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7Логические исследования 2021. Т. 27. № 1. С. 31-45 УДК 16+510.6
Logical Investigations 2021, Vol. 27, No. 1, pp. 31-45 DOI: 10.21146/2074-1472-2021-27-1-31-45
ÂRNON ÂVRON
Implication, Equivalence, and Negation
Arnon Avron
School of Computer Science, Tel Aviv University. Tel Aviv 6997801, Israel. E-mail: [email protected]
Abstract: A system HCL — in the language of {—, o} is obtained by adding a single negation-less axiom schema to HLL — (the standard Hilbert-type system for multiplicative linear logic without propositional constants), and changing ^ to o. HCL — is weakly, but not strongly, sound and complete for CL — (the {—, o}-fragment of classical logic). By adding the Ex Falso rule to HCL — we get a system with is strongly sound and complete for CL —. It is shown that the use of a new rule cannot be replaced by the addition of axiom schemas. A simple semantics for which HCL — itself is strongly sound and complete is given. It is also shown that LHCL — , the logic induced by HCL —, has a single non-trivial proper axiomatic extension, that this extension and CL — are the only proper extensions in the language of {—, o} of Lhcl — , and that Lhcl — and its single axiomatic extension are the only logics in {—, o} which have a connective with the relevant deduction property, but are not equivalent to an axiomatic extension of R — (the intensional fragment of the relevant logic R). Finally, we discuss the question whether Lhcl — can be taken as a paraconsistent logic.
Keywords: Implication, Semi-implication, Negation, Equivalence, Biconditional, Classical propositional logic, Deduction theorems, Paraconsistent Logics
For citation: Avron A. "Implication, Equivalence, and Negation", Logicheskie Issledo-vaniya / Logical Investigations, 2021, Vol. 27, No. 1, pp. 31-45. DOI: 10.21146/20741472-2021-27-1-31-45
1. Introduction
The relevant deduction property (RDP) for a binary connective ^ is a weak form of the classical-intuitionistic deduction theorem which has (somewhat implicitly) motivated the design of the intensional fragments (R^ and R) of the relevance logic R ([Anderson, Belnap, 1975; Dunn, Restall, 2002]).1 In [Avron, 2015] we showed that with one exception, in the pure language of ^ has in
xThe RDP is also the key condition that should be satisfied by what is called in [Avron, 2015] 'semi-implication'. The other condition, included to avoid degenerate cases, is that not every case in which an implication holds is a case in which its converse also holds.
© Avron A.
a finitary logic L the RDP iff L has a strongly sound and complete Hilbert-type system which is an axiomatic extension (i.e. an extension by axiom schemas) of HR^ (the standard axiomatization of R^). The only exception is a logic which, like R^, is an axiomatic extension of LL^ (the purely implicational fragment of Linear Logic): CL^, the pure equivalential fragment of classical logic (where the biconditional is denoted by
The fact that o has in CL^ the RDP raises the question to what extent it can actually be used as an implication connective. A crucial criterion here is the richness of the languages in which it might serve as such, that is: what useful connectives can be added to it. It can easily be seen that it is impossible to add to CL^ a 'conjunction' A such that both tp A ^ o <p and <p A ^ o ^ would be valid, since this would immediately trivialize the logic. Similarly, one cannot add to CL^ a 'disjunction' V such that <p o <p V ^ and ^ o <p V ^ would be valid. It follows that among the connectives used in linear logic and in relevance logics, one may add to CL^ only what are called in linear logic 'multiplicative' connectives, and in relevance logics 'intensional' connectives. The most basic such connective is negation (which together with the implication connective of linear and relevant logics suffices for defining the rest of them). Accordingly, the main goal of this paper is to investigate the equivalence-negation fragment of classical logic, and corresponding proof systems.
2. Preliminaries
In the sequel L is a propositional language, p, d vary over its formulas, p, q over its atomic formulas, and T, S over its theories (i.e. sets of formulas).
Definition 1. A (Tarskian) consequence relation for a language L is a binary relation between theories in L and formulas in L satisfying the following three conditions:
[R] Reflexivity: ^ h ^ (i.e. h
[M] Monotonicity: if T h ^ and T C T', then T' h
[C] Cut (Transitivity): if T h ^ and T', ^ h p then T UT' h p.
Definition 2. Let h be a Tarskian consequence relation for L.
• h is structural, if for every L-substitution d and every T and if T h ^ then 0(T) h d(^).
• h is non-trivial if p h q for distinct atomic formulas p, q.
• h is finitary, if for every theory T and every formula ^ such that T h ^ there is a finite theory r C T such that r h
Definition 3. A (propositional) logic is a pair L = (L, hL), where L is a pro-positional language, and hL is a structural and non-trivial Tarskian consequence relation for L.2 A logic L = (L, hL) is finitary if hL is finitary.
Definition 4. Let L = (L, hL) be a propositional logic, and let D be a (primitive or defined) connective of L. L has the relevant deduction property (RDP) for D if it satisfies the following condition:
T,f I~l 0 iff either T I~l 0 or T I~l f D 0.
Remark 1. If a finitary logic L has a connective D with the RDP, then the following holds for every theory T and formula f: T I~l f iff there exist 01 ,..., 0n e T (n > 0) such that hL 0i D (02 D (■ ■ ■ (0n D f) ■ ■ ■)).
Definition 5. Let L^ = L^ = {o}.
1. HLL^ is the system in L^ presented in Figure 1.
Axioms:
[Id] f ^ f (Identity)
[Tr] (( ^ 0) ^ ((0 ^ 9) ^ (( ^ 9)) (Transitivity)
[Pe] (f ^ (0 ^ 9)) ^ (0 ^ (f ^ 9)) (Permutation)
Rule of inference:
[MP]
( ^ 0
0
Fig. 1. The proof system HLL^
2. HR^ is the extension of HLL^ by the following axiom:
[Ct] (f ^ (f ^ 0)) ^ (f ^ 0) (Contraction)
3. HLL^ is the system in L^ which is obtained from HLL^ by using 'o' instead of HCL^ is the extension of HLL^ by the following axiom:
[Eq] (f o (f o 0)) o 0 (Equivalence)
2This is the notion of propositional logic which has been used in [Avron, 2015], as well as in [Avron et al., 2018].
The following three theorems have been proved in [Avron, 2015].
Theorem 1. A logic L is finitary and has a connective ^ which has in L the RDP iff L has a strongly sound and complete Hilbert-type system which is equivalent to an extension by axiom schemas of either HR^ or HCL^.3
Theorem 2. HCL^ is strongly sound and complete for the equivalence fragment of classical logic (i.e. T \~hcl„ P iff by interpreting ^ as the classical biconditional we get that every assignment that satisfies T also satisfies p).
Theorem 3. CL^ has no proper extension in its language.4
3. The Logic CL^ and the System HCL^
Definition 6. CL — is the equivalence-negation fragment of classical logic. Mc^ is the two-valued matrix which induces CL - .
Proposition 1. CL — does not have the RDP.
Proof. Although —p, p /cl^ q, neither —p /cl^ q, nor —p /cl^ P o q. ®
Definition 7. Let L = (o, —}. The Hilbert-type proof system HCL — is presented in Figure 2. Lhcl^ is the logic induced by HCL —.
Remark 2. The axioms given in Figure 2 are actually not independent, since [N2] can be dropped. This can be seen by substituting —^ for p in both [Id] and [N1]. By applying [MP] to the resulting formulas we get ^ o ——-0. From this we can get [N2] by using HCL^ and Theorem 2.5
Remark 3. In Chapter 11 of [Avron et al., 2018] a notion was introduced of a negation associated with a given binary connective that has the RDP. It was shown there that a logic possesses a binary connective ^ that has the RDP together with a negation — associated with it iff it is induced by some axiomatic extension of the system HLL —. The latter is the standard Hilbert-type system (given in [Avron, 1988]) for the multiplicative fragment (without the multiplicative constants) of linear logic ([Girard, 1987]). It is obtained from HCL — by deleting the axiom [Eq], and then changing o in all axioms and rules to By adding to HLL — the contraction axiom, we get the standard Hilbert-type system for , the intensional fragment of the relevance logic R.
3It seems that the RDP was first raised as being of interest to relevance logic in [Diaz, 1980].
4 A weaker result, that is Post-complete in the sense that one cannot (consistently)
add any new axiom to it in its language, had already been shown by Prior in [Prior, 1962].
51 am indebted to an anonymous referee for this observation.
Axioms:
[Id] f o f
[Tr] (f o 0) o ((0 o 9) o (f o 9))
[Pe] (f o (0 o 9)) o (0 o (f o 9))
[Eq] (f o (f o 0)) o 0
[N1] (f o -0) o (0 o -f )
[N2] --f o f
(Identity)
(Transitivity)
(Permutation)
(Equivalence)
(Contraposition)
(Double neg.)
Rule of inference:
[MP],
f o 0
0
Fig. 2. The proof system HCL^
Theorem 4. Let the logic L be induced by some axiomatic extension (i.e. extension by axiom schemas) of HCL —. Then L has the RDP for o.
Proof. Immediate from Theorem 1. ■
Theorem 5. HCL — is strongly sound for CL—. (I.e., if T\~hcl^ f then by interpreting o as the classical biconditional o, and — as the classical negation, we get that every assignment that satisfies T also satisfies f.)
Proof. Obviously, [MP]^ is a valid rule of inference for the classical biconditional o. It is also easy to check that every axiom of HCLbecomes a classical tautology if o is interpreted as the classical biconditional. Hence HCLis strongly sound for the equivalence-negation fragment of classical logic. ■
Corollary 1. Let f be a formula in the language of CL.
1. There is a formula 0 in the language of CL^ such that:
• If the number of negations in f is even then ICL^ f o 0.
—
• If the number of negations in f is odd then ICL^ f o —0.
—
2. f is a classical tautology iff the number of negations in f is even, and for each atomic p the number of occurrences of p in f is even too.6
6According to [Church, 1956], the second item of this Corollary has independently been observed by McKinsey and Mihailescu. As far as I know, the first item was first proved in [Mihailescu, 1937].
Proof. Obviously, HCL — has the replacement property. (This is true for every axiomatic extension of HLL —.) It is also easy to show that d1 o -02 and o 62 are both equivalent in HCL — to o 02). Using these two equivalences and axiom [N2], we can constructively find for p a formula ^ in the language of CL— such that \~hcl — P o ^ or \~hcl — P o according to the parity of p's number of negations. Hence Theorem 5 implies item 1.
For the second item, note that if ^ is in the language of CL—, then is not a tautology, since we can refute it by assigning t to all atomic formulas. Therefore the second part follows from the first, using Lesniewski's famous criterion for being a tautology in o. (See e.g. Corollary 7.31.7 in [Humberstone, 2011].) ■
Theorem 6. No Hilbert-type system which has [MP] for o as its sole rule of inference can be strongly sound and complete for CL —.
Proof. Suppose that such a system H exists. From Theorem 5 it follows that we may assume that H is an axiomatic extension of HCL^. Hence Theorem 1 implies that H has the RDP. Therefore it follows from the strong soundness and completeness of H that so does CL —, This contradicts Proposition 1. ■
Corollary 2. Lhcl — = CL—, i.e. HCL— is not strongly complete for CL— .7
Next we present semantics for HCL — for which this system is strongly sound and complete.
Definition 8. Let L = {o, -}. CL{—,id} is the two-valued logic which is obtained by interpreting o as the classical biconditional o, and - as the identity connective. id) is the two-valued matrix which induces CL{—,id}.
For the next proof, we need the following easy lemma, which is directly proved in [Avron, 2015] (but also easily follows via an appeal to Lesniewski's criterion, mentioned at the end of the proof of Corollary 1).
Lemma 1.
1. \rcl^ (p o o (^ o p)
2. \rcl^ P o (^ o (p o
Theorem 7. Lhcl — = CL — n CL{—,id}. In other words: T \~hcl — P iff T \cl — p and also T \cl{„—} P- "
7This fact was first noticed in [Avron, 2020] (Corollary 14).
Proof. The strong soundness of HCLfor follows from Theorem 2.
This and Theorem 5 imply the strong soundness of HCL — for CL—nCL{^,id}. To prove strong completeness, assume that T l/HCL— 0. Extend T to a maximal theory T* such that T* l/HCL— 0. Obviously, p G T* iff T* bHCL— p,
and p G T* iff T*, p bHCL^ 0. Therefore the RDP implies that
—
(*) p GT* iff p о 0 gT* Now define a valuation v as follows:
f t if p gT*
v(p) = < (p) \ f if p GT*
Obviously we have:
(**) v(p) = t for every p G T*, while v(0) = f. Next we show that v respects the truth table of the classical biconditional.
• Suppose v(p) = v(0) = t. Then p G T* and ф G T*. Therefore it
follows from the second item of Lemma 1 that T* bHCL^ p о ф. Hence
—
p о ф gT*, and so v(p о ф) = t.
• Suppose v(p) = t and v^) = f. Then p G T*, while ф G T*. Because of the presence of [MP]^, these facts immediately imply that p о ф G T*, and so v(p о ф) = f in this case.
• Suppose v(p) = f and v(ф) = t. By the previous item this implies that ф о p G T*. Therefore the first item of Lemma 1 implies that p о ф G T*, and so v(p о ф) = f in this case too.
• Suppose v(p) = v^) = f. Then p G T* and ф G T*. By (*) above, it follows that p о 0 G T* and ф о 0 G T*. By the first item of Lemma 1, the second fact implies that 0 о ф G T*. Using [Tr] this last fact and the fact that p о 0 G T* together imply that p о ф G T*, and so v(p о ф) = t in this case.
To determine the behavior of v with respect to - we have two cases to consider.
-0 G T*
• Suppose v(p) = t. Then p G T*. By the second item of Lemma 1, this implies in this case that p о -0 G T*. It follows by [N1] that 0 о -p G T*, and so -p о 0 G T* by the first item of Lemma 1. Hence (*) entails that -p G T*, implying that v(-p) = f in this case.
• Suppose v(p) = f. Then p e T*. Hence (*) entails that p о в e T*. By [N1] and [N2], this implies that о -p e T*, and so -p e T* (since we are assuming that -в e T*). Therefore v(-p) = t in this case.
It follows that v is a legal valuation of the classical equivalence-negation
matrix Mc^ . Hence T I/cl^ в in this case.
— —
-в eT *
• Suppose v(p) = t. Then p e T*. Suppose that -p e T*. Then by (*), -p о в e T*, and so the first item of Lemma 1 implies that в о -p e T*. Hence [N1] entails that p о-в e T*. Since p e T*, -в e T* too. A contradiction. It follows that -p e T*, and so v(-p) = t.
• Suppose v(p) = f. Then p e T*, implying by (*) that p о в e T*. Hence the first item of Lemma 1 implies that в о p e T*. Using [N1], [N2], and Lemma 1, we get from this that -p о-в e T*. Since -в e T*, this means that -p e T* too, and so v(-p) = f.
We have shown that v is a legal valuation of the classical equivalence-
identity matrix MCL{H id}. Hence T* /cl{„ id} в in this case.
It follows that if T I/HCL в then either T /cl в or T /cl • л в. ■
4. Proof-theoretical Characterizations of CL— Theorem 8. T/cl^ p iff T, -p /hc^ p.
-H- -H-
Proof. Suppose T/cl- p. Then T, -p /cl- p. Obviously, T, -p /cl{„,id} p as well. Hence Theorem 7 implies that T, -p /HCL— p.
For the converse, let T, -p /HCL^ p. Suppose that T I/CL^ p. Then
— —
there is a a valuation v in MCL — such that v(^) = t for every ^ e T, while v(p) = f. The latter fact implies that v(-p) = t, and so v is a model in MCL — of Tu {-p} which is not a model in MCL — of p. Since T, -p /HCL — p, this contradicts Theorem 5. ■
Theorem 9. HCL - is weakly complete for CL - : /hcl^ p iff /cl^ p-^ ^ — —
Proof. Theorem 5 implies the 'only if' part. For the converse, let /cl — p. By Theorem 8, it follows that -p /hcl — p. Hence Theorem 4 implies that either
/hcl^ -p о p, or /hcl^ p. The first option is impossible by Theorem 5,
— —
since v(-p о p) = f for every valuation v in MCL — . Hence /HCL — p. ■
Remark 4. Theorem 9 easily follows also from Corollary 1 and Theorem 2.8
Theorem 10. Let HCLbe obtained from HCL— by adding to it the Ex Falso rule —- as a rule of inference. Then HCL* is strongly sound and complete for CL.
Proof. That HCL* is strongly sound for CLfollows from Theorem 5, and the strong soundness in classical logic of the special rule of HCL* .
To show strong completeness, let T —cl — p. Then T, —p i~hcl — p by
Theorem 8. By Theorem 4, either T —HCL^ —f o f, or T —HCL^ p. In the
— —
second case we are done. Assume the first. Since —cl^ — (—p o p), it follows
—
by Theorem 9 that T—HCL — — (—p o p) as well. Hence an application of the Ex Falso rule of HCL* yields that T* —cl^ p. ■
i— —
Corollary 3.
1. If T —HCL*_ p, then there is a proof of p from T in HCL* that includes at most one application, made at the end of that proof, of its extra rule.
2. T — cl-, p iff either T—hcl^ p or T is inconsistent in HCL —.
— — 1
Proof. The first part easily follows from the proof of Theorem 10. The second one follows from that theorem together with the first part. ■
Turning to a corresponding Gentzen-type System, we note that in [Avron, Lev, 2005] an algorithm has been given for finding a cut-free sound and complete Gentzen-type system for every logic which has a two-valued characteristic matrix (or even non-deterministic matrix). By applying that algorithm to CL, we get the system GCL — presented at Figure 3. In this presentation r and A vary over finite sets of formulas.
Theorem 11.
1. GCL — is strongly sound and complete for CL.
2. The cut-elimination theorem obtains for GCL — : if —gcl-, r ^ A, then r ^ A has a cut-free proof in GCL —.
Proof. This is a special case of Theorem 4.7 of [Avron, Lev, 2005] and its proof. ■
8Theorem 9 was essentially first proved in [Mihailescu, 1937]. (See also [Bennett, 1937].) The system used there is easily seen to be equivalent to HCL - .
Axioms: p ^ p
Rules: cut, weakening, and the following logical rules:
r,p,0 ^ A r ^ A,p,0 r,0 ^ A, p r,p ^ A,0
r, p o 0 ^ A
r ^ A, p o 0
r ^ A, p r, — p ^ A
r,p ^ A r ^ A, — p
Remark 5. A model of a sequent r ^ A in a two-valued matrix M is usually taken to be a valuation v in M such that v(p) — f for some p G r, or v(p) — t for some p G A. Let S U {s} be a set of sequents. Define: S — m s if every model of S in M is also a model of s. The first item of Theorem 11 means that S —cl — s iff S —GCL — s.
5. The Expressive Power of CL^—
To make our treatment of CLcomplete, we include also a characterization of the set of two-valued connectives that are definable in the language of CL. For this it would be convenient to use 1 and -1 as our truth-values.
Definition 9. Let H : {1, — 1}n ^ {1, -1}, and let x\,...,xn be n variables.
• For 1 < i< n, Xi is a dummy variable of H (xi, ...,x,n) if
H(x1 , ... , Xi— 1 , xi , Xi+1 , ... , Xn) - H(x1, . . . , xi— 1 , Xi-, xi+1, . . . , xn)
for every x1,x2,... ,xn G {—1,1}.
• For 1 i n, Xi is a flipping variable of H (x1,.. .,Xn) if
H (x1 , ... , xi— 1, xi, Xi+1 , ... , Xn) - H (x1 , ... , xi— 1, xi, Xi+1 , ... , xn)
for every x1,x2,... ,xn G {—1,1}.
Theorem 12. A function H : {1, —1}n ^ {1, —1} is definable by a formula of CLiff for every 1 < i < n, xi is either a dummy variable or a flipping variable of H(x1,..., xn) .9
9This theorem can most probably be extracted from Post's discussion in [Post, 1941].
Proof. That any function which is definable by a formula p of CL — satisfies the condition is easily proved by induction on the structure of p.
For the converse, suppose H : {1, — 1}n ^ {1, —1} satisfies the condition. Let 0 = 01 o ■ ■ ■ o where — = pi o pi if Xi is a dummy variable of H(x1,... , xn), — = pi if x is a flipping variable of H(x1,... ,xn). It is not difficult to see that 0 defines H in case H(1,1,..., 1) = 1, while defines H incase H(1,1,..., 1) = —1. ■
Corollary 4. The classical conjunction, disjunction, and (material) implication are not definable by a formula of CL - .10
6. Extensions of Lhcl^
—
Theorem 13. Let HCL— be obtained from HCL- by adding to it p o -p as an axiom. Then HCL^ is strongly sound and complete for CL{—,id}.
Proof. The proof is almost identical to that of Theorem 7. We only need to observe that in the presence of the additional axiom, the case in which -0 e T* is impossible. Hence we remain with the case in which T* l/CL{„ id} 0. ■
Theorem 14. CL — and CL{—are the sole non-trivial proper extensions
of Lhcl^ in its language.
—
Proof. Let L be a logic in the language of HCL — which is a proper extension of Lhcl — . Then there is a theory T and a formula 0 such that T lL 0,
but T I/hcl^ 0. By Theorem 7 there is an assignment v in either Mc^ or
— —
MCL{^ id} such that v(p) = t for every p eT, while v(0) = f. Pick some atomic formula p, and define a substitution S by:
S(q) = ( p 0 p if V(q)= t [ p if v(q) = f
Let T* = {S(p) | p eT}, 0* = S(0). Since L is a logic, T* lL 0*. On the other hand, the obvious fact that v(S(p)) = v(p) for every formula p implies
that v |= T*, while v |= 0*. It follows by Theorem 7 that T* I/hcl^ 0*.
—
Now from Theorem 9 it follows (and it can also easily be shown directly) that every formula which has p as its sole atomic subformula is equivalent in HCL— to one of the formulas in {p, -p,p o p, -(p o p)}. Hence T* U {0*} is a subset of this set. Moreover, the fact that lHCL — p o p implies that 0* = p o p, and
10This result has been shown directly in [Massey, 1977]. It is also proved there that there is no single truth-function that generates precisely the functions definable by formulas of CL - .
that we may assume that p o p G T*. Hence T* U {0*} C {p, —p, —(p o p)}.
However, since — hc^ p o — p o —(p o p) by Theorem 9, any element of
—
{p, — p, —(p o p)} follows in HCL — from the other two. Therefore we remain with the following six cases.
p —l —(p o p): Substituting p o p for p, we get that —l —((p o p) o (p o p). But by Theorem 9, —((p o p) o (p o p) is equivalent in HCL— to —p o p, and so —l — p o p. Hence L extends CL{^,id}.
p —l —p: Substituting p o p for p, we again get that CL{^,id} C L.
—p —l —(p o p): Substituting —(p o p) for p, we again get that CL{^,id} C L.
—p —l p: Substituting —(p o p) for p, we again get that CL{^,id} C L.
—(p o p) —l p: Since — hcl^ q o — q o (—(p o p)) by Theorem 9, we get
—
that q, —q —l p. Hence L extends CLin this case (by Theorem 10).
—(p o p) —l —p: Here we get by a similar argument that q, —q —l —p. Substituting —p for p, it follows (using [N2]) that q, —q —l p. Hence L extends CL — in this case too.
We have shown that L extends either CLor CL{^,id}. However, since these are two-valued logics, neither of them has a proper non-trivial extension, by a general theorem in [Rautenberg, 1981]. (These two cases can also be shown directly using an analysis which is very similar to — though shorter and easier
than — that given above for Lhcl^ .) Hence L is necessarily one of them. ■
—
Corollary 5. HCLi<d is the sole non-trivial proper axiomatic extension of HCL — n
Remark 6. Recall that theorems 2 and 3 imply Prior's result that HCL^ is Post-complete (footnote 4). Corollary 5 means that in contrast, HCL — is not Post-complete.12 However, the difference from HCL^ is small: HCL— has just one proper axiomatic extension.
7. Is Lhcl^ a Paraconsistent Logic?
In Chapter 2 of [Avron et al., 2018] a propositional L for a language with a unary connective — is defined to be —-paraconsistent if p, —p —L q whenever p
and q are distinct variables, and — is a negation of L. Lhcl^ certainly satisfies
—
nThis fact was first proved in [Avron, 2020] (Theorem 16).
12That CL — has no Post-complete axiomatization has already been noted in [Prior, 1962].
the first condition.13 Hence the question whether it is --paraconsistent depends on whether its connective - can be viewed as a negation. This, in turn, depends of course on the definition of negation that one adopts.
In the literature one can find many different definitions of "negation" in L. Some make very weak demands. The minimal ones might be that if p is atomic then p |l -p and -p |L p. A more extensive set of negative conditions of this sort (divided into two groups, called 'verificatio' and 'falsificatio') is given in [Marcos, 2005]. It is also possible to add some positive conditions, like that p lL --p and --p lL p. All these conditions are satisfied in Lhcl —. So according to weak definitions of this sort, Lhcl — is indeed a --paracon sistent logic.
In [Avron et al., 2018], a more restrictive definition of negation has been used. - is called there a negation for L if it is possible to define in the language of L a binary connective o which is either a disjunction for L, or a conjunction for L, or a semi-implication for L, such that the {-, o}-fragment of L is contained in the corresponding fragment of classical logic. In order to see
that no such connective o is available in Lhcl^ , we do not need to repeat
—
the definitions given in Chapter 1 of [Avron et al., 2018] of these notions. It suffices to recall the following facts about them. (They all easily follow from the definitions.)
• If a connective A is a conjunction for a logic L, then for every p and 0: p A 0 lL p, p A 0 lL 0, and p lL p A p. From Theorem 7 it easily follows that such a connective should have in MCL — the truth-table of the classical conjunction. Hence Corollary 4 implies that no such connective
is available in Lhcl^ .
—
• If a connective V is a disjunction for a logic L, then for every p and 0: p lL p V 0, 0 lL p V 0, and p V p lL p. From Theorem 7 it easily follows that such a connective should have in MCL — the truth-table of the classical disjunction. Hence Corollary 4 implies t fiat no such connective
is available in Lhcl^ .
—
• If a connective D is a semi-implication for a logic L, then it has in L
the RDP. Suppose now that D is such a connective in Lhcl^ . Then
—
lHCL — p D p for every atomic p. Hence lHCL — (p D q) D (p D q) (by structurality). Therefore two applications of the RDP for D yield
p,p D q Ihcl^ q. By two other applications of the RDP, this time for o,
—
13In fact, Lhcl ^ satisfies the stronger condition p, —p \fL —q. So if we accept its connective — as a negation, then according to [Avron et al., 2018] it would even be strongly —-paraconsistent.
we get that —HCL — (p D q) o (p o q). It follows that <p D 0 and <p o 0
are equivalent in Lhcl^ . Therefore — hcl^ (p D (p D q)) D q. Hence
— —
the {—, D}-fragment of Lhcl — is not contained in the corresponding fragment of classical logic.
It follows from the above considerations that Lhcl^ is not a paraconsistent
—
logic according to the definition used in [Avron et al., 2018].
Acknowledgements. I am very grateful to Lloyd Humberstone for many most helpful
comments, suggestions, and pointers to the literature. This research was supported
by The Israel Science Foundation (grant no. 817-15).
References
Anderson, Belnap, 1975 - Anderson, A.R., Belnap, N.D. Entailment: The Logic of Relevance and Necessity. Vol. I. Princeton University Press, 1975.
Avron, 1988 - Avron, A. "The Semantics and Proof Theory of Linear Logic", Theoretical Computer Science, 1988, Vol. 57, pp. 161-184.
Avron, 2015 - Avron, A. "Semi-implication: A chapter in universal logic", The Road to Universal Logic, Volume I., ed. by A. Koslow and A. Buchsbaum. Studies in Universal Logic, Birkhauser, 2015, pp. 59-72.
Avron, 2020 - Avron, A. "A Note on Semi-implication with Negation", Abstract Consequence and Logics: Essays in Honor of Edelcio G. de Souza, ed. by A. CostaLeite. Tributes 42, College Publications, 2020, pp. 221-226.
Avron et al., 2018 - Avron, A., Arieli, O., Zamansky, A. Theory of Effective Proposi-tional Paraconsistent Logics. volume 75 of Studies in Logic, (Mathematical Logic and Foundations), College Publications, 2018.
Avron, Lev, 2005 - Avron, A., Lev, I. "Non-Deterministic Multiple-valued Structures", Journal of Logic and Computation, 2005, Vol. 15, pp. 241-261.
Bennett, 1937 - Bennett, A.A. Review of [Mihailescu, 1937], Journal of Symbolic Logic, 1937, Vol. 2, No. 4, p. 173.
Church, 1956 - Church, A. Introduction to Mathematical Logic. Princeton University Press, 1956.
Diaz, 1980 - Diaz, M.R. "Deductive Completeness and Conditionalization in Systems of Weak Implication", Notre Dame Journal of Formal Logic, 1980, Vol. 21, pp. 119-130.
Dunn, Restall, 2002 - Dunn, J.M., Restall, G. "Relevance logic", Handbook of Philosophical Logic, Second edition, Vol. 6, ed. by D. Gabbay and F. Guenthner. Kluwer, 2002 pp.1-136.
Girard, 1987 - Girard, J.Y. "Linear logic", Theoretical Computer Science, 1987, Vol. 50, pp. 1-102.
Humberstone, 2011 - Humberstone, L. The Connectives. The MIT Press, 2011.
Marcos, 2005 - Marcos, J. "On negation: Pure local rules", Journal of Applied Logic, 2005, Vol. 3, pp. 185-219.
Massey, 1977 - Massey, G.J. "Negation, Material Equivalence, and Conditioned Non-conjunction: Completeness and Duality", Notre Dame Journal of Formal Logic, 1977, Vol. 18, pp. 140-144.
Mihailescu, 1937 - Mihailescu, E.Gh. "Recherches sur la negation el l'equivalence dans le calcul des proposition", Annales scientifiques de l'Université de Jassy, 1937, Vol. 23, pp. 369-408.
Post, 1941 - Post, E. The Two-Valued Iterative Systems of Mathematical Logic. Annals of Mathematics Series, Vol. 5, Princeton University Press, Princeton, NJ, 1941.
Prior, 1962 - Prior, A.N. Formal Logic, second edition. Oxford, 1962.
Rautenberg, 1981 - Rautenberg, W. "2-Element Matrices", Studia Logica, 1981, Vol. 40, pp. 315-353.
