Apologetics to the Glory of God

Pat Mefford on Multi-Valued Logic as an Objection to the Impossibility of the Contrary


I will be responding to this post – http://servileconformist.typepad.com/servile-conformist/2012/12/can-presuppositional-apologists-account-for-logic-.html#

Atheist Pat Mefford offers a rather ingenious means of getting around the transcendental method as used in covenantal apologetics. Now, I know Pat, so let me begin with a bit of friendly ad hominem. The argument of Pat’s post strikes me as illustrating the dangers of familiarity with a little bit of philosophy and a lot more sin. Pat proposes non-classical views of logic (in some cases held by an extreme minority of philosophers) in an attempt to overturn a presuppositional apologetic argument. Frankly, if that is the best one can do, then I feel pretty good about my apologetic.

For the sake of context, Pat should probably know two other things going into his argument. First, more sophisticated presuppositional apologists, including Greg Bahnsen, recognize that classical logic is not the only game in town. There are many logics. Second, not all presuppositionalists affirm the so-called “impossibility of the contrary” as a part of their apologetic. Some third generation Van Tilians in particular have advocated “attenuated Van Tilianism” instead. These two points do not constitute an earth-shattering reply to Pat, but they are worth considering. In any event Pat has quoted me extensively in such a way that the second option is not available to me. Thus I will attempt to address his concerns below.

Impossibility of the Contrary (IoC)

Suppose that the structure of the IoC claim looks something like this:

(1)   A v ~A, ~~A, .:. A

An opponent of the argument might object that the apologist is actually proposing something like the following in the first premise of the argument instead of what appears above:

(2)   A v B (C, D,…n)

And that leads to the opening of a massive can of worms. But Pat is suggesting something a bit different. Pat thinks (1) is problematic, but not because he believes the first premise of (1) is better construed as (2). Rather, Pat objects in virtue of multi-valued logic.

Multi-Valued Logic (MVL)

Referring to (1), Pat writes, “I think what is being smuggled in here is an implicit premise that Chris’ opponent accepts the 2-value logic that Chris’ argument rests upon.”

But why should I simply give that away? What if I insisted on a multi-value logic? Such   as Kleene’s 3-valued logic that has a third value that is an intermediate between true and false. There is a large amount of deviant logics to pick from that allow for theorems or valid inferences not available in the classical logic of 2 value truth functions and first order logic (or as I read one smug British logician quip “the tools of standard reasoning”).

Pat writes as though there is some great conspiracy behind presuppositional apologists using classical propositional two-valued logic in their argument. Those sneaky Van Tilians have “smuggled” in binary logic! Of course, virtually everyone else in the world embraces the same logic. And the Christian worldview certainly allows for it. So there is no conspiracy. There is no smuggling. Suggesting that MVL should serve as a default position is probably stretching it. However, I am thankful that Pat is pointing out the assumptions behind arguments. He would make a good presuppositionalist.

In any event I did not immediately follow Pat’s reasoning. I mentioned Pat is a smart guy, and I need him to dumb down his question for me. I specifically asked if Pat is claiming A v ~A is false. Then I asked if he is claiming that it is undefined. I was not sure. I’m still not sure. Pat replied as follows:

A multi-value logic is looking to assign values to a statement beyond just true or false. For example the proposition “Chris will go to Church on Sunday or he will not go to Church on Sunday” is a true proposition. I don’t have a problem with this but someone who is a presentist (like William Lane Craig) who thinks that only the present is real will have concerns that there is no truth maker in the future that assigns value to either side of the disjunction (because the future doesn’t exist) and so the proposition is neither false nor true, it is indeterminate.

The strategy behind multi-value logics is going to different for each, so the other values are going to range from indeterminate, paradoxical, meaningless, as third values, or there could be six values with three different strengths of “true” and three different strengths of “false”.

Which is all very good. But I have to agree with the sentiment of others who commented on that post. For example, B.C. Askins, who, as far as I know, doesn’t have any strong motivation to rescue the IoC from its allegedly inevitable doom, contributed the following:

Without more context its hard to know what he means, but Kleene’s ternary logic doesn’t have “an intermediate between true and false.” It contains a third value described as “undefined,” not “intermediate.” Insisting on such a logic would probably have zero net effect on a discussion of the IOC, but (as I said) without context it’s hard to know what’s going on. There are other multi-valued logics which propose “truth degrees” (between false(0) and true(1)) but I don’t think Kleene’s system (referring to partial recursive function contexts) falls into that category. Two cents…

Right. So like I asked, is Pat claiming A v ~A is false? Or that it is undefined? Or that it has some other value? I’m not sure. And I’m not alone.

The curious reader will turn Pat’s question back on him. Especially given the intuitive plausibility of classical logic, why on earth should anyone reject it (at least in this instance) in favor of MVL? Well, Pat has an answer. He asks his readers to, “take this proposition as an example: (P1)  P1 is false.”

Right away I’m sure most readers recalled this semantic paradox, but to those not yet initiated, this statement is contradictory. How? (P1) is a proposition that implies its own negation.

He continues:

(P1a) If P1 is true, then it is not true

(P1b) If P1 is false, then it is not false.

Classical logic is telling us is telling us that (P1) is true and that (P1) is false. This means that Chris (and every Presuppositional/Covenantal apologist I’ve ever seen/heard/read) is  stuck in a true dilemma.

If I am not mistaken, that bears a striking resemblance to the sort of argument a dialetheist might make. Apparently Pat isn’t ready to commit to MVL just yet. He’s playing with hypotheticals. So let’s have some fun. Assume, for the sake of argument, that Pat gets really wild and trades his Keene in for some Priest. Keene is for kids. Priest is where it’s at. Assume Pat is a dialetheist. He appears to think there are such things as true contradictions. Most likely, he accepts paraconsistent logic to supplement his new-found faith in dialetheism. Of course he does not spell any of this out. Nor does he argue for his position. He merely assumes it, and thinks it is a difficulty for presuppositionalists. That is more or less what he did with respect to MVL. Pat implies that MVL reigns supreme, but I have not seen any argument from him for that position. Nor do I know how it is relevant to the IoC. I have no reason to think he will behave differently in the current hypothetical.

Christian View of Dialetheism

Pat mentions I might attempt to defend my, “use of classical logic, but this strikes me as a losing proposition since all these multi-value logics were created to resolve these kinds of paradoxes in the first place.” Perhaps so, but Pat is jumping the gun. There have been a number of proposed means of dealing with the Liar Paradox he presents above. The same is true concerning propositions that are similar to it. (By the way, Bahnsen addresses them in his lectures.)

But Pat comes closer to my primary response to dialetheism when he wonders, “how this kind of proposition stands in relation to God.” Consider dialetheism within the context of the Christian worldview. If there are true contradictions, then God believes falsehoods. God does not believe falsehoods. Therefore, there are no true contradictions. (See the relevant section in Dr. James Anderson’s Paradox in Christian Theology.) Now on to Pat’s hypothetical affirmation of dialetheism.

Non-Christian View of Dialetheism

Pat suggests that I can, “do away with classical logic and replace it with some a multi-value logic, but this will require a major revision of the whole concept of the thesis and antithesis, Christianity versus Not Christianity.” (As a quick aside, Pat needs to be careful about reading Hegel into Van Til. Van Til was notorious for using terms differently from the way the Continental philosophers used them.) Suppose, for the sake of argument, that I accept dialetheism.

Then what is the problem? Pat cannot persuade me to forsake Christianity, presuppositionalism, or even the IoC. Why? Because if contradictions are true, then I can accept anything he states in contradiction to these positions while simultaneously holding them! It works out quite nicely. If Pat wins the argument, then he loses. He works his logicks and I remain a Christian presuppositionalist affirming the IoC.

Similarly, one could argue as follows:

Q ^ ~Q



Q v R


Or to put it another way, anything follows from a contradiction ((Q ^ ~Q) -> R). However, if Pat has rejected the principle of explosion through paraconsistent logic then the aforementioned argument will not carry much weight. The point could be argued, but I am not about to do it here.

It is probably more important to ask how Pat discerns a dialethia. Even assuming that a dialethia exists, it does not follow that everything is a dialethia. Further, arbitrarily ascribing the status of dialethia to random propositions seems entirely unhelpful to the dialetheist’s case. Yet Pat appears to have done exactly that in this instance. Hypothetically anyway. A similar situation arose with respect to MVL.


Pat is thinking about MVL. Possibly he’s considered dialetheism. He is doing so in order to respond to the IoC. Given the current status of dialetheism and other deviant logics, I feel as though presuppositionalists are in good shape. It is a testimony to the power of the transcendental method that atheists feel the need to run about preaching the virtues of dialetheism (and there are a number who have). However, Pat has a lot of work to do. He needs to respond to the numerous solutions to philosophical difficulties like the Liar Paradox. He needs to be explicit about his thoughts regarding paraconsistent logic. He needs to provide some sort of criteria for distinguishing between an actual contradiction and a dialethia. He needs to provide a good argument that the IoC involves a dialethia. He has not done any of this, and I am not terribly worried that he will. Consider the analogue for other non-classical logics.

There are much more important things to be concerned about. What happens to a person when he dies? Who is Jesus Christ? These are the questions Pat should start taking more seriously.



14 responses to “Pat Mefford on Multi-Valued Logic as an Objection to the Impossibility of the Contrary”

  1. Patrick Mefford Avatar

    Thanks for the full reply Chris, I’ll have to think on this for a few days.

  2. B.C. Askins Avatar

    Mefford’s argument is a strange one. To summarize: “Either Chris must defend classical logic (described as a “losing proposition” and being “stuck in a true dilemma”) OR embrace multi-valued logic.”

    It is odd to argue by way of disjunction in order to deny the Law of the Excluded Middle (the basis for most MVLs). And, to be clear, by “odd” I mean “self-refuting.”

  3. David Byron Avatar

    A formal logic is a tool for modeling thoughts and the relationships among them.

    FWIW, at least one advocate of an “attenuated” adaptation of Van Til has argued *for* considering the employment of non-classical logics to model some aspects of the Christian Theistic conceptual space.

    See, for example, http://www.baroquepotion.com/vantil/archive-Jul-1999/msg00142.html
    and especially http://www.baroquepotion.com/vantil/archive-Jul-1999/msg00147.html

    In particular, I endorse in http://www.baroquepotion.com/vantil/archive-Jul-1999/msg00141.html the usefulness of free logic as an alternative way to usefully model apparent paradoxes when our goal is to operate over them logically rather than merely rhetorically.

    For this reason, I do not consider the introduction of multivalent logic to pose any particular threat to the conceptual apparatus at issue. Instead, I consider the belief that it would do so to be a symptom of misunderstanding (either of the apparatus or the logic or both)! Choosing the right tool for the right job presupposes(!) adequate familiarity with both the job and the tools.



    1. C.L. Bolt Avatar
      C.L. Bolt

      Good to see you Dr. Byron. Thanks for the comment and links!

  4. David Byron Avatar

    “usefulness of free logic as an alternative way to usefully…” should be “suitability of free logic as an alternative way to usefully…”

  5. […] impossibility of the contrary claim found in covenantal/presuppositional apologetics. I responded here. Pat responded here and […]

  6. […] 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, […]

    [WORDPRESS HASHCASH] The comment’s server IP ( doesn’t match the comment’s URL host IP ( and so is spam.

  7. […] 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 […]

  8. […] 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 […]

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  9. James Avatar

    The argument you presented with regards to the IOC:
    (1) A v ~A, ~~A, .:. A
    Isn’t this question begging, because ~~A and .:. A are logically equivalent? The conclusion of the argument is found in the second premise because ~~A = A.
    It doesn’t feel like anything has been proved?
    Please can you speak into this issue, I am not sure how to overcome it and is causing me some issues as it is the sort of argument which forms the basis of any t.argument.


    1. C. L. Bolt Avatar
      C. L. Bolt

      The conclusion of a deductively valid argument is contained in the premises of the argument.

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