Two-sided Sequent Calculi for FDE-like Four-valued Logics

We present a method that generates two-sided sequent calculi for four-valued logics like first degree entailment (FDE). (We say that a logic is FDE-like if it has finitely many operators of finite arity, including negation, and if all of its operators are truth-functional over the four truth-values ‘none’, ‘false’, ‘true’, and ‘both’, where ‘true’ and ‘both’ are designated.) First, we show that for every n-ary operator ⋆ every truth table entry f⋆(x1,…,xn) = y can be characterized in terms of a pair of sequent rules. Secondly, we use these sequent rules to build sequent calculi and prove their completeness. With the help of two simplification procedures we then show that the 2 ⋅ 4n sequent rules that characterize an n-ary operator can be systematically reduced to at most four sequent rules. Thirdly, we use our method to investigate the proof-theoretical consequences of including intuitive truth-functional implications in FDE-like logics.

1 A sequent calculus is canonical if (i) its axioms include the standard axioms, (ii) its rules include the standard ones including cut and weakening, and (iii) all its other rules and axioms are canonical. Anpremise rule is canonical if its conclusion is of the form 1 or 1 where and are finite and if for all of its premises (0 ) it holds that 1 [4, pp. 121-122]. (Note that we use a slash ( ) where most proof theorists would use an arrow ( ).) 2 A hypersequent is a finite sequence 1 1 of sequents , where and are finite multisets of formulas. See Avron [3] for an accessible introduction to hypersequent calculi. An -valued sequent is an -tuple 1 of finite multisets of formulas. See Zach [27, § 3.1] for a solid review of -valued sequent calculi. A labelled sequent is a finite multiset of labelled formulas of the type , where the label is usually interpreted as a truth-value or a set of truth-values. See Carnielli [10], Baaz et al. [6], and Caleiro et al. [9] for general methods for obtaining labelled calculi for -valued logics. defined in terms of FDE's negation, disjunction, and conjunction. This is not what we will do however: we directly provide sequent calculi for FDE-like four-valued logics that are not functionally complete. To do so, we use the correspondences between single truth table entries and proof rules that were studied by Kooi and Tamminga [14]. 3 Accordingly, we first characterize any truth-functional operator in terms of a set of sequent rules. We then add, operator by operator, the operator's characterizing sequent rules to a basic sequent calculus that deals only with negation. This method is modular: no matter how many operators are thus included, the resulting sequent calculus will always be complete with respect to its particular semantics.
Our paper proceeds as follows. First, we present a semantical definition of the kind of truth-functional four-valued logics to be studied and characterize any truth table entry of any -ary operator with two sequent rules. Second, we use these characterizing sequent rules to define cut-free sequent systems and show that each of the resulting systems is complete with respect to its particular four-valued semantics. With the help of two simplification procedures we then show that the 2 4 sequent rules that characterize any -ary operator can be systematically reduced to at most four sequent rules. We illustrate our simplification procedures by applying them to FDE's truth tables for disjunction and conjunction and show that we thus obtain standard sequent rules. Third, we use our method to investigate the proof-theoretical consequences of including intuitive truth-functional implications in FDE-like logics. Lastly, we discuss related work by Baaz et al. [6] and Wintein and Muskens [26] that shares some of our theoretical objectives but proposes different ways to realize them. A short summary and an open question conclude the paper.

FDE -Like Four-Valued Logics
Belnap [8, p. 43] discusses the concept of validity for first degree entailment (FDE). His remarks imply that an argument from a set of premises to a set of conclusions is FDE-valid (notation: FDE ) if it preserves both truth and nonfalsity. An argument preserves truth if for every valuation it holds that if is 'true' or 'both' for all in , then is 'true' or 'both' for some in . An argument preserves non-falsity if for every valuation it holds that if is 'none' or 'true' for all in , then is 'none' or 'true' for some in . Because an argument preserves non-falsity if and only if the argument preserves truth, and because every four-valued logic that we consider in this paper has negation, we can characterize FDE-validity by focusing on truth-preservation alone.
Accordingly, we say that a logic is FDE-like if it has finitely many operators of finite arity, including negation, and if all of its operators are truth-functional over the four truth-values 'none', 'false', 'true', and 'both', where 'true' and 'both' are designated. FDE-like logics form an infinitely large family whose members share a common trait: they all have the same negation. (Hence, if a truth-functional fourvalued logic does not have negation, it is not FDE-like.) To be specific, an FDE-like logic L4 evaluates arguments consisting of formulas from a propositional language L built from a set P of atomic formulas, using negation ( ) and finitely many additional truth-functional operators of finite arity. (To obtain the propositional language of FDE, just add two binary operators: disjunction ( ) and conjunction ( ).) Moreover, for any FDE-like logic L4, a valuation is a function from the set P of atomic formulas to the set 0 1 0 1 of truth-values 'none', 'false', 'true', and 'both'. We use the following boldface shorthands: n abbreviates , 0 abbreviates 0 , 1 abbreviates 1 , and b abbreviates 0 1 . A valuation on P is extended recursively to a valuation on L by the truth-conditions for negation and the truthconditions for the finitely many additional operators of finite arity. The truth table gives the truth-conditions for negation: An argument from a set of premises to a set of conclusions is L4-valid (notation:

L4
) if and only if for every valuation it holds that if 1 for all in , then 1 for some in . Consequently, 1 and b are the designated values: these are the values that are preserved in L4-valid arguments. 4  (it preserves truth) and

L4
(it preserves non-falsity). Two remarks on terminology and notation: when the context does not give rise to ambiguities, we also say that an argument is valid if it is L4-valid and write ' ' rather than '

L4
'. In addition, the text includes two small abuses of notation (the one is not the negation of the other): we write '1 ' instead of 'for all it holds that 1 '; and '1 ' instead of 'for all it holds that 1 '. 4 An interesting and important variant of L4-validity is Pietz and Rivieccio's [20] exactly true logic (ETL). This four-valued logic is also truth-functional over n, 0, 1, and b, but its only designated value is 1. Accordingly, an argument is ETL-valid if and only if for every valuation it holds that if 1 for all in , then 1 for some in . Wintein and Muskens [26] provide an elegant sequent calculus for ETL. A direct application of our method to ETL-like logics is impossible (the ETL-counterpart of Lemma 1 holds, but the ETL-counterpart of Lemma 3 fails). Nonetheless, our method can be used to study single conclusion ETL-like logics, because the following implication holds: if both 1 L4 and L4 1 , then 1 ETL .
Note that the omnipresence of negation makes a two-sided representation of FDElike logics possible. In fact, by sorting out the negated formulas left and right of the slash, any two-sided sequent can be seen as a four-sided one 1 2 3 4 . If we think of the four indices in a four-sided sequent as labels, any four-sided sequent 1 2 3 4 can be seen as a single set of labelled formulas.

Correspondences
Any truth table entry of any truth-functional -ary operator can be characterized with a pair of sequent rules. To prove this, we first give a definition that generates a pair of sequent rules from any truth table entry 1 of any truth-functional -ary operator , where all the 's and are truth-values in n 0 1 b . We then show that the pair of sequent rules thus obtained characterizes the truth table entry from which it was generated.
Let us first fix a language L built from a set P of atomic formulas, using negation ( ) and finitely many operators of finite arity. Our characterization result quickly follows from three modest observations:

Lemma 1 Let
L and L . Then iff for every such that 1 and 1 it holds that 1 iff for every such that 1 and 1 it holds that 1 .
Proof Immediately from the definition of L4-validity.
Our second straightforward observation is that the truth-value of a formula under a valuation can be characterized by specifying, first, whether is designated and, secondly, whether is designated:

Lemma 2 Let
L and let be a valuation. Then n iff 1 and 1 0 iff 1 and 1 1 iff 1 and 1 b iff 1 and 1 .
Proof Immediately from the truth table for negation.
To state the third observation, we introduce two operations and , where is a formula and is a truth-value. Given a sequent , the operator places the formula , depending on the truth-value , on the left-hand side or the right-hand side of . The operator does the same thing to the negation of : Consequently, every -ary operator can be characterized in terms of 2 4 sequent rules. We use these characterizing sequent rules to generate cut-free sequent calculi and to prove the completeness of these calculi with respect to their particular semantics.

Sequent Calculi
We now define sequent calculi for each and any four-valued logic L4 with negation and finitely many truth-functional additional operators of finite arity. Sequents are of the form , where per usual both and are finite, possibly empty, multisets of formulas. If there is a proof of the sequent in the sequent system G4, then we write

G4
. 5 The core of each of the calculi in this paper is the calculus G4 , a basic sequent calculus for four-valued negation, consisting of an axiom, left and right rules for negation, left and right contraction rules, and left and right weakening rules.
The rules of G4 are the following: 6 Logical axiom: Logical rules: Structural rules: Note that G4 has no cut. 5 The family of sequent calculi for four-valued logics that we refer to with the generic name G4 is not to be confused with Kleene's ( [13], p. 306) calculus 4 or the calculus G4ip (Troelstra and Schwichtenberg [25], p. 102; Negri and von Plato [17], p. 122). 6 The notational conventions are similar to the ones in Negri and von Plato [17].
To obtain a sequent calculus for a four-valued logic L4 based on negation and finitely many additional truth-functional operators of finite arity, just add to G4 for each additional -ary operator the 2 4 rules that are generated from its truth table by Definition 2.

Completeness
Let G4 be a sequent system for a four-valued logic L4 based on negation and finitely many additional truth-functional operators of finite arity. To prove the completeness of G4 with respect to L4, we suppose that G4 and then, to show that L4 , construct on the basis of our supposition a valuation such that 1 and 1 . Our completeness proof is similar to Priest's [21] completeness proof for a tableau system for FDE.
The valuation is constructed as follows: we first build a proof tree with root and apply, going upwards, all the available logical rules (these are L , R , and the 2 4 characterizing -rules of any additional -ary operator ) until every applicable logical rule has been applied. Because there might be different -rules that apply to the same conclusion, we require that our proof tree be monotonic. A proof tree is monotonic if and only if it holds that if is a child of , then and . Note that every proof tree can be transformed into a monotonic one by repeated applications of the contraction rules LC and RC. (The two weakening rules LW and RW do not play a role in the completeness proof -in fact, our logical axiom suffices to simulate them.) If every applicable logical rule has been applied, there must be an open and complete branch of our proof tree with root and leaf , because otherwise we would have, contrary to our supposition, that

G4 . A branch with leaf is open if and only if
is not an axiom (that is, ). A branch with leaf is complete if and only if every rule that is applicable to has already been applied. We use this leaf of our open and complete branch to define a valuation from to n 0 1 b :

Two Simplification Procedures
Definition 2 generates a clutter of different sequent rules with the same conclusion. For instance, if 1 and 1 such that 1 , then the positive rules 1 and 1 have the same conclusion 1 . We prove that the 2 4 sequent rules that characterize any -ary operator can be systematically reduced to at most four sequent rules. 7 To do so, we first define the product of two rules with the same conclusion and show that the product rule thus obtained is exactly as powerful as the two rules together. Because the sequent rules generated by Definition 2 have at most four different conclusions, product rules systematically reduce the 2 4 sequent rules that characterize an -ary operator to at most four sequent rules. The at most four sequent rules thus obtained might have at most premise sequents. Second, we define an ordering relation on sequents and use it to minimize the number of premise sequents of a sequent rule. We show that the minimized sequent rule is exactly as powerful as the sequent rule that was minimized. Third, we illustrate our two simplification procedures by applying them to the truth tables and of FDE and show that we thus obtain standard sequent rules for disjunction and conjunction.
In the remainder of this section, we have to be somewhat pedantic about our use of variables: we use the 's and 's as variables over multisets of formulas (this is just usual practice) but the 's and 's as variables over sets of formulas.
Definition 4 (Product) Let 1 be an -premise rule and 2 an -premise rule with the same conclusion , where 1 and 1: Then the product of 1 and 2 , denoted by 1 2 , is the following premise rule: If either 1 or 2 is a zero-premise rule, then 1 2 is the zero-premise rule with the same conclusion .
Lemma 5 Let G4 1 2 be a proof system that contains two rules 1 and 2 with the same conclusion. Let G4 1 2 be the proof system that contains the product rule 1 2 instead of the rules 1 and 2 . Let L . Then Proof ( ) We show that any application of the rule 1 can be replaced by repeated applications of the weakening rules and an application of the product rule 1 2 . Suppose the rule 1 has been applied to the premises (1 ) to obtain the conclusion . By repeated applications of the weakening rules LW and RW to the premises of the rule 1 , we obtain the premises (1 and 1 ) of the product rule 1 2 . Finally, we apply the product rule 1 2 to these new premises and obtain the conclusion . Similarly, any application of the rule 2 can be replaced by repeated applications of the weakening rules and an application of the product rule.
( ) We show that any application of the product rule 1 2 can be replaced by repeated applications of the rules 1 and 2 . Suppose the product rule 1 2 has been applied to the premises (1 and 1 ) to obtain the conclusion . For every with 1 we have premises (1 ), to which we apply the rule 1 to obtain the conclusion . In this way, we obtain conclusions (1 ), to which we apply the rule 2 to obtain the sequent . By repeated applications of the contraction rules LC and RC we obtain the desired conclusion .
The product rule of an premise rule and an premise rule has premises. This number can be reduced significantly by minimizing the product rule's premises. We define minimization on the basis of the following ordering: if and only if and . (Note that the 's and 's are variables over sets of formulas.) Definition 5 (Minimization) Let be a sequent rule. Then the minimization of , denoted by min , is the sequent rule with the same conclusion as and a premise set that exactly contains those -minimal elements of the premise set of that are not axioms.
To illustrate this definition, consider the sequent rules n11 and nb1 : n11 nb1 .
By multiplying these two rules we obtain the product rule n11 nb1 , which has sixteen premises. After minimization only three premises survive: 8 min n11 nb1 .

Lemma 6
Let G4 be a proof system that contains the rule . Let G4 min be the proof system that contains the rule min instead of the rule . Let L . Then Proof ( ) We show that any application of the rule can be replaced by an application of the rule min . Suppose the rule has been applied to obtain a conclusion . Because every premise of min is a premise of , we obtain the conclusion also by applying min to those premises of that survive minimization. Moreover, we discard every subtree issuing in a premise of that did not survive minimization.
( ) We show that any application of the rule min can be replaced by additional axioms, repeated applications of the weakening rules, and an application of the rule . Suppose the rule min has been applied to obtain a conclusion . For every premise of it holds that either (i) is a premise of min , (ii) is not a premise of min , but , or (iii) is not a premise of min , but there is a premise of min such that and . The premises of type (ii) are logical axioms. The premises of type (iii) are obtained from the relevant premises of min by repeated applications of the weakening rules LW and RW. Finally, we obtain the conclusion by applying to the premises of the types (i), (ii), and (iii) thus obtained.
We use minimized product rules to characterize any -ary operator with at most four sequent rules. . If there are no rules of a particular kind, then the relevant minimized product rule is undefined.
To conclude this section, let us illustrate our two simplification procedures by applying them to the truth tables and of FDE and show that we thus obtain standard sequent rules for disjunction and conjunction. FDE's truth tables for disjunction and for conjunction are the following:  . Proof Analogous to the proof of Lemma 7.
Finally, we add the new rules for disjunction and the new rules for conjunction to the basic sequent calculus G4 that only contains a logical axiom, logical rules for negation, and the structural rules of contraction and weakening. We use G4 to refer to the resulting sequent calculus. 9 If the language L is built from atomic 9 Note that G4 for FDE is similar to the sequent calculi given by Pynko [ formulas using only , , and , the preservation of truth and the preservation of non-falsity coincide. 10 Consequently, G4 is sound and complete for FDE in Belnap's original sense:
In sum, our method not only generates two-sided sequent calculi for FDE-like four-valued logics, but also generates sequent calculi with at most four sequent rules for each -ary operator. It does so in two steps. (Recall that a logic is FDE-like, if it contains finitely many -ary operators, including negation, and if all of its operators are truth-functional over the four truth-values n, 0, 1, and b, where 1 and b are designated.) In the first step, each truth table entry of each of the relevant -ary operators is characterized by a pair of sequent rules. In the second step, we add these sequent rules to the basic sequent calculus G4 and simplify them systematically using (i) our product rule that combines two given rules with the same conclusion, and (ii) our minimization of the premise set of a given rule. In this way, we systematically reduce the sequent calculus we found first to a sequent calculus with at most four sequent rules for each -ary operator. We illustrated our method by automatically producing a sequent calculus for FDE that is very similar to the ones that have been presented in the literature. Accordingly, finding an elegant sequent calculus for a given FDE-like four-valued logic is a process that can be fully automated.

Implications
We now apply our method to investigate the proof-theoretical consequences of including intuitive truth-functional implications in FDE-like four-valued logics. To do so, we start from Dunn's [11] observation that the conditions under which a formula is false can be distinguished from the conditions under which that formula is not true, and that these conditions are logically independent. Dunn uses this observation to present an intuitive semantics for FDE by giving both a truth-condition and a falsity-condition for its operators of negation, disjunction, and conjunction. We use Dunn's observation to provide intuitive semantics for implication by giving both a truth-condition and a falsity-condition. There are four natural ways to give a truth-condition for implication: is true iff if is true, then is true.
is true iff if is true, then is not false.
is true iff if is not false, then is true.
is true iff if is not false, then is not false. 10 See Dunn [12,Proposition 4] for an accessible proof.
Likewise, there are four natural ways to give a falsity-condition for implication: ( 1 ) is false iff is true and is false.
is false iff is true and is not true.
is false iff is not false and is false.
is false iff is not false and is not true.
Let us say that a formula is true if its truth-value is either 1 or b, that a formula is not true if its truth-value is either n or 0, that a formula is false if its truth-value is either 0 or b, and that a formula is not false if its truth-value is either n or 1. We use the following boldface shorthand notations: T abbreviates 1 b , F abbreviates  0 b , T abbreviates n 0 , and F abbreviates n 1 . (Note that any intersection of an element of T T and an element of F F is a singleton.) Using the boldface shorthand notations, the four truth-conditions for implication give rise to the following four generalized truth tables: 1 T T

T T F T T F T T 4 F F F T T F T T
Likewise, the four falsity-conditions for implication give rise to the following four generalized falsity tables: . For instance, we get 2 3 0 b 1, because 0 T L 2 and b F R 2 and 0 F L 3 and b F R 3 and 2 T F 3 F F T F 1 . In this way, we obtain sixteen different truth tables. Obviously, the method we developed in the previous sections can be used to find the sequent rules that characterize any of these sixteen truth tables. With a slight adaption, however, it can do much more.
We first note that our boldface shorthand notations T, T, F, and F are obviously related to the truth-values 0 and 1:

Lemma 9 Let
L and let be a valuation. Then Another small abuse of notation: we write ' T' instead of 'for all it holds that T', and similarly for the other boldface shorthands.

Definition 7
Let L and L . Let T T F F . Then is the following operation: Then it must be that T. ( ) Suppose for every such that T and T it holds that T. Then it must be that . The other three cases are proved similarly.

Definition 8 Let
with be one of our eight generalized tables and let 1 2 T T F F be such that 1 2 . Then 1 2 is the following sequent rule: Proof ( ) Assume that 1 2 . Suppose 1 and 2 are valid. Then, by Lemma 10, for every such that T and T it holds that 1 and 2 . Then, because 1 2 , it must be that for every such that T and T, it holds that . Then, again by Lemma 10, is valid. Therefore, 1 2 is validity-preserving.
( ) Assume that 1 2 is validity-preserving. Let 1 and 2 be logically independent atomic formulas, let 1 2 T T F F , and let be an arbitrary valuation such that 1 1 and 2 2 . Given the valuation and 1 2 , we define two sets of literals, and , as follows: Two facts about and are to be noted. First, T and T. Secondly, 1 1 and 2 2 are valid. By assumption, 1 2 is validity-preserving. Hence, 1 2 is valid. By Lemma 10, for every such that T and T it holds that 1 2 . Hence, 1 2 . Because was arbitrary, we conclude that for every such that 1 1 and 2 2 it holds that 1 2 . Therefore, 1 2 .

Lemma 11
Let with 1 2 3 4 be one of our four generalized truth tables. The tables are characterized by the following rules: The results from the present section help us to establish the proof-theoretical consequences of including a truth-functional implication into our FDE-like four-valued logic. First, we know which pairs of positive sequent rules characterize the four natural truth-conditions for implication and which pairs of negative sequent rules characterize the four natural falsity-conditions for implication. Second, we know that the positive pair and the negative pair can be chosen independently. Third, we know that both the positive rules and the negative rules come in pairs. We cannot choose a positive left rule from one pair and a positive right rule from another (or a negative left rule from one pair and a negative right rule from another), because doing so would make our proof system unsound. Accordingly, choosing a truth-functional implication for an FDE-like four-valued logic can now be one on the basis of much more complete information.

Baaz, Fermüller, Salzer, and Zach (1998)
Baaz et al. [6] present a general method that, although it was not designed to do so, can also be applied to obtain two-sided sequent calculi for FDE-like logics with at most four sequent rules for each -ary operator. Their method makes use of sets of truth-values as labels in labelled sequent calculi. Given the set n 0 1 b of truth-values, we first need to fix a system of signs, that is, a set of subsets of such that for every there is a subset such that . A labelled sequent is a finite multi-set of formulas of the type where and L . If we choose the system of signs 1 2 3 4 with 1 1 b , 2 n 0 , 3 0 b , and 4 n 1 , 11 any labelled sequent can be rewritten as a twosided sequent , because any FDE-like logic contains the negation and its truth table . Any 1 in is represented by an to the right of the slash, any 2 in by an to the left of the slash, any 3 in by a to the right of the slash, and any 4 in by a to the left of the slash. Accordingly, the labelled sequent 1 2 3 4 is represented by the two-sided sequent . Conversely, any two-sided sequent can be rewritten as a labelled sequent with labels 1 through 4 .
Given the system of signs 1  cut: 3 4 Propositional rules: An introduction rule for an -ary operator labelled with is of the form 1 1 where the formulas in each (1 ) are of the form for some 1 and . It is easy to see that, if we rewrite the labelled sequents in these axioms and structural rules as two-sided sequents, then we obtain special cases of the logical axiom and of the structural rules LC, RC, LW, and RW of our calculus G4 . Note again that G4 does not have a cut rule.
To compare the introduction rules that are obtained by Baaz  . They then define a partial disjunctive normal form, each conjunct of which corresponds to a 1 in . This disjunctive normal form is then converted into a partial conjunctive normal form from which they read off the introduction rule . If we apply their method to the truth table for the unary operator , we find the following four rules (note that we leave out the axioms): Rewriting the labelled sequents as two-sided sequents, we find the following four two-sided rules, respectively: In their two-sided formulations, the first two rules are redundant, the rule 3 is our logical rule , and the rule 4 is our logical rule . Likewise, if we apply their method to the truth table for the binary operator , we find the following four rules (note again that we leave out the axioms and that we do some serious trimming): Rewriting the labelled sequents as two-sided sequents, we find, respectively, the rules , , , and of Lemma 7.

Wintein and Muskens (2016)
Another way to obtain two-sided sequent calculi for FDE-like logics is via a two-sided sequent calculus for a functionally complete set of operators that is truthfunctional over n 0 1 b . Muskens [16, p. 49] showed that is such a set of operators, where the truth tables for and are the ones given above and the truth tables for and are the following: In the fashion of Section 6, we can see the truth table  as obtained from the  generalized truth table  and the generalized falsity table  . Likewise, the  truth table  can be seen as obtained from the generalized truth table  and the  generalized falsity table  : F T F T

Definition 8 and Theorem 4 imply that the truth table for
is characterized by the following four rules: . Likewise, the truth table is characterized by the following four rules (use the product rule and the minimization rule repeatedly): . As a consequence, the sequent calculus G4 that is obtained from our basic sequent calculus G4 by adding the rules for , , and is sound and complete for the FDE-like logic that is built from a set P of atomic formulas, using the functionally complete set of operators . In turn, this sequent calculus could be used to find sequent rules for any -ary operator that is truth-functional over n 0 1 b , given a translation of the formula 1 into a formula in which only operators from occur. Wintein and Muskens [26,Definition 7] present signed sequent rules for and . In their approach, a signed formula of the form ' ' consists of a sign from the set 1 1 0 0 and a formula from a propositional language. 12 Accordingly, a signed sequent is simply a finite set of signed formulas. Their signed sequent rule for is the following: , for 0 1 0 1 1 0 1 0 .
As a matter of fact, any signed sequent can be rewritten as a two-sided sequent and vice versa: iff .
Using these rewriting rules, it is easy to see that the two-sided sequent rules for and that we obtained using our method coincide with the signed sequent rules presented by Wintein and Muskens.

Conclusion
We presented a method for generating two-sided sequent calculi for FDE-like four-valued logics and showed that any truth-functional -ary operator can be characterized with at most four sequent rules. To compare the sequent calculi generated by our method with existing sequent calculi, we applied our method to the four-valued truth tables for , , , and , and found that it gives us rules that are similar or even identical to the ones that have been proposed in the literature.
Moreover, we used our method to study intuitive truth-functional implications in FDE-like four-valued logics. We found that we must make a principled distinction between the pair of positive sequent rules that characterizes the truth-conditions of an implication and the pair of negative sequent rules that characterizes the falsityconditions of an implication. Although any pair of positive sequent rules can be combined with any pair of negative sequent rules, we showed that both positive and negative sequent rules come in pairs: if we choose one sequent rule from a pair, we must also choose the other sequent rule from that pair. We submit that this might be one of the senses of the elusive concept of harmony in proof theory.
We close the paper with an open question. Our method of characterizing single truth table entries with proof rules has been applied to a wide range of truth-functional two-, three-, and four-valued logics. 13 These results can be summarized in a conditional: if a truth-functional many-valued logic has such and such properties, then our method of characterizing single truth table entries is applicable. The open question is about the conditional's converse: if our method of characterizing single truth table entries is applicable to a truth-functional many-valued logic, then what properties does such a logic have?