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).
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.
(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.