A Unifying Double-Reference Approach to Semantic Paradoxes: From the White-Horse-Not-Horse Paradox and the Ultimate-Unspeakable Paradox to the Liar Paradox in View of the Principle of Noncontradiction

3.1.1 A Double-Reference Approach to Dissolving the White-Horse-Not-Horse Paradox

The alleged “White-Horse-Not-Horse” Paradox was historically presented as the following pair of statements (or the “White-Horse-Horse” [WHH] and “White-Horse-Not-Horse” [WHNH] sentential pair), in terms of its incomplete natural language expressions:

(WHH) The white horse is the horse.

(WHNH) The white horse is not the horse.

which, given that they are used to state the way things are, they can be presented in a semantic-ascent way via the truth predicate as follows:

(WHH)_T “The white horse is the horse” is true.

(WHNH)_T “The white horse is the horse” is not true.

The aforementioned presentation pairs, the pair of (WHH) and (WHNH) or the pair of (WHH)_T and (WHNH)_T, seem to present contradictory or inconsistent statements and thus bring about a contradiction. When employing the word “seem” in the aforementioned saying, I mean this: whether the aforementioned presentation pair really presents contradictory or inconsistent statements would depend on, or need to be sensitive to, whether the aforementioned statements are (considered to be) the semantically completely or incompletely given sentences;^[22] if they are taken to be semantically completely given, they do constitute a pair of contradictory statements; if they are taken to be short for semantically completed given sentences, the contradictory appearance can be dissolved if they undergo completion to fill out the missing propositional elements.^[23]

With the foregoing narrative explanation in place, the alleged “White-Horse-Not-Horse” Paradox, as presented earlier in terms of (incomplete) folk way (WHH) and (WHNH), can be rephrased in a complete way showing its complete semantic proposition of a sentential token of people’s basic employment of language, the semantically incompletely given sentences above, (WHH) and (WHNH), need to undergo completion to fill out the missing propositional elements. For example, the explicit description of the addressed specific-part referent as what is sought at the referential stage is missing in the originally given incomplete sentence:

(WHH): The white horse is [identical to] horse.

(WHNH) The white horse is not [identical to] the horse.

After undergoing completion to fill out the missing description of the specific-part referent (i.e., the description of the specific “white-color” attribute of the white horse), the resulting complete sentence that explicitly gives the complete propositional content is this:

(WHH)* The white horse with its specific white-color attribute in referential focus is (identical to) the horse regarding (being relative to) their shared general attribute of being a horse [that is shared by all horses].

(WHNH)* The white horse with its specific white-color attribute in referential focus is not (identical to) the horse regarding (being relative to) the distinct “white color” attribute [that is shared by all white horses but not by all horses].

Through the double-reference analysis, it turns out that the presentation pair, the pair of (WHH) and (WHNH) or the pair of (WHH)* and (WHNH)*, does not present contradictory or inconsistent statements but talk about distinct referentially identified objects with their distinct “double-reference” identities (i.e., distinct double-referent complexes) respectively in the further formulated (WHH)_DR and (WHNH)_DR as given below: one referentially identified object with its “double-reference” identity, which brings about one double-referent complex, is the white horse (as the semantic-whole-referent simplex) plus its specific-part attribute “being a horse” or “having horseness” (as the specific-part-referent simplex) in focus in (WHH)_DR; the other referentially identified object with its “double-reference” identity, which brings about the other double-referent complex, is the white horse (as the semantic-whole-referent simplex) plus its specific-part attribute “being white” or “having whiteness” (as the specific-part-referent simplex) in focus in (WHNH)_DR.

The alleged White-Horse-not-Horse Paradox is thus dissolved from a holistic vantage point of double-reference approach, which can be highlighted and symbolically formulated as follows:

(WHH)_DR WH_[h] =_[H] H

(WHNH)_DR 〜 (WH_[w] =_[W] H)^[24]

or, given that “α _[γ] ≠_[Γ] β” is short for “〜 (α _[γ] =_[Γ] β)

(WHNH)_DR WH_[w] ≠_[W] H

(WHH)*_DR “WH_[h] =_[H] H” is true

(WHNH)*_DR “WH_[w] =_[W] H” is not true

In this way, the addressed “double-reference” treatment, as implicitly shown in Gongsun Long’s “White-Horse-Not-Horse” argumentation and as explicitly elaborated in the philosophical interpretation presented here, dissolves the paradoxical appearance of the alleged “White-Horse-Not-Horse” paradox. The addressed double-reference treatment constitutes one exemplar case of how, given an object, we can effectively and consistently talk about the object while pointing to its distinct aspects at the same time in a non-self-contradictory way, although thus subsequently saying different things might bring about seemingly contradictory predicative comments (via seemingly contradictory sentential predicates). The double-reference approach essentially points to the referential-sensitivity core of the principle of noncontradiction as captured by Aristotle’s classical characterization,^[25] rather than being against the genuine point of the principle of noncontradiction as one fundamental semantic principle in capturing the relation between language/thought and the world.^[26]

3.1.2 A Double-Reference Approach to Resolving the Ultimate-Unspeakable Paradox

The so-called Ultimate-Unspeakable Paradox can be presented as the following pair of statements [or the “Ultimate-Speakable” (US) and “Ultimate-Unspeakable” (UUS) sentential pair) in terms of its incomplete natural language expressions:

(US) The Ultimate is speakable.

(UUS) The Ultimate is unspeakable.

which, given that they are used to state the way things are, they can be presented in a semantic-ascent way via the truth predicate as follows:

(US)_T ‘The Ultimate is speakable’ is true.

(UUS)_T ‘The Ultimate is unspeakable’ is not true.

In classical Chinese philosophy, the ultimate reality, such as the metaphysical Dao in classical Daoism, is sometimes labeled “the unspeakable,” which is considered “unspeakable” by some commentators in the sense that the Dao itself can be neither talked about nor described; we thus have to ponder it in silence. Such an understanding results from a widely circulated but problematic interpretation of certain sayings and passages in the classical Daoist text Dao-De-Jing; one of the most oft-cited textual passages is the opening message that consists of the first two statements in Chapter 1 of the Dao-De-Jing, which I have examined in my previous writings.^[27]

In this section, through interpreting Lao Zi’s treatment of the point of the opening message of the Dao-De-Jing text in accordance with his fundamental insight in this classical Daoist text, I explain how the suggested “double-reference” perspective approach can resolve the Ultimate-Unspeakable Paradox by talking about a referentially identified contradictory aspect of the ultimate as a whole in a noncontradictory way.

As explained in my previous writing, instead of indiscriminately giving a negative claim against any language engagement with the ultimate concern, in the first six-character statement, Lao Zi reveals a two-sided transcendental insight: on the one hand, it positively affirms the role of the language-engaged vantage point of view in capturing the Dao; on the other hand, it alerts us to the limitation of the finite dimension of the addressed vantage point of view and emphasizes the transcendental dimension of the Dao as the ultimate concern. More specifically, Lao Zi’s fundamental insight concerns how it is possible to designate the Dao as a whole while simultaneously denoting a certain aspect of it without exhausting it: The ultimate reality (the Dao as a whole) can be rigidly designated (via a certain communication link), its distinct manifestations (distinct aspects of the way the ultimate reality is) can be contextually denoted, and they can be thus referred to simultaneously in a nonmysterious way. Consequently, the Dao can be talked about and captured both in the sense that the Dao as a whole can be designated and that its manifestations can be denoted contextually (in certain specific sentential contexts). In this way, the attempt to mystify the Dao into something “unspeakable” would not be in accordance with the genuine point of Lao Zi’s fundamental insight in the Dao-De-Jing on the relation between the ultimate concern with the eternal Dao and the truth-pursuing language engagement in the text.

Now, even given that the two commentary predicates “speakable” and “unspeakable” narrowly mean “able to give definite description” and “unable to give definite description,” respectively, Lao Zi’s approach suggests the following line of resolving the alleged “ultimate-unspeakable” paradox from his holistic vantage point in the context of the classical Daoist text Dao-De-Jing. In the two statements of the opening message, the same Dao as a whole is rigidly designated but its distinct dimensions are specifically pointed to and denoted, that is, the so-far-descriptively denoted finite dimension as the collection of all so-far-descriptively denoted specific parts, and the eternal or infinite dimension as the collection of all the dynamic never-ending parts; they are predicatively commented respectively as “is speakable” (in the senses both of rigid designation and of definite description) and “is unspeakable” (in the sense of definite description): even in the same sense of definite description, the two seemingly contradictory predicative comments are about different (or even contradictory) “double-reference” identities of the ultimate Dao in this context. It is important to note that the law of identity does not indiscriminately exclude or prohibit talking about a contradiction as referentially identified contradictory aspects of an object, as illustrated in Lao Zi’s treatment of the addressed “ultimate-unspeakable” paradox examined here. In this case, Lao Zi talks about the ultimate reality Dao as a whole while at the same time respectively focusing on two distinct dimensions, its finite dimension and its infinite dimension (as two distinct specific attributes relative to the Dao as a general whole) as essentially addressed in the opening statements of the Dao-De-Jing:

(DS)_[f] The Dao [with its finite dimension in focus] is speakable [relative to […]]

and

(DUS)_[i] The Dao [with its infinite dimension in focus] is unspeakable [relative to […]].

In this context of inference, (DS) and (DUS) jointly illustrate the point of one refined principle concerning the referential sensitivity of the law of identity for logical inferences (“RSLILI” for short) that can be presented and formulated in a more complete and accurate way as follows:

(RSLILI) ↓ α _[f] =_[F] α, …, rather than α _[f] =_[g][G] α (g ≠ f, G ≠ F).

where “↓” means a vertical inference context from its premise statement(s) to its conclusion statement on an object under examination; in this inference context, one needs to maintain the law of identity that is referentially sensitive to which aspect, f, of an object α is specifically in focus besides the object as a whole under examination and remains in the same track from f to F: α _[f] =_[F] α; one cannot, and is not supposed to, change the track half way from f to another distinct specific aspect g and thus to G: α _[f] =_{[g] [G]} α. That is, it is the same (instead of another different) specific aspect, f, of the same object α (with f generalized into F) that is double-referentially talked about in the “premise” and “conclusion” statements in a logical inference. It is noted that, in the logical-inference context, the relative identity between the object talked about in the “premise” statement and the same object talked about in the “conclusion” statement is also nonsymmetric in the following sense: from the logical point of view, the conclusion statement of a (valid) logical inference is to be inferred from the premise statement(s), rather than the reverse; the aspect in focus of the object talked about in the conclusion statement in the valid logical inference needs to be consistently kept the same as that of the same object talked about in the premise statement: the latter is logically prior to the former; in a valid truth-keeping logical inference, the former is to be kept identical to the latter. (In this case, (RSLILI), α _[f] = _[F] α and α _[g] = _[G], rather than α _[f] = _[g][G] α and α _[g] = _[f][F] α, where g and f are referentially identified distinct [or even contradictory] aspects of the Dao, G ≠ F).)

In this way, as illustrated by Lao Zi’s treatment, the double-reference approach explains how it is possible for the same Dao as a whole to be talked about while having its distinct (or even contradictory) aspects/dimensions also talked about in a non-contradictory way.

The Ultimate-Unspeakable Paradox is thus resolved from a holistic vantage point of double-reference approach, which can be highlighted and symbolically formulated as follows:

(US)_DR U_[f] =_[F] S

(UUS)_DR 〜 (U_[i] =_[I] S)

or, given that “α _[γ] ≠ _[Γ] β” is short for “ 〜 (α _[γ] =_[Γ] β),

(UUS)_DR U_[i] ≠_[I] S

(US)*_DR “U_[f] =_[F] S” is true

(UUS)*_DR “U_[i] =_[I] S” is not true

3.2.1 A Double-Reference Perspective in Treating the Simple Version of the Liar Paradox

The simple version of the Liar Paradox via the simple (directly self-referring) Liar sentence is given as follows:

(LS) This sentence is false.

or

(LS) (LS) is false.

The Liar sentence is supposed to say of itself that it is false. One can derive an apparently paradoxical consequence. On the one hand, suppose that it is true; then it is as it says it is (i.e., it is false), that is, if (LS) is true, then it is as it says it is: that is, it is false. On the other hand, suppose that it is false; it being false is just what it says it is; so it is true. At this moment, let me give a bit more explanation of how the former alternative’s and the latter alternative’s inferences are actually based on people’s pre-theoretic “way-things-are-capturing” understanding of truth. According to the following bi-conditional which delivers people’s pre-theoretic “way-things-are-capturing” understanding of truth and on which Tarski relies to develop his account of the material adequacy of the truth definition:^[28]

A sentence is true if and only things are as it says they are.

that is,

if a sentence is true, then things are as it says they are;

if things are as a sentence says they are, then the sentence is true.

In this way,

If (LS) is true, then it is false and true.

If (LS) is false, then it is false and true.

At this point, can we say that the above is paradoxical? There is something tricky here. Unless we have some independent reason to maintain some principle of bivalence to the effect that every sentence, thus including (LS), is either true or false, we cannot automatically exclude the possibility that (LS) lies in a gap between being true and being false (being neither true nor false). Now, quite naturally, or it is arguably correct to recognize, (LS) per se creates a linguistic object as already given above on the paper with its literal senses together with people’s pre-theoretic “way-things-are-capturing” understanding of truth regarding the included truth predicate or its cognates (“is true,” “is false (not true),” etc.); it is also noted that, in such a pre-theoretic understanding, the understanding of identities of things or objects is inclusive and minimally ontologically loaded, which is compatible with and open to being identified as the existence of linguistic objects. In this way, in view of the existence of the linguistic object that is created by the Liar sentence, we do have independent reason to maintain that (LS) is either true or false; that is, (LS) does imply or bring about the alleged contradiction associated with the linguistic object created by (LS) and designated by its name “this sentence” as follows:

(LS-F) (LS) is false.

(LS-T) (LS) is true.

It is assumed that anything that is false is not true, and anything that is true is not false; the aforementioned (LS-F) and (LS-T), respectively, mean, or can be presented as, “(LS) is not true” and “(LS) is not false.” In the following, “is false” and “is not true” are taken to be the equivalent truth/falsity-predicate expressions as predicative comments that point to the same specific attribute (that of failing to capture the way it is) possessed by the linguistic object designated by the referring name “(LS)”; “is true” and “is not false” are taken to be the equivalent truth/falsity-predicate expressions as predicative comments that point to the same specific attribute (that of capturing the way it is) possessed by the linguistic object designated by the referring name “(LS).”

In view of the double-reference phenomenon/feature of people’s basic employment of language as explained before, in (LS-F), the referring name “(LS)” designates the created linguistic object (as a whole) while at the same time denoting its specific attribute of being false or being not true (i.e., that of failing to capture the way it is) in focus. In contrast, in (LS-T), the referring name “(LS)” designates the created linguistic object (as a whole) while at the same time denoting its specific attribute of being true (i.e., that of capturing the way it is) in focus.

Now, through a double-reference characterization of the referentially identified contradictory aspects of the addressed linguistic object designated by its referring name “(LS),” it turns out that the presentation pair characterized in terms of double-reference, i.e., the pair of (LS-F)_DR and (LS-T)_DR:

(LS-F)_DR (LS-F)_[f] =_[F] F,

(LS-T)_DR 〜 [(LS-T)_[t] =_[T] F],

or, given that “α _[γ] ≠_[Γ] β” is short for “ 〜 (α _[γ] = _[Γ] β),”

(LS-T)_DR (LS-T)_[t] ≠_[T] F,^[29]

does not present contradictory or inconsistent statements but talks about distinct referentially identified objects with their distinct “double-reference” identities (i.e., distinct double-referent complexes) in a nonself-contradictory way: one referentially identified object with its “double-reference” identity (as one double-referent complex), designated by the double-reference referring name “(LS-F)_[f],” is the designated linguistic object as a whole (as the semantic-whole-referent simplex) indicated by “(LS-F)” in “(LS-F)_[f]” together with its specific-part attribute “being not true” or “being false” (as the specific-part-referent simplex) indicated by “_[f]” in “(LS-F)_[f],” which is in referential focus in (LS-F)_DR; the other referentially identified object with its “double-reference” identity (as one double-referent complex), designated by the double-reference referring name “(LS-T)_[t],” is the designated linguistic object as a whole (as the semantic-whole-referent simplex) indicated by “(LS-T)” in “(LS-T)_[t]” together with its specific-part attribute “being true” or “being not false” (as the specific-part-referent simplex) indicated by “_[t]” in “(LS-T)_[t],” which is in referential focus in (LS-T)_DR.

3.2.2 A Double-Reference Perspective in Treating the Devious Version of the Liar Paradox

Although the Liar Paradox via the simple (directly self-referring) Liar sentence addressed earlier seems to play a merely linguistic or intellectual game, the addressed “paradoxical” character of the Liar Paradox is neither just a piece of philosophical exotica nor merely occurs in the simple Liar sentence but might be given in its devious version. It is arguably the case that it can arise in reali-life situations involving almost any ascription of truth or falsity, as Kripke correctly points out and illustrates as follows:

The versions of the Liar paradox which use empirical predicates already point up one major aspect of the problem: many, probably most, of our ordinary assertions about truth and falsity are liable, if the empirical facts are extremely unfavorable, to exhibit paradoxical features. Consider the ordinary statement, made by Jones:

(1) Most (i.e., a majority) of Nixon’s assertions about Watergate are false.

Clearly, nothing is intrinsically wrong with (1), nor is it ill-formed. Ordinarily the truth value of (1) will be ascertainable through an enumeration of Nixon’s Watergate-related assertions, and an assessment of each for truth or falsity. Suppose, however, that Nixon’s assertions about Watergate are evenly balanced between the true and the false, except for one problematic case,

(2) Everything Jones says about Watergate is true.

Suppose, in addition, that (1) is Jones’s sole assertion about Watergate, or alternatively, that all his Watergate-related assertions except perhaps (1) are true. Then it requires little expertise to show that (1) and (2) are both paradoxical: they are true if and only if they are false.

The example of (1) points up an important lesson: it would be fruitless to look for an intrinsic criterion that will enable us to sieve out—as meaningless, or ill-formed—those sentences which lead to paradox. (1) is, indeed, the paradigm of an ordinary assertion involving the notion of falsity; just such assertions were characteristic of our recent political debate. Yet no syntactic or semantic feature of (1) guarantees that it is unparadoxical. Under the assumptions of the previous paragraph, (1) leads to paradox. Whether such assumptions hold depends on the empirical facts about Nixon’s (and other) utterances, not on anything intrinsic to the syntax and semantics of (1).^[30]

Kripke gives a right diagnosis here regarding the fact that the addressed (at least seemingly) “paradoxical” character can arise in many real-life situations of what speakers say that involve an ascription of truth or falsity. The sample case given in the aforementioned cited passage from Kripke 1975, though its understanding involves some background knowledge of the Watergate event in the US politics, can be rephrased in a plain way as follows. Suppose that B-Biden and T-Trump are two opposing candidates in an election and that the only thing B-Biden says about T-Trump is this sentence:

〈1〉 A majority of what T-Trump says about me is false.

Now suppose that T-Trump says only the following three things about B-Biden:

〈2〉 B-Biden is soft on illegal immigrants.

〈3〉 B-Biden is a big spender.

〈4〉 Everything B-Biden says about me is true.

If the empirical facts are that 〈2〉 is true and 〈3〉 is false, then, according to the standard analysis that treats the truth predicate (T) and the falsity predicate (F) as sentential predicates about sentences/propositions (having the function of assigning the truth values, truth and falsity [respectively indicated by “1” and “O”] to sentences as specified in the standard propositional logic [sentential logic]), both 〈1〉 and 〈4〉 are paradoxical: each is true if and only if it is false. That is, if 〈1〉 is true, then 〈1〉 is false; if 〈1〉 is false, then 〈1〉 is true; and if 〈4〉 is true, then 〈4〉 is false; if 〈4〉 is false, then 〈4〉 is true. In this way, even ordinary sentences like 〈1〉 that involve indirectly self-referring in real-life situations (via 〈4〉 in the aforementioned case) might become paradoxical, depending on “contingent” empirical facts. Such sentences like 〈1〉 that lead to “paradox” in the aforementioned sense (assigning the truth values, truth and falsity, to sentences as specified in the standard propositional logic) might as well be called “contingent” devious Liar sentences.

However, from the vantage point of the referentially based “way-things-are-capturing” understanding of truth with its vertical “double-reference-based” dimension, the referentially identified “truth” attribute and “falsity” attribute (as what are captured respectively by “is true” and “is false”) are two distinct specific attributes (if any) that are possessed by the truth-bearer object as the collection C of what a speaker S says: some part(s) of the collection C of what S says is(are) to possess the specific “truth” attribute of capturing the way things are, while some other part(s) is(are) to possess the specific “falsity” attribute of failing to capture the way things are. In this way, given the referentially based “way-things-are-capturing” understanding of truth with its vertical “double-reference-based” dimension, adding one more predicative comment “is true” to “a majority of what T-Trump says about me [B-Biden] is false” does not amount to affirming that a majority of what T-Trump says about B-Biden is false but ascribes the referentially identified “truth” attribute to a majority of what T-Trump says about B-Biden that possesses the attribute of capturing the way things are; adding one more predicative comment “is false” to “a majority of what T-Trump says about me [B-Biden] is false” does not amount to saying that “a majority of what T-Trump says about me [B-Biden] is not false” or “a majority of what T-Trump says about me [B-Biden] is true” but ascribes the referentially identified “falsity” attribute to a majority of what T-Trump says about me [B-Biden] that possesses the attribute of failing to capture the way things are.

In this way, the alleged paradox can be dissolved in the following way. From the vantage point of the referentially based “way-things-are-capturing” understanding of truth with its vertical “double-reference-based” dimension), if 〈1〉 is true (that is, the further commentary predicate “is true” is added ascribing the referentially identified “truth” attribute to a majority of what T-Trump says about B-Biden that possesses the attribute of capturing the way things are), then, in view of the given empirical facts (〈2〉 is true and 〈3〉 is false), 〈4〉 is true, and thus 〈1〉 (i.e., “A majority of what T-Trump says about me is false”) is true, which means ascribing the referentially identified “truth” attribute to a majority of what T-Trump says about B-Biden that possesses the attribute of capturing the way things are. If 〈1〉 is false (that is, the commentary predicate “is false” is added ascribing the referentially identified “falsity” attribute to a majority of what T-Trump says about B-Biden that possesses the attribute of failing to capture the way things are), then, in view of the given empirical facts (〈2〉 is true and 〈3〉 is false), 〈4〉 is false, and thus, 〈1〉 (i.e., “A majority of what T-Trump says about me is false”) is false, which means ascribing the referentially identified “falsity” attribute to a majority of what T-Trump says about B-Biden that possesses the attribute of failing to capture the way things are. In sum, if 〈1〉 is true, then 〈1〉 is true; if 〈1〉 is false, then 〈1〉 is false. That is, for those instances of the devious version of the Liar Paradox that are given via the addressed “contingent” devious Liar sentences (as illustrated in Kripke’s sample case cited earlier), it is arguably the case that the alleged “paradoxical” appearance can be dissolved in the suggested way of “double-reference” treatment in view of the referentially based “way-things-are-capturing” understanding of truth.

At this point, I intend to make some engaging comments on Kripke’s judgment to the effect that “no syntactic or semantic feature of (1) [i.e., the addressed devious Liar sentence as illustrated by 〈1〉 in the rephrased Kripke’s sample case] guarantees that it is unparadoxical.”^[31] In my view, this judgment is questionable to the following extent in two connections. First, (directly or indirectly) self-referring feature of a sentence, either directly (as shown in the aforementioned simple version of the Liar sentence) or indirectly (as shown in the aforementioned devious version of the Liar sentence), can be one semantic-syntactic feature of the possible Liar sentence, although in the devious case whether it would be a Liar sentence might depend on relevant empirical facts.^[32] Second, if the addressed “double-reference” feature is one across-the-board semantic-syntactic feature of people’s basic employment of language (saying something about an object) (in the way as explained in Section 2.1), if the deep semantic-syntactic structure of the truth-value-assigning sentences is referentially based (in the way as explained in Section 2.2), the “paradoxical” appearance of the addressed subtypes of the Liar Paradox (type) can be dissolved by addressing the referentially based semantic-syntactic feature of the truth-value-assigning sentences through the suggested way of “double-reference” treatment from the vantage point of the referentially based “way-things-are-capturing” understanding of truth.^[33]

3.2.3 Some Remarks on How the Double-Reference Approach Agrees and Disagrees with Dialethist Approach

In this subsection, I make some explanatory engaging remarks on how the suggested double-reference approach to the Liar Paradox would partially agree and partially disagree with (the Priest-style) dialetheist approach, especially in view of their distinct treatments of the principle of noncontradiction. The explanatory remarks here are relatively brief as this author has given a detailed explanation of how a suggested refined characterization of the principle of noncontradiction can constructively engage with dialetheism on the issue.^[34]

On the one hand, the double-reference approach agrees with the dialetheist approach in this connection: acknowledging the existence of contradictions (either the contradictions in this natural world or the contradictions that are addressed by some manifestation tokens of the Liar paradox type); in this sense and to this extent, the double-reference approach endorses the basic “contradiction-acknowledging” point of the “perspective” dimension of the dialetheist approach while treating its “predicatively-identified” contradictory statements regarding the addressed “predicating-identified” contradictions (which dialetheist treatment typically resorts to and focuses on) as secondary, which is compatible with (and in some case complementary with) the former’s treatment of “referring-identified contradictory aspects of the referred object as primary, although the refined characterized principle of noncontradiction is to “push back” the “predicating-identified” contradiction to where it is primarily located, i.e., the referring-identified contradictory aspects of the object under examination.

On the other hand, the double-reference approach disagrees with the dialetheist approach in the following connection. The former approach does not indiscriminately reject the core point of the principle of noncontradiction as one basic semantic guiding principle in capturing the relation between human language/thought and the world; we can and need to maintain the core point of the principle of noncontradiction as one basic semantic principle whose due characterization is to guide us to talk about the world (including its contradictions) and objects (including linguistic contradictions associated with some linguistic objects created by such linguistic presentations as the simple version of the Liar sentence) in a nonself-contradictory way, as specifically shown by the analysis of the Liar paradox earlier and as more generally explained in a recent cross-tradition engaging examination of the principle of noncontradiction by this author.^[35]