The Quantified Argument Calculus with Two- and Three-valued Truth-valuational Semantics

We introduce a two-valued and a three-valued truth-valuational substitutional semantics for the Quantified Argument Calculus (Quarc). We then prove that the 2-valid arguments are identical to the 3-valid ones with strict-to-tolerant validity. Next, we introduce a Lemmon-style Natural Deduction system and prove the completeness of Quarc on both two- and three-valued versions, adapting Lindenbaum’s Lemma to truth-valuational semantics. We proceed to investigate the relations of three-valued Quarc and the Predicate Calculus (PC). Adding a logical predicate T to Quarc, true of all singular arguments, allows us to represent PC quantification in Quarc and translate PC into Quarc, preserving validity. Introducing a weak existential quantifier into PC allows us to translate Quarc into PC, also preserving validity. However, unlike the translated systems, neither extended system can have a sound and complete proof system with Cut, supporting the claim that these are basically different calculi.


Introduction
The Quantified Argument Calculus (Quarc), introduced in [2], has since been the subject of several publications, extending and applying it in a variety of ways [3,4,15,21,[23][24][25]30], and a number of researchers are currently exploring it in additional directions. Still, a direct completeness proof for the Lemmon-style Natural Deduction system used in the original paper hasn't been published in any journal. This is one aim of the present work. Our proof, adapting Lindenbaum's construction to a new formal system with truth-valuational semantics, is of some interest in its own right. In addition, as has been discussed elsewhere [4,15], it is natural to have a three-valued version of Quarc. We shall develop such a three-valued system, proving that Presented by Heinrich Wansing; Received May 25, 2021 its consequence relation coincides with that of the two-valued system, but unlike [15], we shall do it with a truth-valuational semantics and not a modeltheoretic one, and with a different conception of validity than the one used in that paper. We shall then explore the relation of that three-valued Quarc to the Predicate Calculus (PC).
Quarc was developed with the aim of being closer than PC to natural language, primarily in the way it incorporates quantification but also in other, related ways: inclusion of modes of predication, of reordered relation terms, anaphora, and more. As the calculus has by now been motivated and informally introduced in several publications, we shall do neither here, apart from concisely presenting its approach to quantification.
Consider the sentences, 1. Alice is prudent 2. Every student is prudent.
While 'Alice' occupies the subject or argument position in (1), this position is occupied by 'every student' in (2). Namely, the quantifier 'every' followed by the unary predicate 'student' form the quantified argument of that sentence. Quarc follows this analysis of quantification. With the argument written to the left of the predicate, it formalises these two sentences as: 3. (a)P 4. (∀S)P .
Let us next consider particular or specific quantification. The sentence, 5. Some students are prudent will be formalised,

(∃S)P
where ∃ is read 'some', not 'there is' or 'there exists'. It has been argued in several works [1,2,4] that particular quantification in natural language is not related to existence (see also [27]chap. 18), and this claim has been used to address several philosophical puzzles [4]. Although the use of an existential quantifier is legitimate if we are not interested in being faithful to the logic and concepts of natural language, Quarc is interested in the latter. We accordingly read particular quantification as having no ontological commitment and as unrelated to existence or to a formal existential quantifier. To emphasise this, we could follow a suggestion found in some works mentioned above and formalise it by an inverted P, from Particular or the Latin Particularis, and not an inverted E, from Existence [26,47]. But as this distinction is not essential for formal work in this paper, and as we investigate below the formal relations between Quarc and PC, we decided not to introduce it here.
We turn to the language of Quarc. 1

Quarc: Syntax
The syntax of Quarc is a little simplified compared to that of [2], making less and somewhat different use of parentheses and commas. The formula rules are also somewhat different, in order to achieve unique parsing (as will be proved below), absent from the 2014 version. However, there is a straightforward bidirectional translation between the formulas of both versions (a fact we shall not prove here).

Language
Definition 2.1.1. (Language) A language of Quarc consists of: • singular arguments: a non-empty, countable set of symbols, disjoint from the set of all other symbols listed below and strings thereof.

Formulas
Definition 2.2.1. (Quantified Arguments) Let P be a unary predicate. Then ∀P and ∃P will be called (universally and particularly) quantified arguments.
For example, if R is a binary predicate, then R 2,1 is a reordered (binary) predicate, and is the reordered form of R (as 2, 1 is the only non-identity permutation of 1, 2); if S is a ternary predicate, then S 2,3,1 is one of the five reordered forms of S. Definition 2.2.3. (Labels and Sources) Let a be a singular argument, QP a quantified argument and x an anaphor. Then a x and QP x are called xlabelled (singular and quantified) arguments, where the x (written as a subscript) is considered not an anaphor but a label. In a string of symbols, the source of an occurrence of an anaphor x is the closest occurrence of an xlabelled argument to its left. If, in a string, α is the source of (an occurrence of) x, we say that (the occurrence of) x is an anaphor of α. A label of a singular or quantified argument is not part of the argument it is attached to. (a) Let a 1 , . . . , a n be unlabelled singular arguments and P a non-reordered n-ary predicate. Then a 1 . . . a n P is a formula, which is also called a basic formula.
(c) Let a 1 . . . a n P be a formula where none of a 1 , . . . , a n is labelled (but P may be reordered). Then a 1 . . . a n ¬P is also a formula.
(d) Let φ be a formula. Then so is ¬φ.
For example, (aaR ∧ abR) is a Quarc formula where no anaphor or quantifier occurs, from which we can construct (aa x R ∧ xbR), which is a formula led by a x . From the latter we can in turn construct (a∀P x R ∧ xbR), which is a formula governed by (that occurrence of) ∀P . We may also write this formula without parentheses, a∀P x R ∧ xbR. (b) χ is a 1 . . . a n P τ , where a 1 , . . . , a n are unlabelled singular arguments and P τ is a reordered predicate.
(c) χ is a 1 . . . a n ¬P , where a 1 , . . . , a n are unlabelled and P may be reordered.
(e) χ has exactly one of the forms (φ∧ψ), (φ∨ψ), (φ → ψ), and it is neither led nor governed. Proof. It follows from 2.2.4 that at least one of (a) -(g) is the case. So what remains is the uniqueness claims. There are two sorts of uniqueness claim here. One is 'exactly one of (a) -(g) is the case' for χ, the other is about the uniqueness of the 'predecessor(s)' of χ in each of (a) -(g). We omit the proofs of such claims as they are straightforward.
With parsing being unique, we can proceed to define the complexity of formulas, which will later facilitate inductive proofs. (b) If χ is a 1 . . . a n P τ , then comp(χ) = comp(a 1 . . . a n P ) + 1(= 1).

Quarc: Truth-valuational Semantics
In [2], a truth-valuational semantics (TVS) was used, as was done in [3]. We shall use such semantics here too, and not the more familiar model-theoretic one. Truth-valuational semantics is intuitive, powerful and elegant, and of much philosophical interest. Significant formal work has been done on it, especially around the seventies [11,14,[17][18][19], and it deserves more attention than it currently receives. It has met some philosophical criticisms, which we hope to address elsewhere (but see (Ben-Yami manuscript)), and a formal one [20], which was addressed in [3]. Although our purpose here is not to defend this approach, its successful application below may serve this aim as well. -Our main results, however, can also be translated into modeltheoretic semantics.
As was mentioned above, we shall explore both a two-valued version of Quarc and a three-valued one. The three-valued version follows Strawson's claim that, when the predicate in the quantified argument of a subjectpredicate statement has no instances, it is natural to take that statement as lacking a truth value ( [31] sec. V.c; [32], sec. 6.III.7). One who asserts, 'Some/all/most/seven students are present', or even 'No student is present', presupposes, on this analysis, that there are students. This presupposition supports the classification of statements of the form, All/some S are P , as lacking a truth value when S has no instances. And although the common approach in contemporary logic is to make the formal version of the universal 'Every S is P ' vacuously true when S is as above, it is doubtful that this reflects truth in natural language: few would take the sentence, 7. All my children love their mother to be true, when uttered by a childless person [15,553]. Attempting to remain closer to this analysis of the semantics of natural language, both ∀SP and ∃SP will be defined below as neither true nor false in case S has no instances.
We shall generalise this gap approach to any formula of the form, α 1 . . . α n P , in which any of the α i is a quantified argument. This, we emphasise, is a regimentation of what is found in natural language (as Strawson himself later claimed [33]). Some sentences of this form may ordinarily be taken as false, while others as neither true nor false. However, some such regimentation seems unavoidable when applying exact formal tools to ordinary language; even if one doubts Strawson's remark, that 'ordinary language has no exact logic' [31,344], it is enough that ordinary language has no logic capturable by an extensional system comparable in its simplicity to Quarc. All the same, the unavoidability of regimentation makes other alternatives also worth exploring, as was done for example in [24].
We shall make Quarc two-valued by imposing an instantiation rule, forcing each unary predicate to have instances, as was done in [2,130]. The three-valued system will be obtained by eliminating this rule. We shall introduce a third value, 'undefined' or u, to capture the cases discussed above and others dependent on them.
We proceed to the definitions.
(d) Connectives. Let φ and ψ be formulas. Then: where φ(a) is the formula from which φ(a x ) is immediately generated. (h) Instantiation. For every unary predicate P in L, there is a singular argument a in L for which v(aP ) = 1.
We sometimes write 'true' and 'false' instead of 1 and 0.
Remark. It follows from this definition and unique parsing (Proposition 2.2.5) that a valuation for L is uniquely determined by its assignment to basic formulas of L. Similarly for the three-valued semantics introduced later.
Validity on the truth-valuational approach is defined while allowing the addition and elimination of individual constants to and from a language; namely, validity is independent of a specific individual constant list [2,131], [11,183], [14,19]. Such a definition will be used below.
Definition 3.1.2. (2-validity) Let L be a Quarc language. An argument whose premises constitute the set Γ of L-formulas and whose conclusion is the L-formula φ is 2-valid, written Γ 2 φ, iff for any language which contains all the singular arguments occurring in either Γ or φ, every 2-valuation that assigns all formulas in Γ 'true' assigns φ 'true' as well.
Remark. If an argument in a Quarc language is valid, it is valid as an argument in every Quarc language which contains all the singular arguments involved. For this reason, we do not define validity as relative to specific languages.

Three-valued TVS
As mentioned above, the three-valued version is obtained by eliminating the Instantiation rule. As basic formulas remain either true or false and as the truth conditions specified in the valuation rules Reorder, Negative predication, and Anaphora are also not affected, we have to modify only the rules Connectives, where we shall use Kleene's strong tables, and Particular and Universal quantification: For a Quarc language L, a 3-valuation is a function, v, from the set of L-formulas to {0, 1, u} that satisfies the following rules (in addition to those same as in Definition 3.1.1): (d) Connectives. Let φ and ψ be formulas. Then: (f ) Particular quantification. Let φ(∃P ) be a formula governed by an occurrence of ∃P . Then: (g) Universal quantification. Let φ(∀P ) be a formula governed by an occurrence of ∀P . Then: Notice that a gap, u, can be introduced only by the two quantification rules. Rules (a), (b) and (c) never assign u to a formula and rules (d) and (e) assign u to a formula φ only if one is already assigned to the formula or formulas used to determine φ's truth value. Consequently, if every unary predicate does have an instance, no gap will be introduced at any stage and the rules will coincide with the 2-valuation rules. Hence,

Proposition 3.2.2. Any 2-valuation is also a 3-valuation.
While the conception of validity for a two-valued system is clear, namely, truth of the premises leads to truth of the conclusion, we face several formal options when we move to a three-valued system. We may keep the conception of truth leading to truth -strict-to-strict (SS) validity, as used in [15]; we may also consider an argument valid just in case, if its premises are not false, its conclusion isn't false either -tolerant-to-tolerant (TT) validity [12,27]; or we may require that truth does not lead to falsity -strict-to-tolerant (ST) validity; and other options are also possible. 3 Each of these options has formal advantages and disadvantages. For instance, while SS validity is transitive, in the sense that if φ ψ and ψ χ then φ χ, the Deduction Theorem does not generally hold in it: if we use Kleene's strong tables, as above, then p, q q but p q → q. (All the claims in this paragraph are easy to verify.) On TT validity with strong Kleene tables, modus ponens is invalid; while ST validity is not transitive. Formal considerations might weigh for or against a certain choice, but they cannot decide between them.
In this paper we adopt strict-to-tolerant (ST) validity. While Quarc with SS validity has already been researched [15], no publication has explored Quarc with ST validity. Moreover, as we shall see starting with the next subsection, Quarc with ST validity has several interesting formal properties. We therefore define: (3-validity) Let L be a Quarc language. An argument whose premises constitute the set Γ of L-formulas and whose conclusion is the L-formula φ is 3-valid, written Γ 3 φ, iff for any language which contains all the singular arguments occurring in either Γ or φ, no 3-valuation assigns 'true' to all formulas in Γ and 'false' to φ.
From a conceptual point of view, ST validity allows one to represent possible failures of truth-preservation that do not lead to falsity as not being features of invalid argumentation. In a system in which false presupposition causes a truth value gap, as in the three-valued Quarc developed here, ST validity allows us to distinguish between invalid arguments and presupposition failure. This might reflect to some extent intuitive classifications: the argument, 8. All children love their mother; so, all Bob's children love their mother. might seem 'alright', although on the approach developed above to presupposition, in case Bob has no children the conclusion isn't true even if the premise is. However, not all cases in which an argument is ST-valid despite presupposition failure are also intuitively 'alright'. For instance, the following two arguments, suggested by a reviewer, 9. Bob has no children; so, some of Bob's children love their mother. 10. Bob has no children; so, either some of Bob's children love their mother, or grass is green. seem both objectionable. Accordingly, whether such a systematisation of the relations of validity to presupposition failure reflects anything intuitive, or whether it is just a formal way of distinguishing them, requires further consideration. But since the formal interest in ST-validity does not depend on the verdict on this question, we shall not pursue it any further here.

Coincidence of the Consequence Relations
In this subsection we prove Coincidence: any Quarc argument is 2-valid iff it is 3-valid. The basic idea behind this proof was already noted by [10], comparing classical consequence (our 2 ) with strict-to-tolerant consequence (our 3 ): Obviously, a classical countermodel to the entailment from Γ to Δ is an st-countermodel. But conversely, any st-countermodel can be turned into a classical countermodel, basically because reassignments of the values 1 or 0 to subsentences with value 1 2 in the original model do not alter the value 1 or 0 assigned to the sentences in which they appear. [10,21] However, that there is a classical model, or in our case a 2-valuation, which coincides with a given 3-valuation on the value it assigns to any formula which isn't assigned u on that 3-valuation, should be shown. We shall soon do that for Quarc, but we shall later consider a calculus in which this is not the case ( §6). Namely, we shall later show that for some calculi, the classical consequence relation does not coincide with the strict-to-tolerant one. Here we proceed with the proof for Quarc.
Proof. We prove the contrapositive. Suppose Γ 2 φ. Then, for some language L , there is a 2-valuation v on which all formulas in Γ are true and φ is false. Since v is also a 3-valuation (Proposition 3.2.2), Γ 3 φ. Proposition 3.3.2. For every Quarc language L, for every 3-valuation w for L, there is a language L extending L (in the sense that every L-formula is an L -formula) and there is a 2-valuation v for L such that for every L-formula φ: Proof. For each language L, let L = L∪{e}, where e is a singular argument new to L. For each 3-valuation w for L, let v be a 2-valuation for L such that: (b) for every unary predicate P such that w(aP ) = 0 for every a in L, v(eP ) = 1; (c) for every other basic L -formula ϕ (i.e. those not decided in (a) or (b)), v(ϕ) = 0.
Given the definition above, every unary predicate has instances, and therefore v is indeed a 2-valuation. Then, the proposition is proved by induction on the complexity of L-formulas.
Base case: φ has complexity 0, in which case φ is a basic formula, then by definition v(φ) = w(φ).
Induction step: φ has complexity n + 1, assuming that the proposition holds for all formulas of complexity up to n. Then the case is divided into subcases according to Proposition 2.2.5. Here we consider only two of them, the proofs of the rest adding nothing of interest.
(2) φ is governed by an occurrence of ∀P . Suppose w(φ) = 1. Then w(cP ) = 1 for some c in L and w(φ(c)) = 1 for every such c; hence, by IH, v(cP ) = 1 for some c in L and v(φ(c)) = 1 for every such c. In this Proof. We prove the contrapositive. Suppose Γ 3 φ. Then for some language L there is a 3-valuation on which all formulas in Γ are true and φ is false; hence, by Proposition 3.3.2, for some language L = L ∪ {e}, there is a 2-valuation on which all formulas in Γ are true and φ is false. Hence,  order. The formula in the last line of the proof is its conclusion. If there is a proof with the formula φ as conclusion, depending only on formulas from the set Γ, then φ is provable from Γ, written Γ φ.

Lemmon-style Natural Deduction
Remark. Since Quarc languages differ only in their singular arguments, any proof is a proof in any Quarc language that contains all the singular arguments occurring in it.
We shall usually write to the right of the formula the name of the derivation rule which justifies the line, possibly followed by line numbers, according to the conventions specified below.

Sentence negation to predicate negation (SP)
. . a n ¬P SP i 6 In this and the next rule, P may be reordered.
In this and the next rule, φ(a x ) is a formula led by a x ; φ(a) is the formula 12 from which φ(a x ) is immediately generated.
In this and the next three rules, φ(QP ) is a formula governed by an 18 occurrence of QP , and φ(a) is the formula got from φ(QP ) by replacing 19 that occurrence of QP with a. 20 Universal Introduction (∀I) 21 i In this rule, a must not occur in φ(∀P ) or any formula in lines L − i.
Instantial Import (Imp) 26 In this rule, a must not occur in φ(QP ), ψ, any formula in lines L 1 , or 28 any formula in lines L 2 − j − k. 29

Completeness of Quarc
The soundness of the closely related proof system of [2] was proved in that paper, and the adaptation of that proof to the two-valued system of this paper is straightforward, so we do not provide it here. And since, by Coincidence, if Γ 2 φ then Γ 3 φ, the proof system is also sound with respect to the 3-valued semantics. An indirect proof of the completeness of Quarc with natural deduction is found in [24,25], taken together: a Gentzen-style proof theory is developed in these papers, in the former the authors show it to be equivalent to the proof system of [2], and in the latter they prove its completeness. However, a direct proof of the above hasn't been published in any article. A Henkin-style proof is found in [22] and in [6]. 5 In this section we provide a direct proof of the completeness of Quarc with natural deduction. First, we prove the completeness theorem for the two-valued Quarc: for any set Γ of formulas and formula φ, if Γ 2 φ then Γ φ. The proof is an adaptation of Lindenbaum's construction to Quarc with truth-valuational semantics, and it is close to Leblanc's proof of the completeness of the first-order Predicate Calculus, where he also uses truthvaluational semantics [18, §2.3]. Then, by Coincidence, we will have the completeness of three-valued Quarc: if Γ 3 φ then Γ φ. All the semantic concepts mentioned in this section, if not otherwise specified, are those of the two-valued Quarc.

Satisfiability and Consistency
We start with a couple of definitions: Definition 5.1.1. (Satisfiability) A set Γ of formulas is satisfiable iff, for some language which contains all the singular arguments occurring in Γ, there is a valuation on which all formulas in Γ are true; we say of such a valuation that it satisfies Γ.
Definition 5.1.2. (Consistency) A set Γ of formulas is consistent iff, for any formula φ, at most one of φ and ¬φ is provable from Γ. A set of formulas is inconsistent if it is not consistent.
In the rest of this section, we provide the proof of the proposition below, from which the completeness theorem follows.
Proposition 5.1.3. If a set of formulas is consistent then it is satisfiable.

Maximal Consistent Set
Definition 5.2.1. (Maximal consistent set) Δ is a maximal consistent set of L-formulas iff Δ is consistent and for every L-formula ψ / ∈ Δ, Δ ∪ {ψ} is inconsistent.
The following five propositions are consequences of this definition, whose proofs are straightforward and not provided here.
Proposition 5.2.5. Let Δ be a maximal consistent set of L-formulas. Let a 1 , . . . , a n be singular arguments in L, and P an n-ary predicate in L. Then: . . a n ¬P ∈ Δ iff a 1 . . . a n P / ∈ Δ.
Proposition 5.2.6. Let Δ be a maximal consistent set of L-formulas, and φ(a x ) an L-formula led by a x . Then: is immediately generated.

Instance and Witness
Definition 5.3.1. Δ is an instance-complete set of L-formulas iff for each unary predicate P in L, aP ∈ Δ for some singular argument a in L.   Suppose Δ is a consistent set of L-formulas. We construct a sequence Δ 0 , Δ 1 , Δ 2 , . . . of sets of L * -formulas in the following scheme.

Lindenbaum's Lemma
Let Δ 0 = Δ; and for each n ∈ N: (1) in case Δ n ∪ {φ n } is inconsistent, let Δ n+1 = Δ n ; (2) in case Δ n ∪{φ n } is consistent and φ n is not governed by any occurrence of a particularly quantified argument, let Δ n+1 = Δ n ∪ {φ n }; (3) in case Δ n ∪ {φ n } is consistent and φ n is governed by an occurrence of a particularly quantified argument ∃P , let Δ n+1 = Δ n ∪{φ n }∪{dP }∪ {φ(d)}, where d is the first item in the sequence d 0 , d 1 , . . . that does not occur in Δ n or φ n . φ(d) is the formula which results from φ n by replacing that occurrence of ∃P with d.

Proposition 5.4.2. Each set Δ i in the sequence is consistent.
Proof. By induction on i in Δ i . Base case: Since Δ 0 = Δ and Δ is consistent, Δ 0 is consistent. Induction step: Assuming that Δ n is consistent, we show that Δ n+1 is also consistent whichever case it falls into: (1) Δ n+1 = Δ n . Since Δ n is consistent, so is Δ n+1 .
It is easy to see that Also, the construction ensures Δ * is instance-and witness-complete, as proved respectively below.

Proposition 5.4.4. Δ * is an instance-complete set of L * -formulas.
Proof. For every unary predicate P in L * , the formula ∃P P is φ n for some n. We already saw in Example 4.2.1 that ∃P P for any unary predicate P , so Δ n ∪ {∃P P } is consistent if Δ n is consistent; and since ∃P P is governed by that occurrence of ∃P , Δ n+1 = Δ n ∪ {∃P P } ∪ {dP } for some d. Hence, for every unary predicate P in L * , there is some d for which dP ∈ Δ * . Proposition 5.4.5. Δ * is a witness-complete set of L * -formulas.
Now that Δ * is maximal consistent, instance-and witness-complete, we have proved Lindenbaum's Lemma.

Proposition 5.5.1. Let L be a Quarc language and B an instance-complete set of basic L-formulas. If a 3-valuation v is such that, for any basic
Proof. Since B is instance-complete, v complies with the Instantiation rule and is thus a 2-valuation.

Proposition 5.5.2. (Truth lemma) Let Δ be a maximal consistent, instanceand witness-complete set of L-formulas.
Let v * be the valuation for L such that: for every basic formula φ of L, v * (φ) = 1 iff φ ∈ Δ. Then, for every Proof. By induction on the complexity of L-formulas.
Base case: φ has complexity 0, in which case it is a basic formula, then by definition v * (φ) = 1 iff φ ∈ Δ.
Induction step: φ has complexity n + 1, where we assume (IH) that the proposition holds for every formula of complexity up to n. We consider the following subcases, the proofs of the rest adding nothing of interest.

Summary
As we noted earlier, the completeness theorem follows from the proposition that every consistent set is satisfiable, i.e. for any consistent set Δ of Lformulas, there exists a valuation on which all formulas in Δ are true. (The valuation in question can be a valuation for any Quarc language which contains all the singular arguments occurring in Δ). We have shown by Lindenbaum's construction that every consistent set Δ of L-formulas can be extended to a maximal consistent, instance-and witness-complete set Δ * of L * -formulas (where L * is L plus a list of new singular arguments). And, by the Truth Lemma, every such set Δ * is satisfied by a valuation for L * . Since Δ is a subset of Δ * , the valuation also satisfies Δ. Thus, we have proved the completeness of two-valued Quarc:

PC-to-Quarc Translation
The relations of Quarc and PC have been a topic of research since the first publication of a Quarc's precursor in [16], and several additional results have been established since [15,25,29]. We are readdressing the question here because of the specific properties of the Quarc system of this paper. First, it has a three-valued truth-valuational semantics, unlike any system considered so far: all were two-valued, apart from the one of [15], which however used model-theoretic semantics. Moreover, unlike the three-valued system of [15], the system of this paper uses ST validity and, again unlike that system, it is not extended with defining clauses, which played an essential role in the PC-to-Quarc translation of that paper.
This section is dedicated to a general method for translating formulas of PC into formulas of Quarc. It also investigates whether such a translation preserves truth values and (in)validity.

PC: Syntax
In this subsection we introduce a version of the syntax of the Predicate Calculus. • a non-empty countable set of constants.
Remark. We will use a, b, c, . . . (possibly with subscripts) for arbitrary constants, x, y, z, . . . for arbitrary variables, and P , R, S, . . . for arbitrary predicates. We will sometimes use Q for either of ∀ and ∃.  P is an n-ary predicate and a 1 , . . . , a n are constants, then P a 1 . . . a n is a formula, which is also called a basic formula. Complexity of PC formulas can be defined along the same lines as that of Quarc formulas (Definition 2.2.6).

Two-valued TVS for PC
Since we work with TVS for Quarc, it is simpler to work with TVS for PC as well. The set of truth-valuationally valid PC arguments can be shown to coincide with that of model-theoretically valid ones, along the lines of proofs found in [18, §4.2]. Accordingly, the preservation of validity under translation proved below applies also to PC with model-theoretic semantics.  (2-validity) Let L be a PC language. An argument whose premises constitute the set Γ of L-formulas and whose conclusion is the Lformula φ is 2-valid, written Γ 2 φ, iff for any PC language which contains all the constants occurring in either Γ or φ, every 2-valuation that assigns all formulas in Γ 'true' assigns φ 'true' as well.

The Translation Manual
As in some previous versions, the PC-to-Quarc translation involves the introduction into Quarc of a special unary predicate, T (for Thing), which is used to reflect in Quarc the fact that quantification in PC is not restricted by means of any predicate. Here we use T as a logical predicate, so it occurs as a constant in both semantics and proof system. 6 We add a valuation rule to Definitions 3.1.1 (2-valuation) and 3.2.1 (3valuation), which specifies the semantic behaviour of T : (i) Thing. For any singular argument a, v(aT ) = 1.
We proceed to correlate the PC and Quarc languages, where all Quarc languages we consider are enriched by the logical unary predicate T . Definition 6.3.1. (Language correlation) With each PC language L p we correlate the unique Quarc language L q that satisfies the following requirements: (a) The singular arguments of L q are the constants of L p .
(b) The non-logical predicates of L q are those of L p , and arities are preserved.
Clearly, in this way every Quarc language is correlated with a unique PC language.
We define the PC-to-Quarc translation as a function that maps each and every formula of L p to a formula of L q .
(3) Quantified formulas: The following proposition is easy to prove and will be used in later proofs.

Truth Value Preservation
We move on to show that the PC-to-Quarc translation is adequate, in the sense that there is a bijection between PC valuations and Quarc 3-valuations (where again, all Quarc languages we consider are enriched with T ), such that the truth value of a formula is preserved by the translation under this bijection. It will follow that the (in)validity of PC-arguments is also preserved by the translation.
Definition 6.4.1. (Valuation correlation) With each PC valuation v p for a PC language L p we correlate a Quarc valuation v q for L q , the Quarc language that we correlate with L p , such that for every basic formula φ of The following proof shows that the PC-to-Quarc translation, together with the valuation correlation, preserves truth values. Proposition 6.4.2. Let χ be a formula of a PC language L p and f (χ) its translation in L q , the Quarc language that we correlate with L p . Let v p be a valuation for L p and v q the correlated valuation for L q .
Remark. We need to check both 'true' and 'false' cases, because Quarc formulas can also have the third value.
Proof. By induction on formulas of L p .

Validity Preservation
Proposition 6.5.1. Let Γ be a set of formulas of a PC language L p . We write f (Γ) for the set of the translations of all the members of Γ. Let φ be a formula of L p and f (φ) its translation. Then: Proof. Let L q be the Quarc language correlated with L p . Then f (Γ) is a set of L q -formulas and f (φ) is an L q -formula. Suppose f (Γ) 3 f (φ). Then, by the definition of validity, for some Quarc language L q there is a valuation v q on which all formulas in f (Γ) are true and f (φ) is false. Let L p be the PC language with which L q is correlated, and v p the PC valuation (for L p ) with which v q is correlated. Then, by Proposition 6.4.2, every formula in Γ is true on v p , and φ is false on v p . Hence, Γ 2 φ.
Suppose Γ 2 φ. Then, by the definition of validity, for some PC language L p there is a valuation v p on which all formulas in Γ are true and φ is false. Let L q be the Quarc language correlated with L p , and v q the Quarc valuation (for L q ) correlated with v p . Then, by Proposition 6.4.2, every formula in f (Γ) is true on v q , and f (φ) is false on v q . Hence, f (Γ) 3 f (φ).

Soundness and Completeness in Quarc with T
We next consider the soundness and completeness of Quarc enriched with T . As we shall see, there are important differences between the two-and three-valued systems.
As for the proof system, we add to Definition 4.1.2 the derivation rule ThI (Thing Introduction), as follows: where a is any singular argument.
The soundness preservation of ThI should be straightforward, since aT is true for any a on any valuation, on both the two-and three-valued systems. Given the soundness of the two-valued Quarc without T , it follows that the two-valued Quarc with T is also sound.
However, the soundness of the three-valued Quarc followed from the coincidence theorem above. Yet that proof does not apply to Quarc with T : given a 3-valuation w for a language, to generate a 2-valuation v for an enriched language such that if w(φ) = 1(0) then v(φ) = 1(0), it introduced a singular argument e such that, for any nonempty unary predicate P , v(eP ) = 0. Since T is nonempty (i.e., for some a, w(aT ) = 1), this would make v(eT ) = 0, which would violate the semantic rule for Thing, namely, that for any singular argument a, v(aT ) = 1. As we shall soon see, the proof system introduced above and enriched with ThI is unsound with respect to the three-valued Quarc with T .
The proof for the completeness of two-valued Quarc with T is essentially the same as in Sect. 5, noticing that any maximal consistent set of a language contains aT for any singular argument a in the language: Proposition 6.6.1. Let Δ be a maximal consistent set of L-formulas. Then aT ∈ Δ for any a in L.
Proof. Suppose a is a singular argument in L. Then, by ThI, Δ aT ; and hence, by Proposition 5.2.2, aT ∈ Δ.
And since aT is a basic formula, the valuation induced by a maximal consistent set (the one employed in proving the Truth Lemma) complies with the valuation rule 'Thing'. Accordingly, the two-valued Quarc with T is sound and complete.
Consider now the formula, ∃T P . On the two-valued Quarc, due to Instantiation, it is true on any valuation, so 2 ∃T P . Since two-valued Quarc with T is complete, it follows that ∃T P , which is also simple to show directly. By contrast, since on the three-valued Quarc with T , while aT is true for any a on any valuation, aP can be false for every a on a valuation, 3 ∃T P . It follows, first, that 2 and 3 do not coincide on Quarc with T , and secondly, that the proof system developed above is unsound for three-valued Quarc with T .
Whether a sound and complete proof system for three-valued Quarc with T can be developed, we leave an open question. Notice, however, that Cut will not be admissible in such a system: since ∃P P cannot be false on a valuation, 3 ∃P P ; since if ∃P P is true on a valuation, so is ∃T P , ∃P P 3 ∃T P ; but as explained above, 3 ∃T P . If a proof system for Quarc with T is sound and complete, the following should therefore hold on it: ∃P P , ∃P P ∃T P , ∃T P . This would necessitate modifying the derivation rules for connectives introduced above. These issues arise in the translating system, the three-valued Quarc with T , despite their inexistence in the translated system, PC.
Ever since the first works of [8,9], the literature in this area has focused on formal systems which preserve their set of valid arguments when their two-valued semantics is replaced by a three-valued one and validity is defined as strict-to-tolerant (ST). This coincidence holds for the Propositional Calculus, the Predicate Calculus, and as we proved in Sect. 3.3, for Quarc (without T ). The exceptions found in the literature for which there is no such coincidence are cases of paradox, specifically of the Predicate Calculus augmented with a truth predicate, or with a similarity predicate in the presence of vague concepts. While the system cannot then have a consistent 2-valuation because of, for instance, paradoxes generated by liar sentences, that is not the case with 3-valuations; in that case, Kripke's fixed point construction shows that there are 3-valued consistent models [13]. However, as we have just seen, a different result obtains for Quarc with T : the system has consistent 2-and 3-valuations, neither is paradoxical, but their inference relations do not coincide, for while 2 ∃T P , ST ∃T P . 7

Quarc-to-PC' Translation
While in the previous section, we translated PC into an extended version of Quarc, Quarc plus T , we shall here translate Quarc into an extended version of PC. We need to introduce gaps into the valuations of the Predicate Calculus, and for that purpose we add a quantifier symbol ∃ to Definition 6.1.1 (Language), and, correspondingly, in Definition 6.1.2 we add the following formation rule: If φ is a formula containing one or more occurrences of a, and x is a variable new to φ, then ∃ xφ[x/a] is a formula.
We will call this extended system PC'. The version of Quarc considered below is the three-valued one without T .

Three-valued TVS for PC'
(c) Let φ be a formula containing c, and x a variable new to φ. Then: Definition 7.1.2. (3-validity) Let L be a PC' language. An argument whose premises constitute the set Γ of L-formulas and whose conclusion is the Lformula φ is 3-valid, written Γ 3 φ, iff for any PC' language which contains all the constants occurring in either Γ or φ, no 3-valuation assigns 'true' to all formulas in Γ and 'false' to φ.

The Translation Manual
Definition 7.2.1. (Language correlation) With each Quarc language L q we correlate the unique PC' language L p that satisfies the following requirements: (1) Basic formulas: t(a 1 . . . a n P ) = P a 1 . . . a n (2) Reorder: t(a τ 1 . . . a τ n P τ ) = t(a 1 . . . a n P ) (3) Negative predication: t(a 1 . . . a n ¬P ) = ¬t(a 1 . . . a n P ) (4) Truth functional compounds: (5) Anaphora: Let φ(a x ) be a formula led by a labelled argument a x , and φ the formula from which φ(a x ) is immediately generated. Then t(φ(a x )) = t(φ) (6) Quantification: Let φ(QP ) be a formula governed by an occurrence of QP , and φ the result of replacing the governing occurrence by c, a singular argument new to φ(QP ). Let x be a variable new to t(φ). Then The following proposition is easy to prove and will be used in later proofs.

Truth Value Preservation
We move on to show that the translation is adequate, in the sense that there is a bijection between Quarc valuations and PC' valuations, such that the truth values of Quarc formulas are preserved by the translation under this bijection. It will follow that the (in)validity of Quarc arguments is also preserved by the translation. language that we correlate with L q , such that: for every basic formula φ of Quarc, v p (t(φ)) = v q (φ).
The following proof shows that, with valuations correlated, the Quarcto-PC' translation preserves truth values. Proposition 7.3.2. Let χ be a formula of a Quarc language L q and t(χ) its translation in L p , the PC' language that we correlate with L q . Let v q be a valuation for L q and v p the correlated valuation for L p . Then: v p (t(χ)) = v q (χ).
Proof. By induction on the complexity of L q -formulas.
Base case: Suppose χ is a basic formula of L q . Then by the valuation correlation v p (t(χ)) = v q (χ).

Provability in PC'
Similar issues to those of provability in the three-valued Quarc with T arise also for PC'. Let φ be a PC' formula containing a, and x a variable new to φ.

Conclusion
In this paper, we developed both a two-valued and three-valued truthvaluational semantics for the Quantified Argument Calculus (Quarc). The two-valued version followed closely that in [2], which included an Instantiation rule, forcing unary predicates to have instances. The three-valued semantics eliminated this rule, taking formulas governed by either ∀P or ∃P to presuppose that P has instances and making them truth-value-less otherwise. The elimination of the Instantiation rule in this way creates a richer and in a sense a semantically more natural system, which is therefore of much interest. This approach was also followed in [15], but unlike that paper, this one adopted strict-to-tolerant and not strict-to-strict validity. We then proved a coincidence result: Γ 2 φ iff Γ 3 φ. This result does not hold on the SS validity approach. For example, since ∃P P is never false, hopefully of some additional interest, as the adaptation of such proofs to truth-valuational semantics is not found in later literature. We believe these results contribute to the interest in the three-valued Quarc with ST validity. Lastly, we investigated the relations between the three-valued Quarc and PC. (The relation of Quarc as a two-valued system to PC has been addressed in several works [16,25,29].) The unrestricted nature of PC quantification was imitated in Quarc by adding to it a logical predicate T , which applies to all singular arguments of all languages. The gappy nature of three-valued Quarc quantifiers was imitated in PC by adding to it a weak existential quantifier, ∃ , for which ∃ xφ(x) is truth-value-less in case it has no instances. We managed in this way to incorporate a semantic image of each calculus in the extension of the other, in the sense that an argument in the one is valid just in case so is its translation into the other. It should be emphasised that neither extension is justified by internal considerations on its system (unlike the elimination of Instantiation), but was done only in order to imitate features of the other system. In fact, as we saw, either extension creates difficulties for a proof system for its calculus, forcing it not to admit Cut if sound and complete, difficulties inexistent either in the unextended version or in the translated calculus. Although we have not proved that a different, less problematic extension of either system which allows for an incorporation as above is impossible, other options we have tried had similar drawbacks and so far none has been suggested in the literature. To this extent, our results support a claim already made in several works, that although related, Quarc and PC are essentially different calculi.
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