A Gentzen Calculus for Nothing but the Truth
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Abstract
In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the fourvalued logic for firstdegree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbertstyle axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the BelnapDunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multipletree character—two proof trees may be needed to establish one proof.
Keywords
BelnapDunn logic Gentzen calculi Multiple tree calculi Exactly true logic1 A Signed Sequent Calculus for Exactly True Logic
In Belnap and Dunn’s wellknown fourvalued semantics for the logic of firstdegree entailment FDE (Belnap [3, 4], Dunn [5]) the classical principles of Bivalence (every sentence is true or false) and Noncontradiction (no sentence is both true and false) are given up. This leads to four possible combinations of truth values, as sentences can now be either true and not false (T), false and not true (F), neither true nor false (N), or both true and false (B). These four combinations can be thought of as subsets of the set {1, 0} of classical truth values, so that T can be identified with {1}, F with {0}, N with \(\varnothing \), and B with {1, 0}.

i. ¬φ is true if and only if φ is false, ¬φ is false if and only if φ is true;

ii. φ ∧ ψ is true if and only if φ is true and ψ is true, φ ∧ ψ is false if and only if φ is false or ψ is false;

iii. φ∨ψ is true if and only if φ is true or ψ is true, φ∨ψ is false if and only if φ is false and ψ is false.
A more compact way to characterise the semantics of conjunction and disjunction in the BelnapDunn logic is to say that they correspond to meet and join in the following lattice, called L4 in [3, 4]. Negation corresponds to a topbottom swap—leaving the other two values as they are.
As the authors further point out [12, p130], in particular sequent calculi that enjoy cutelimination and the subformula property qualify as nice.Presumably, this will not make it easy to find a nice sequent calculus for this logic. [12, p129]
But in fact an analytic and cutfree sequent calculus that can characterise ETL already exists, be it that a small modification must be made in order to tailor it to this logic. In Wintein & Muskens [16] we have given a calculus whose rules (restricted to the {∧, ∨, ¬} fragment of the logic considered there) are essentially those of the \(\mathbf {PL}_{\mathbf {4}}^{\mathbf {t}}\) calculus presented in Definition 2 below.^{1} The calculus is based on foursided sequents,^{2} but instead of writing sequents as 4tuples Γ_{1}∣Γ_{2}∣Γ_{3}∣Γ_{4} of sets of sentences, we represent them in an equivalent but more convenient way as finite sets of signed sentences x : φ, where x ranges over a set of four signs and φ is a sentence of \(\mathcal {L}_{t}\).
The four signs we will use are 1, \(\mathsf {\overline {1}}\), 0, and \(\mathsf {\overline {0}}\). While their role in the sequent calculus is purely syntactic, they also have an informal interpretation that is obtained by letting 1 correspond to {T, B}, \(\mathsf {\overline {1}}\) to {F, N}, 0 to {F, B}, and \(\mathsf {\overline {0}}\) to {T, N}, i.e. 1 : φ can be read as ‘ φ is true’ (or, ‘1 is an element of the value of φ’), \(\mathsf {\overline {1}}:\varphi \) as ‘ φ is not true’ (‘1 is not an element of the value of φ’), 0 : φ as ‘ φ is false’, and \(\mathsf {\overline {0}}:\varphi \) as ‘ φ is not false’.
Definition 2 (\(\mathbf {PL}_{\mathbf {4}}^{\mathbf {t}}\)calculus)
It is worth noticing that there is a tight connection between the rules for connectives presented here and Dunn’s evaluation scheme mentioned above. For example rule (∧^{2}) corresponds to the rule that φ ∧ ψ is false (not true) iff φ is false (not true) or ψ is false (not true). Other rule schemes here can be explained similarly. The following is an example of a sequent proof obtained using the \(\mathbf {PL}_{\mathbf {4}}^{\mathbf {t}}\) calculus.
Example 1
And here is an example of a (failed) proof attempt. We will use this and the previous example for an analysis of the antiprimeness of ETL a bit further on.
Example 2
An easy but useful lemma shows that there is a duality between truth and nonfalsity and between falsity and nontruth in this calculus.
Lemma 1
Proof
By an inspection of the sequent rules. □
Let us turn to the connection between the calculus \(\mathbf {PL}_{\mathbf {4}}^{\mathbf {t}}\) and the semantics of the logic. The following definition makes the informal interpretation of the four signs given above explicit by connecting sequents and the valuations refuting them.
Definition 3
The next observation fleshes out the tight connection between the \(\mathbf {PL}_{\mathbf {4}}^{\mathbf {t}}\) calculus and the BelnapDunn truth conditions a bit further.
Lemma 2
For every instantiation of a rule scheme, the bottom sequent is refuted by a valuation V iff one of the top sequents is refuted by V. In case of the (R) rule, which has no top sequents, this boils down to the statement that its bottom sequent is irrefutable.
Proof
By inspection of each of the rule schemes. □
This brings us to the completeness theorem. It already follows from the results in [10] and [16], but for the convenience of the reader we provide a short direct proof here.
Theorem 1 (Soundness, Completeness)
A sequent is provable iff it is irrefutable.
Proof
The ⇒ direction follows easily by an induction on proof depth plus the observation in Lemma 2. For the ⇐ direction, assume that Θ is not provable. We use induction on the total number n of connectives occurring in Θ. If n = 0, Θ is a set of signed propositional constants that is not a conclusion of the (R) rule. This means that Θ does not contain a pair 1 : α, \(\mathsf {\overline {1}}:\alpha \), or a pair 0 : α, \(\mathsf {\overline {0}}:\alpha \). The valuation V such that 1 ∈ V(α)⇔1 : α ∈ Θ and 0 ∈ V(α)⇔0 : α ∈ Θ, for all propositional constants α, refutes Θ.
If n > 0, Θ can be written as a sequent Σ, 𝜃, where 𝜃 is some signed sentence containing at least one connective and 𝜃 ∉ Σ. Inspection of the rules shows that in this case Θ follows from a sequent Θ_{1} or from a pair of sequents Θ_{1} and Θ_{2}, each containing fewer than n connectives. One of these top sequents must be unprovable and hence, by induction, refuted by some valuation V. Lemma 2 gives that Θ is refuted by the same V. We conclude that a sequent is refutable if it is unprovable. □
Remark 1
Proposition 1
Proposition 2
A natural first reaction to the antiprimeness of ETL might be to assume that the feature must be due to the fact that disjunction has an unusual meaning in the logic. In particular it may seem that ETL must assign a different meaning to ∨ than FDE does, as the latter is not antiprime. We do not think that this is a correct analysis of the phenomenon, however. What is it that determines the meaning of a logical connective? Two traditional answers suggest themselves: truth conditions and inferential rules. But the truth conditions of ∨ are the same in the two logics and correspond to join in the L4 lattice, while the inferential rules are likewise the same in our analysis—they are given by the (∨^{1}) and (∨^{2}) rules.
The only point where ETL and FDE differ is in their definitions of entailment^{3}—FDE takes T and B as designated values, while ETL uses only T. It is this feature, we like to argue, that is solely responsible for the difference in behaviour.^{4}
2 Adding Expressivity
ETL was defined in terms of a consequence relation on the language of classical propositional logic \(\mathcal {L}_{t}\), but as is well known this language cannot express all truth functions on 4:={T, B, N, F}. In this section we will consider extensions of ETL to more expressive languages, one containing an extra implication and one that is functionally complete. We will also consider some extensions of FDE to the functionally complete language and compare the logics thus defined.
2.1 Adding an Appropriate Implication Connective
The fact that \(\mathcal {L}_{t}\) does not allow for an appropriate implication connective provides a motivation^{6} to consider an extension of \(\mathcal {L}_{t}\). Let \(\mathcal {L}^{\supset }_{t}\) be the result of extending \(\mathcal {L}_{t}\) with the connective ⊃, whose semantics is given by the following truth table.
Definition 4
Proposition 3
Proof
By inspection. □
Let us also give a syntactic characterisation of this entailment relation. It can be done by first adding rules for ⊃ to the signed sequent calculus.
Definition 5 (\(\mathbf {PL}_{\mathbf {4}}^{\supset }\)calculus)
Proposition 4
2.2 A Functionally Complete Extension
Adding ⊃ to ETL turned out to be plain sailing. What about other connectives? ETL also lacks a strong negation connective and a connective expressing that two sentences are logically equivalent, so one may consider adding those and other operators as well. We will add all possible operators in one fell swoop and define a functionally complete extension \(\mathcal {L}_{ti}\) (the t is for ‘truth’, the i for ‘information’) of \(\mathcal {L}_{t}\), studying ETL entailment on it. This mirrors a move in the literature, where FDE consequence has been studied on functionally complete extensions of \(\mathcal {L}_{t}\) (see e.g. Muskens [9, 10], Arieli and Avron [2], Ruet [15], Pynko [13], or Omori and Sano [11]).
Definition 6
Furthermore, while ¬ corresponds to a topbottom swap in L4, − corresponds to a topbottom swap in A4. Thus, ⊗, ⊕ and − are dual to ∧, ∨ and ¬ in a sense that is aptly explained by referring to the lattices L4, associated with the “truth order on 4” and A4, associated with the “information order on 4”.
Proof
Note that a ≤ _{ t } b iff a ∈ {T, B} ⇒ b ∈ {T, B} and a ∈ {T, N} ⇒ b ∈ {T, N} both hold. □
The expressibility of B and N in \(\mathcal {L}_{ti}\) depends on the presence of ⊗ or ⊕ and it are these connectives that make Open image in new window and Open image in new window come apart. The two relations are coextensional on the {¬, ∧, ∨, −} fragment of \(\mathcal {L}_{ti}\).
Proposition 5
Let φ be a sentence of \(\mathcal {L}_{ti}\) that only contains the connectives ∧, ∨, ¬ and − and let Γ be a set of such sentences. Then Open image in new window .
Proof
For any valuation V, let V ^{′} be the unique valuation such that, for all propositional constants p, V ^{′}(p) = V(−p) (V ^{′} is the valuation we have earlier called the dual of V). By inspection of the truth tables it is easily seen that, for any φ in the {¬,∧,∨,−} fragment of the language, V ^{′}(φ) = V(−φ). This means that if V is a counterexample to Open image in new window , V ^{′} Open image in new window is a counterexample to and vice versa. □
So which of the three entailment relations for extended FDE considered above is the ‘right’ one? Our vote goes to Open image in new window for reasons of symmetry and because it is based on the L4 lattice that is so central to the logic. A definition based on L4 is also the one explicitly endorsed by Belnap [3, 4] for standard FDE.^{10} (See also [10, 16]).
Proposition 6
Proof
The first claim follows from an inspection of the definitions. The second from the observation that Open image in new window but not Open image in new window . The third from the fact that Open image in new window but not Open image in new window , where n is defined as −b. □
Readers familiar with Arieli and Avron [1] will be interested to verify that the pair 〈F O U R ^{−},{T}〉, which induces Open image in new window , is not an ultralogical bilattice (the reason being that the constraint a∨b ∈ {T} ⇔ a ∈ {T} or b ∈ {T} is violated). This means that Arieli and Avron’s sequent calculus for ‘ultralogical bilattice logic’ does not characterise Open image in new window .
All consequence relations discussed until now can be captured syntactically by the P L _{ 4 } calculus below, which is (a notational variant of) the propositional part of the calculus considered in [16]. Sequents are redefined in the obvious way—as sets of pairs x : φ, where x is a sign and φ now is an \(\mathcal {L}_{ti}\) sentence.
Definition 7 (PL_{4}calculus)
The definitions of derivation, proof tree, and provability of a signed sequent given in Definition 2 are extended in the obvious way again.
The good news at this point is that Theorem 1 generalises to the present setting. If Definition 3 is generalised in the obvious way, inspection of the additional rule schemes presented above will tell that Lemma 2 still goes through, so that the proof of Theorem 1 can remain unchanged.
These relations indeed characterise the entailment relations they were meant to.
Proposition 7
Open image in new window , Open image in new window , Open image in new window
Proof
From Theorem 1, the equivalence in Eq. 15, and the relevant definitions. □
And again we have:
Proposition 8
Proof
Using Theorem 1. □
3 Conclusion
We have given a characterisation of Pietz and Rivieccio’s [12] Exactly True Logic by means of (a notational variant of a fragment of) the analytic and cutfree signed sequent calculus presented in [16]. This calculus was originally devised in order to characterise a generalisation of the logic of firstdegree entailment in a functionally complete language. That it can also be used for ETL rests on the fact that there is some leeway in mapping unsigned sequents to the signed sequents that need to be proved. One signed calculus can therefore model more than one logic.
The antiprimeness of ETL, the fact that, as Pietz and Rivieccio observe, Eq. 5 does not hold while Eq. 4 does, will only be a stumblingblock for characterising the logic if it is insisted upon that the calculus for doing that is a traditionally twosided (or twosigned) one. Our proofs for ETL are based upon a foursigned calculus in which premises are marked with two distinct signs. In this setup all mystery about why Open image in new window and Open image in new window have proofs while Eq. 5 leads to a failed proof attempt and hence to refutability has disappeared.
We have also extended ETL to a functionally complete language and have compared it with FDE in this setting. Proving ETL consequence in the presence of the ‘consensus’ operator ⊗, or its ‘gullibility’ dual ⊕, in general requires the development of two proof trees in our system, as does proving FDE entailment in this context.
Footnotes
 1.
\(\mathbf {PL}_{\mathbf {4}}^{\mathbf {t}}\) stands for ‘4valued propositional logic for \(\mathcal {L}_{t}\)’. The signs considered in the calculus in [16] are pairs whose first element is either n (north) or s (south) and whose second element is w (west) or e (east). The 1 sign of the present calculus corresponds to the northwest sign of the calculus in [16] , 0 to southwest, \(\mathsf {\overline {1}}\) to northeast, and \(\mathsf {\overline {0}}\) to southeast.
 2.
For the idea of nsided sequents, see Rousseau [14], where classical (twosided) sequents Γ_{1}⊩Γ_{2} are generalised to sequents of the form Γ_{1}∣⋯∣Γ_{ n }.
 3.
If entailment is understood syntactically, the difference is reflected by the initial assignments of signs to the premisses and conclusion of an argument whose validity is under consideration.
 4.
In a similar vein, Hjortland [8] argues that Strong Kleene Logic and the Logic of Paradox agree on the meaning of the logical connectives but disagree on the meaning of logical consequence.
 5.
In contrast, FDE does not have a resolution theorem (nor a deduction theorem) and there is no \(\mathcal {L}_{t}\) definable connective that allows FDE to enjoy modus ponens. In particular, modus ponens expressed in terms of \(\rightsquigarrow \) is equivalent to disjunctive syllogism, which is invalid in FDE (but valid in ETL).
 6.
In the context of FDE, an entirely similar motivation leads Arieli and Avron [1] to consider an extension of \(\mathcal {L}_{t}\) with an appropriate implication connective. In fact, the semantic definition of our ⊃ as given in Definition 4 can be considered to be the ETL counterpart of the connective considered by Arieli and Avron.
 7.
However, the set is not a minimal functionally complete set of connectives of 4 as it has proper subsets that can also express all the truth functions over 4. The set of connectives {∧, ¬, ⊗, −} is minimal in this sense though, as can easily be shown.
 8.
We use the same notation for connectives of \(\mathcal {L}_{ti}\) and the truth functions that they denote here, trusting that this does not cause any confusion.
 9.
Wherever we superscript a syntactic or semantic consequence relation with ti, the relation is to be understood as defined between sets of sentences and sentences of \(\mathcal {L}_{ti}\).
 10.See, e.g., [4, p. 15] while on page 43 of [3] we find:
But I agree with the spirit of a remark of Dunn’s, which suggests that the False really is on all fours with the True, so that it is profoundly natural to state our account of “valid” or “acceptable” inference in a way which is neutral with respect to the two.
Notes
Acknowledgments
We would like to thank the referee for encouraging words and helpful comments. Stefan Wintein wants to thank the Netherlands Organisation for Scientific Research (NWO) for funding the project The Structure of Reality and the Reality of Structure (project leader: F. A. Muller), in which he is employed. Reinhard Muskens gratefully acknowledges NWOs funding of his project 36080050, Towards Logics that Model Natural Reasoning
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