simply an internal critique of that position?”
      No, it’s not simply an internal critique of a particular position. However, demonstrating that the adoption of an MVL is non-problematic for a VanTilian was part of an internal critique of Pat Mefford’s dilemma for Chris Bolt’s apologetic.

      2.) “Are you a staunch defender of Law of Non Contradiction which cannot be assailed?”
      If it “cannot be assailed” why would I need to staunchly defend it? (You’re asking a loaded question here.)

      3.) “If the LnC doesn’t always obtain then what do we make of that?”
      I take this to be a rhetorical question, introducing the following questions, right?

      4.) “Would you say that we simply have a human view of LNC but God knows there are no contradictions?”
      As humans, we only have a human view of anything, including the LNC. Even with respect to divine revelation, the Holy Spirit’s illumination of Scripture is the illumination of a human view of it.
      If there is such a thing as a “true contradiction” (in the dialetheist sense) then it is an object of God’s omniscience – like everything else which is knowable.

      5.) “If yes, then what is the unbeliever accounting for?”
      All that is necessary for the sort of argument which Chris Bolt was using is the existence of necessary truths, though not necessarily the LNC. The LNC is a commonly used example of a necessary truth because of its familiarity to most people and its near universal acceptance as an absolute rule of inference. However, there are alternative options available to the apologist, which may be necessary in responding to the sorts of objections which might be raised by a dialetheist. So the unbeliever must account for necessary truths, of one kind or another (i.e. moral, modal, physical, logical, ontological, etc.). If he says the LNC isn’t a necessary truth, then I’ve presented several lines of argument available to the apologist. Pick your poison.

      6a.) “If he is accounting for the instances where the LNC seems to hold, how do we know which instances those are…”
      There’s some truth tables in this article which may help you begin to sketch an idea about when the LNC would and wouldn’t hold in dialetheism (and its logical sibling ‘analetheism’): http://homepages.uconn.edu/~jcb02005/papers/analetheism.pdf

      6b.) “…and that it is absolute since we just said it doesn’t always obtain?”
      If it doesn’t always obtain, it’s not absolute – as I think you recognize. It can be generally valid without being absolute, though.

      Thanks again for the questions. Let me know if you have any more or need further clarification. Cheers!

  3. Hi B.C Askins,

    Thanks for the reply is was helpful. A few loose ends that I am trying to tie together. As you said defending classic logic is probably the direction I would go. So in the Burgis dissertation he says he is not grounding it out in extra semantic reality, what does he mean by that and why does he mention it?

    Also, supposing we did adopt a robust dialetheism view I don’t quite understand when we know its dialetheia or not. If it results in some sort of liar paradox where you have a “true contradiction” then its dialetheia is that the route, but if it doesn’t give a “true contradiction” then its not?
    If we have these dialetheia in our system then those “true contradictions” really on Christianity are unknowable to us creatures but fully known by God via creator / creature distinction, would that be accurate? So with dialetheia we would still be calling them to account for all the non dialethia to make them intelligible? Does dialtheism do any harm to the view that logic is universal? Thanks.

    • Burgis says that the Liar Paradox is not “grounded out in extra-semantic reality,” meaning that the sentence “This sentence is not true” does not refer to any realities outside of the signs and signifiers that we use to designate and communicate meaning. The sentence “It is snowing outside” is grounded out as true or false based on the extra-semantic realities of the weather outside (i.e. either it is or is not snowing). The Liar Sentence is meaningless (he argues) because it is only about truth without any corresponding reality upon which to “ground” that truth. Starting to make sense?

      Your following questions raise a whole host of issues which I (unfortunately) don’t have enough time to track down with you. Apologies in that regard. Essentially, a dialetheia would have certain characteristics (which aren’t fully agreed upon by philosophers – imagine that!). The LNC would still apply to non-dialetheic sentences then.

      If you’re interested in further reading in this regard, I’d start with Graham Priest’s book Doubt Truth to be a Liar. It’s less technical than Priest’s other books on the subject, but still gives many insights regarding dialetheism. Thanks again for reading and commenting.


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