Non-transitive Correspondence Analysis

The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic NC3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{NC}}_{\mathbf{3}}$$\end{document}. As a result, for each binary extension of NC3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{NC}}_{\mathbf{3}}$$\end{document}, we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining n-sequent proof systems for any n-valent logic with deterministic or non-deterministic matrices is not applicable to NC3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{NC}}_{\mathbf{3}}$$\end{document} and its binary extensions.

and outputs definitions of each n-ary (mostly but not limited to, unary or binary ones) extensions of the logic in question. Having been obtained thus, definitions are oneto-one correspondences (hence, correspondence in the name of CA) between each entry of the truth-table for a logical connective (that is, an extension of the logic being analysed) and a valid formula or an entailment. These correspondences (which we dub equivalences sometimes), moreover, allow setting up a sound and complete proof system (mostly but not limited to, a natural deduction system) 7 for each extension of the logic under consideration. Lastly, CA might be equipped with an adequate proof searching procedure: it makes CA a powerful and flexible framework to study (fragments of) tabular logics.
CA is not the only general method for obtaining a proof system for some logic given its tabular semantics. For example, Anshakov-Rychkov's method (Anshakov & Rychkov, 1994) allows one to formalize n-valued logics with Rosser and Turquette's J -operators (Rosser & Turquette, 1952) via Hilbert systems. Another example of such a method is presented by  showing how n-sequent calculi as well as analytic tableaux can be presented for n-valued logics. In , these calculi are transformed into labelled natural deduction systems. Section 6 below is particularly addressed to the method for obtaining proof systems by Avron and his co-authors (Avron et al., 2007(Avron et al., , 2013.

Lemmon-Style Natural Deduction
In choosing Lemmon-style natural deduction (Lemmon, 1998), we follow Weir who explains himself as follows: "Since the tree-form architecture tends to spread too much in all but the shortest proofs I will stick with the linear sequent format, in Lemmon-style" (Weir, 2013a, p. 101). The tree-like format or Gentzen-style natural deduction that Weir mentions above goes hand in hand with a linear format or Fitchstyle natural deduction. However, Weir utilises Lemmon-style natural deduction, i. e. another (arguably, less popular) kind of the linear format. 8 The crucial difference is that, basically, a line in a Lemmon-style derivation visually contains a set of the hypotheses (assumptions, premises, etc.) which a formula on this line follows from. 9 In a Fitch-style derivation, a line contains a formula only and does not contain a set of the hypotheses which a formula on this line follows from. Moreover, it is sometimes very difficult to visually determine this set because some lines above might be discarded. Hence, the visualness in question might be obscure, and one needs much more (in the worst case) efforts to scrutinize the previous nondiscarded lines in a derivation in order to detect this set. Let us refer the reader to the monograph by Indrzejczak (2010) which seems to be the most exhaustive up-to-date study of natural deduction. There she finds historical excursuses with a documented claim that the origin of natural deduction should be ascribed to Jaśkowski rather than 7 In Leszczyńska-Jasion et al. (2019b), sequent calculi with invertible rules, i. e. rules which preserve validity in both directions, are obtained via CA for the negation fragment of classical logic. 8 Note that one of the reasons Lemmon-style being less popular is, apparently, Lemmon's premature death by accident in the mountains (Thomas, 1968). 9 Lemmon-style natural deduction and sequent calculus have this visualness in common.
Gentzen as well as a thorough discussion of the pros and cons of different formats of natural deduction that leads to a clear-cut decision pro hybrid systems.
The structure of this paper is as follows. Section 2 contains an application of CA to the trivalent non-transitive logic NC 3 . Section 3 contains the Lemmon-style natural deduction system for NC 3 . Examples of both NC 3 -derivations and derivable rules are in Sect. 4. Section 5 contains Lemmon-style natural deduction systems for the binary extensions of NC 3 as well as proofs of their soundness and completeness. Section 6 discusses related works. Concluding remarks are in Sect. 7.

Correspondence Analysis for the Non-transitive Logic NC 3
We start with the Weirian semantics for NC 3 and then apply CA to it in order to obtain equivalences for each binary extension of NC 3 .

The Weirian Semantics for NC 3
NC 3 is built over a propositional language L with an alphabet P, ⊥, , ¬, ∧, ∨, →, (,) , where P is a denumerable set { p, q, r , s, p 1 , . . .} of propositional variables. The set F of all L -formulae is defined in a standard way. The letters φ, ψ, χ , ρ, ϕ with subscripts run over L -formulae. The letters Γ , Δ, X , Y , Z , Σ, Ξ with subscripts run over finite sets of L -formulae.
Let be a truth-functional operator (or connective) and f be its truth table. Let us remind the reader that an interpretation v of all the connectives of NC 3 , except ⊥, and →, is as in the Kleenian strong logic K 3 : , where 1, 1 /2, and 0 stand, respectively, for 'true', 'indeterminate', and 'false'. 10 Implication of NC 3 , , and ⊥ are the Łukasiewiczian ones: Definition 1 Γ | NC 3 ψ iff, for each valuation v and for every φ such that φ ∈ Γ , it holds that if, for every χ such that The first effect of Definition 1, according to Weir, is that both conjunction and disjunction do not standardly interact with the entailment relation. For example, φ, ¬φ | NC 3 ⊥, but φ ∧ ¬φ | NC 3 ⊥. Note that such behaviour of conjunction is not specific to NC 3 . There might be conjunctions which behave alike in logics, where an entailment relation is standardly definable via one designated value, i. e. as in K 3 . It is easy to check that it holds in logics with one designated value, where conjunction takes a value "1" if both conjuncts take a value " 1 /2". 11 Examples of such conjunctions are conjunction of Sette's logic P 1 (Sette, 1973) and conjunction of 10 Unlike in K 3 , 1 is not really a designated value in NC 3 ; because the conception of entailment combines truth-preservation 'downwards' and falsity-preservation 'upwards' the notion of a designated value drops out (or perhaps one can say that 1 is designated, 0 is 'anti-designated'). 11 We suppose conjunction behaves classically on the classical inputs, as well.
Another effect of Definition 1 is that some classical entailments are neo-classically invalid (Weir, 2013a, p. 101): 12 "My proposal for a restriction on classical logic focuses on 'minimax unsound' rules, where the rule is nonetheless sound (as a one-step natural deduction inference) with respect to | 3 . Examples of such rules are ¬E, → E, ∨E and disjunctive syllogism, e.g. in the latter case with P false and Q gappy, the minimum value of {(P ∨ Q), ¬Q} is 1 /2 whilst the conclusion P takes the false value 0. The (sole) proposed restriction on classical propositional logic is that in such minimax unsound rules any assumption on which both major and minor premiss depends must be 'determinate"'.
The rule of disjunctive syllogism is neo-classically problematic in case both its premisses depend on the same formula(e). Classically, φ | χ follows from both φ | ψ ∨ χ and φ | ¬ψ whereas that is not the case neo-classically. 13 In order to get neo-classically valid entailments from such entailments, Weir proposes so called Determinacy of a sentence φ and Indeterminacy of a sentence φ (abbreviated as D(φ) and I (φ), respectively).
We follow him despite ↔ / ∈ L . ¬D(φ) and I (φ) are used synonymously in this paper. 14 Getting back to the entailment in the previous passage, This trick illustrates the Weirian idea. To be sure, it is not the case that this trick makes neo-classically invalid entailments valid. The trick is that whenever one detects a weak problematic indeterminate point in reasoning one is able to effectively determine it in a special way. Hence, in checking out a classical proof one has an option to improve it. In the simplest cases, to be sure, the premise D(φ) is derivable from initial premisses. So, the proposed determination does not affect the entailment in question. However, in general, one has to pay the price of making a classically valid entailment neo-classically valid by adding the needed determinacy formulae.
Let us conclude with the important features of determinacy highlighted by Weir.
Hence, in adding determinacy formulae to a problematic entailment one could skip negation(s), confine oneself to D(φ), and decrease the complexity of added formulae. 16 Third, v(D . . . (D(φ)) . . .) = 1. Hence, higher-order determinacy is ruled out (this fact will be discussed on page 14 below).

Correspondence Analysis for NC 3
Let L (•) k be an extension of L by binary truth-functional operators • 1 , . . . , • k . The set F (•) k of all L (•) k -formulae is defined in a standard way. The common name of the corresponding logics is NC (•) k 3 .
Definition 2 (Single entry correspondence (Kooi & Tamminga, 2012;Tamminga, 2014) Definition 2 captures the key essence of CA: one-to-one correspondence between entries of the truth table for a binary connective and natural deduction rules for this connective. Below, in Sect. 5 there is a thorough explanation how this one-to-one correspondence works. In the further exposition, we use "| " as an abbreviation for "| NC (•) k 3 ". 16 However, the situation with the other connectives is not so irenic. For instance, A study of this topic is outside the scope the present paper. 17 It is improper to read, say, the first equivalence as follows: for any φ, ψ, χ, the entailment holds iff f • (0, 0) = 0. One could easily find instances of φ, ψ, χ such that this entailment holds regardless of the values of φ, ψ, χ, φ • ψ. The proper reading is natural and easily extractable from the proof of this theorem below. 18 Note 'D(φ • ψ)∧' right to the double corkscrew in each equivalence f • (x, y) = 0 and f • (x, y) = 1. This clause plays an important role in proving the respective cases of Lemma 11.
Then there is a valuation v such that downwards truth-preservation fails or there is a valuation w such that upwards falsity-preservation fails. Let us start with the failure of downwards truth-preservation which implies With regard to upwards falsity-preservation, one must consider four cases depending on which formulae left of the double corkscrew are assumed to take a value 19 Let us clarify the strategy on this equivalence, following the first referee's suggestion, and denote f • (0, 0) = 1/2 by X and ¬φ, ¬ψ, From left to right' is conventionally shown both by contraposition and by reductio, viz., when we want to show "If X , then Y ", we show its contraposition, "If it is not the case that Y , then it is not the case that X ". In order to show the latter, we reason on contrary, viz, we assume "It is not the case that Y and it is the case that X ", arrive at contradiction, and derive "It is not the case that X ". (The specifics of NC 3 lie in the fact that we need to consider two subcases because of its bidirectional entailment relation.) Note that in this paper, 'from right to left' is shown unconventionally via contraposition, i. e. by showing "If it is not the case that X , then it is not the case that Y ". The standard way is to show "If Y , then X " directly. 20 Note that, for any w and φ, w(D(φ)) = 1/2. Hence, In the case (1), The case (2) is considered by analogy with the case (1). The cases (3) and (4) are also considered by analogy with the case (1). However, arriving at contradiction is shorter due to Then there is a valuation v such that downwards truth-preservation fails or there is a valuation w such that upwards falsity-preservation fails. Let us start with the failure of downwards truth-preservation which implies there is a valuation v such With regard to upwards falsity-preservation, one must consider four cases depending on which formulae left of the double corkscrew are assumed to take a value distinct from 0: The case (2) is considered by analogy with the case (1). The cases (3-4) boil down to the subcase that Then there is a valuation v such that downwards truth-preservation fails or there is a valuation w such that upwards falsity-preservation fails. Let us start with the failure of downwards truth- With regard to upwards falsity-preservation, one must consider two cases depending on which formulae left of the double corkscrew are assumed to take a value distinct from 0: (1) w(I (φ)) = 0, w(I (ψ)) = 1 and w(I (φ • ψ)) = 0 and (2) w(I (ψ)) = 0, w(I (φ)) = 1 and w(I (φ • ψ)) = 0. Note that the assumption is that f The case (2) is considered by analogy with the case (1).
We would like to stress that the equivalences above are, certainly, not unique. This fact clearly indicates that obtaining CA, still, remains a creative and nonalgorithmic task. Therefore, it is a future roadmap to make CA a routine and algorithmic task à la (Avron et al., 2007;Avron & Konikowska, 2005).

The Rules of Natural Deduction System for NC 3
The rules of Weir's natural deduction system for NC 3 denoted here by ND NC 3 operate on a line which is a quadruple hs, nl, fr, an , where hs stands for a (possibly, empty) set of hypotheses fr depends on, nl stands for a number of a line which fr occurs on, fr stands for a f ormula which is derivable on a line, an stands for an analysis of fr and specifies whether fr is a hypothesis, or an axiom, or is derived from the previous lines via some ND NC 3 -rule. For a line which has a standard structure of a sequent, then hs, fr, and nl are referred to as the antecedent of, consequent of, and unary corkscrew of a sequent, respectively. For example, the line ψ (1) χ is to mean the sequent ψ χ . The letters α and β with subscripts run over sequents. The capital I or E stands for an 'introduction' and 'elimination', respectively.
Weir notes that the rule of H ypothesis includes thinning. 22 The Weirian notation '(1)' is not to mean that H is applicable to the first line only. 23 Let us kindly thank Alan Weir for the following explication of the notation (personal communication): "Here I indexes all the sentences in common to assumption sets X and Ξ ∪ Z , and so (4.i) can be a large (finite, in the finitary systems) set of sub-proofs, collapsed to one line in this non tree-form format. I use square brackets as in [4.i] to indicate determinacy assumptions used to implement the restrictions which can block transitivity". In derivations themselves, there are no i and the enumeration of lines is traditional. 24 (2) ¬φ 1, ¬I I N T Though Weir utilises the same name ¬I for the two previous rules, we would like to dub them differently: C L and I N T stand for 'classical' and 'intuitionistic' versions of the Weirian ¬I , respectively.
Note that Weir dubs it the 'Mingle' rule in Weir (2013b, p. 3) in order to stress the fact that it is derivable in Anderson and Belnap (1975) via the mingle axiom φ → (φ → φ). For the reason that such a name is not in line with the traditional usage of the term mingle rule (Kamide, 2002), we dub it M rather than Mingle.
Being a Lemmon-style (Suppes-style or a system with dependencies) natural deduction system, 25 ND NC 3 has the distinctive feature of such systems, viz., a device of listing all the hypotheses which a formula follows from (depends on or rests on) in a special leftmost column. It allows for easily checking out the hypotheses on which premises of a rule are dependent on. Hence, an advantage of ND NC 3 is the simplicity and clearness of the notion of a derivation for it (Lemmon, 1998;Suppes, 1957;Weir, 2013a, p. 31, pp. 8-9, p. 5).
Our definition is a slight modification of the one by Pavlović (2015, p. 79).
Definition 3 A derivation in ND NC 3 (ND NC 3 -derivation) is a non-empty finite sequence of lines such that each line results from an application of some ND NC 3rule to lines occurring earlier in the sequence. A derivation in ND NC 3 of a formula φ from a (possibly, empty) set of formulae Γ is a derivation in ND NC 3 such that the last line k of it is Γ , k, φ, an .
If Γ = ∅, then φ is said to be a theorem of ND NC 3 .

Examples of ND NC 3 -derivations and the ND NC 3 -derivability of some new ND NC 3 -rules
This section is devoted to examples of ND NC 3 -derivations. Some of them illustrate that the ND NC 3 -derivability might be classical or neo-classical. The choice of examples is governed by metatheorems in Sect. 5. Note that in the examples below, sets of hypotheses might contain a number of a formula instead of the formula itself or a mix of numbers and formulae. This feature, adopted for space-saving reasons, is characteristic of a Lemmon-style derivation (in contradistinction to, say, a sequent-style derivation) and is employed from the first pages of the textbooks (Lemmon, 1998;Suppes, 1957) and by Weir (2013b). Of course, any number is replaceable with the corresponding hypothesis because every hypothesis has its unique number in a derivation.
Examples 1-2 are used in showing the ND NC 3 -derivability of the rule ¬¬E: Note that there are two ways to show its derivability. If ¬φ / ∈ X , then the way is classical as in Example 3.1 below. Otherwise, it is neo-classical as in Example 3.2 below, where X = {X , ¬φ}.

Example 3.1 ¬¬E is ND
In the same fashion, the ND NC 3 -derivability of the rules ¬¬I , ¬∨E, and ¬∨I is shown.
X (1) φ Given X (2) ¬¬φ ¬¬I : 1 Example 4 shows the ND NC 3 -derivability of the rule ¬I D which is to be read as "introduction of negation inside determinacy". 27 In the same fashion, both ND NC 3theoremhood of D(¬ p) → D( p) and the ND NC 3 -derivability of the rule ¬E D ("elimination of negation inside determinacy") are shown.
Its justification readily boils down to the one for → E. Let D 0 (φ) be φ and D n+1 be D(D n (φ)). The case with D m I (φ) is similar. Then higher-order determinacy, i. e. determinacy of determinacy or determinacy of indeterminacy, for instance, is ruled out by 28 Lemma 1 [Proposition 1 (Weir, 2013b)]. D n (φ), for all n ≥ 2, and D m+1 I (φ), for all m ≥ 0.
Hence, two rules which we dub D(D) and D(I ), respectively, are justified. 29

-(1) D n (φ) D(D), n ≥ 2 -( 1 ) D m I (φ) D(I ), m ≥ 1
Let us remind the reader that in applying → E, ∨E, and ¬E she has to check for an overlap between the premises of these rules and, if found, to determinate each formula from the overlap. However, derived classical, as Weir dubs it, variants of these rules are justified by 30 Lemma 4 [Proposition 2 (Weir, 2013b)]. Where we have proofs of the determinacy of the succedents of a premiss of ¬E and → E rules, and both disjuncts of the major premiss succedent of the ∨E rule, then the full classical rule is neo-classically derivable, that is we do not need to establish the determinacy of the overlapping assumptions.
We denote them → E C L , ∨E C L , and ¬E C L .
Cut, also, has a derived classical version Cut C L whose justification easily follows from the justification of → E C L . Another version of Cut C L is obtainable via replacing The rule of disjunctive syllogism is justified in Weir (2013a, p. 101): It is not difficult to justify rules of thinning 31 and I E : 29 Weir employs the following Lemma 2 [Lemma 1 (Weir, 2013b)]. If ND NC 3 φ ∨ ¬φ, then ND NC 3 D(φ). 30 Weir employs the following Lemma 3 [Lemma 2 (Weir, 2013b)]. If X ND NC 3 D(φ), then X ND NC 3 φ ∨ ¬φ. 31 Note that it is not the thinning that the rule of hypothesis on page 10 above includes.

X
(1) φ Given X , Ξ (2) φ thinning: 1 At last, let us remind the reader of the entailments from footnote 16 above which proofs we do not present here due to space reasons only.

NC 3 for the binary extensions of NC 3
Here we demonstrate the way CA allows one to obtain the rules of a natural deduction system for each binary extension of NC 3 .
In the former entry, we need the equivalence . So, ND NC 3 contains the rule: In the latter entry, we need the equivalence f 1, 1, 0) Thus, we obtain exactly nine rules for (or any other connective) because its truthtable has nine entries and because each rule obtainable via CA is independent. 32 We hope the reader easily manages to obtain the remaining rules for herself. Sure, the rules look clumsy and dissimilar to the conventional ones. Arguably, this phenomenon is effected by the generality of CA. In practice, the number of rules for a connective is (considerably) fewer than the number of entries in its truth-table. 33 At present, the problem of finding the general (and algorithmically implemented) way to combine both conventional rules and the ones obtainable via CA, is still open.
We direct the reader's attention to the difference between the Weirian rules and the rules obtained via CA with respect to introducing determinacy formulae into a derivation. In the former rules, one determines every hypothesis occurring in an overlap of sets in antecedents of premises of a rule. In the latter rules, one needs not check up the sets in question for overlapping. The needed determinacies have already occurred in the antecedents of the rules. To put it another way, the rules obtainable via CA are local rather than global in the Weirian sense (Weir, 2013a, p. 101). We believe this is another effect caused by the generality of CA, where • might be any connective and, hence, has any truth-table. In a situation like this, there seems to be no alternative way to preserve soundness than guaranteeing it locally, at every line of a derivation. 34 32 See Petrukhin (2018) for a standard proof. 33 The Gentzenian harmony is a well-known argument pro. 34 Classical predicate logic got such an alternative in Aguilera and Baaz (2019), where a correct proof is allowed to make unsound inferences in order to construe shorter proofs. Despite the fact that these results depend on rules for quantifiers only, we believe in a perspective of studying unsound inferences that make proofs shorter in propositional logic, too.

Soundness and Completeness of ND
Let us remind the reader that L (•) k is an extension of L by binary truth-functional operators • 1 , . . . , • k , that F (•) k is the set of all L (•) k -formulae defined in a standard way, and NC (•) k 3 is the common name of the corresponding logics.
Proof is by induction on the depth of a derivation defined in the usual way. Note that each ND NC 3 -rule is shown to be valid in Weir (2013a, p. 102) and Weir (2013b, pp. 5-7). The validity of ND (•) k NC 3 -rules is shown in Theorem 1.
We would like to precede the completeness theorem with addressing the first referee's point of view that "the method of prime theories simply cannot work for a logic without generalized transitivity, since this idea of 'containing its own consequences' seems strongly tied up with transitivity" and his (her) kind recommendation to look "at the work of Frankowski, particularly (Frankowski, 2004a), for ideas about related methods that might be more suitable".
Let us start with acknowledging that the first referee's point of view is fair, and the Henkinian method in the form suggested in the embryonic draft of this paper is not applicable to NC 3 . Roughly, the standard condition of closure -Γ φ iff φ ∈ Γthat holds of the Henkin-type sets used in Kooi and Tamminga's completeness proofs (Kooi & Tamminga, 2012;Tamminga, 2014), fails for the equivalent sets in NC 3 , as Weir notes (Weir, 2013b, pp. 8-9). As a result, below we employ the Henkinian method not in the standard form, but in the form that Weir tailors this method to the needs of NC 3 in Weir (2013b, pp. 9-18) following both Weir's and the second referee's amendments to this exposition.
Second, the need for modification of the Henkinian method to non-transitive approaches is not confined to the Weirian one only. For example, Wansing and Skurt wonder "if identity is deprived of transitivity, what does that mean for the Henkin-style completeness proof for classical first-order logic with the impoverished notion" and show how "the Henkin-style completeness proof for classical first-order logic with identity can thus be modified" (Wansing & Skurt, 2019, p. 536, 542).
Lastly, let us agree with the first referee's kind recommendation that it is instructive to apply (see (Barrio et al., 2019) for one of applications) the method of Frankowski (2004a, b) who underlines that his approach is closely related to the one of Malinowski (1990), to the logics discussed in this paper. With all due respect, both the first and the second points above together with the non-embryonic completeness proof below make applying the Frankowskian method a topic of a future paper.
The completeness theorem below is proved via the Henkinian paradigmatic method with ND (•) k NC 3 -specific modifications and discriminations. We base our proof on the Weirian proof of the completeness theorem for NC 3 (Weir, 2013b, pp. 8-18). We start with auxiliary definitions and lemmata adapting the Weirian original terminology for . In the remaining proofs and definitions from this Section, therefore, stands for ND Definition 4 Weir (2013b, p. 9) A set Γ is 1. Weakly consistent iff Γ ⊥; 2. Strongly consistent iff it is not the case that Γ φ and Γ ¬φ; 3. Negation-complete iff φ ∈ Γ or ¬φ ∈ Γ ; 4. Negation-complete 1 iff Γ φ or Γ ¬φ.
Note that Weir employs special 1 and 4 rather than standard 2 and 3, respectively, employed in the Tammingian (in fact, standard) proof. The need for these discriminations is argued with the following Lemma 5 (Weir 2013b, pp. 8-9). It is not the case that Γ φ iff φ ∈ Γ .
Let us show how it works on the example of a weakly consistent Γ = {φ ∧ ¬φ} 35 suggested by the first referee. So, we start with the hypothesis φ ∧ ¬φ φ ∧ ¬φ. By ∧E 1 , we derive Γ φ. (The case with ∧E 2 is the same.) However, due to the failure of closure, this does not mean we should expand Γ to a weakly inconsistent {φ ∧ ¬φ, φ}. 36 On the other hand, we could continue and by ∧E 2 , derive Γ ¬φ. Again, this does not mean we should expand Γ to a weakly inconsistent {φ ∧¬φ, ¬φ}. Moreover, φ ∧ ¬φ φ and φ ∧ ¬φ ¬φ do not allow us to derive φ ∧ ¬φ ⊥, by ¬E because we lack determinacy of φ ∧¬φ that both φ and ¬φ depend on. The failure of closure, nevertheless, does not prevent us from employing the Henkinian method. Instead of assigning truth-values of formulae in a maximally (weakly) consistent set standardly in terms of set-theoretic membership, we will do it in terms of the ND (•) k NC 3derivability following Weir (2013b) (see the beginning of Lemma 11). Now let us get back to the completeness proof and list all the preliminary definitions and lemmata that Weir employs (Weir, 2013b, pp. 8-18).
In what follows, both Γ and Δ will be employed in the sense of Lemma 6. Let us remind the reader that Weir's Lemma 1, Proposition 1, Lemma 2, and Proposition 2 are Lemma 2 on page 14, Lemma 1 on page 14, Lemma 3 on page 14, and Lemma 4 on page 14, respectively.
Lemmata 4, 7, 8, 9, and 10 justify the following Lemma 11 [Proposition 5 (Weir, 2013b)] Let the canonical valuation v for the negation-complete 1 weakly consistent Henkin set Δ be the function v generated from the semantic clauses for the connectives by extending the following assignment of truth values to propositional variables: For the canonical model valuation v for Henkin set Δ the following holds: Now let us remind the reader footnote 18 caused by the second referee's remark that both non-triviality of Δ and the proof above that Δ φ • ψ do not imply that Δ ¬(φ • ψ), i. e. the needed half of the condition for v(φ • ψ) = 1.
By Theorems 2 and 3, we obtain the following Corollary 1.

Related Work
This section is an update to the comparative study of the CFOSC-method 39 and CA in Petrukhin and Shangin (2019), where footnote 4 contains the following hypothesis: "We don't mean here we can provide an example of correspondence analysis being applied to a non-transitive logic. What we mean here is that transitivity property of the logical consequence relation (i. e. validity of cut-rule) is not a necessary part of correspondence analysis". So, the present paper is a confirmation of this hypothesis, and CA is shown to be applicable to the logics which the CFOSC-method is not applicable to. We, again, underline that the CFOSC-method is more general than CA primarily due to the fact, that the former deals with both deterministic and nondeterministic matrices. 40 For non-deterministic matrices, we direct the reader to the lists of references in Avron and Zamansky (2011), Avron and Lev (2001), Avron and Konikowska (2005), Kearns (1981) and, especially, to the emeritus Lomonosov MSU professor Yuri Vasilyevich Ivlev's contribution, who has been proposing nondeterministic matrices (he dubs them quasi-matrices) for propositional and modal logics since the year of 1972 (Ivlev, 1987(Ivlev, , 1988(Ivlev, , 1998(Ivlev, , 2000(Ivlev, , 2005(Ivlev, , 2020. Getting back to the CFOSC-method, we highlight that it sets up an n-sequent calculus for each n-valued logic that is characterizable by any finite-valued semantics based on the above-mentioned types of matrices. As a result, such a n-sequent calculus can be transformed into an ordinary sequent calculus and a natural deduction calculus. Proof systems of the latter type are usually (but dissimilar to the ones by the CFOSCmethod) obtainable via CA, too. There are two crucial differences between the ways both methods set up proof systems. First, the ones by the CFOSC-method contain superficial, dependent rules, and special pruning procedures run in order to get rid of them. The ones by CA have independent rules only. Second, the CFOSC-method is designed to set up cut-free sequent calculi, and this type of proof system is wellknown to be inefficient from the view of computational complexity theory (D'Agostino & Mondadori, 1994). On the other hand, natural deduction calculi are among the 38 Roughly, Weir proves there that if there are at least two formulae ψ 1 and ψ 2 such that Δ ψ 1 and Δ ψ 2 , then Δ ⊥, by the ⊥ rule. 39 The acronym CFOSC, "Cut-Free Ordinary Sequent Calculi", is exclusively our idea. We call it this simply because Avron and his co-authors never call it that themselves. 40 Applying CA to non-deterministic matrices is an open question. However, note that there is nothing in the nature of CA to prevent it. proof systems (in)efficiency of which is, still, unprovable (Cook & Reckhow, 1979;Urquhart, 1995).
The motivational reasons of the two methods are different, too. In the case of CFOSC, one of the reasons is to provide a general proof of the cut-elimination theorem, a corner stone of the modern automated proof theory. Such a general proof allows one to obtain cut-free sequent calculi, where a proof searching procedure becomes a routine and brute-force task [this procedure is highly inefficient, nevertheless: the complexity of a cut-free sequent calculus is shown to be factorial, i. e. it is worse than the exponential one of the truth-table method (Urquhart, 1995)]. 41 Hence, from the beginning CFOSC rests on the presupposition that the entailment relation of a treated logic is the Tarski-Scott one (Scott, 1974), that is, it enjoys the generalized transitivity (see, for instance, Avron and Lev (2001, p. 2) and Avron et al., (2007, 44). To be sure, we are claiming neither that it is a drawback of CFOSC (the general cut-elimination theorem is not expected to hold when the general transitivity fails), nor that CFOSC cannot be adapted to (the above-treated) non-transitive logics. On the other hand, Petrukhin (2018) shows flexibility of CA by generalizing different CAs.

Conclusion
We present CA for the Weirian non-transitive trivalent neo-classical logic NC 3 in order to obtain sound and complete Lemmon-style natural deduction systems for all the binary extensions of it. The future research will be about proof-searching for (the obtained extensions of) NC 3 . An application of Frankowski's method (Frankowski, 2004a, b) to the target logics is another future research.