In this paper I study the dialectical logics DL* and HEGFL. I compare some of the most distinguishable properties of both logics and I present some uknown features of HEGFL with respect to classical logic, weakly connexive logics and two negation-like connectives.
HEGFL Paraconsistent Nega
In this paper I study two dialectical logics, DL* and HEGFL. The former
was introduced in [
Both are contra-classical logics, that is, they validate arguments that are
classically invalid. In particular, they have contradictions as theorems.
Nonetheless, whereas DL* is very standard to a certain extent |it is one of the logics
obtained by the so-called \positive logic plus approach"|, HEGFL is of further
interest as it exhibits many more contra-classical features. For example, versions
of weak-connexivity, in the sense of Pizzi [
Unlike many of the logics developed by da Costa and his collaborators, DL* has not received attention. This investigation can contribute to the formal understanding of DL* and akin logics by relating them to other contra-classical logics using more contemporary approaches. This is just a rst step towards that end. (In spite of their Hegelian origins, I will o er (almost) no comments of Hegel's philosophy.) ? Supported by the PAPIIT project IN403719: \Intensionality all the way down: a new plan for logical relevance". I thank Luis Estrada-Gonzalez for his invaluable help on previous versions of this paper.
The plan of the paper is the following. In the rst section I present the logic DL* in proof-theoretic and model-theoretic terms. In the second section I present the logic HEGFL and some of its more distinguished properties. In the third section I discuss the relation of HEGFL with weakly connexive logics and two \negative" connectives. 1
First, I will present DL* in proof-theoretic terms and then in model-theoretic terms.
Let L be a propositional language with a denumerable set P ROP = fp1; :::; png of propositional variables and with the set of connectives f^; _ !; $g. The set of formulas F ORM is recursively de ned as usual. In this paper, when either new connectives or propositional constants expand L I indicate it in a subscript attached to L. For example, I write Lf ; g to mean the expansion of the set of connectives by f ; g, with the corresponding de nition of the set of formulas F ORMf ; g; I write Lflng to mean the expansion of L by the collection fl1; :::; lng with its corresponding de nition of the set F ORMflng.
Consider positive logic de ned on L, i.e. the logic characterized by the following schemata and rule: A1) A ! (B ! A) A2) (A ! B) ! ((A ! (B ! C)) ! (A ! C)) A3) (A ^ B) ! A A4) (A ^ B) ! B A5) A ! (B ! (A ^ B)) A6) A ! (A _ B) A7) B ! (A _ B) A8) (A ! C) ! ((B ! C) ! ((A _ B) ! C)) A9) A _ (A ! B) MP) A ! B; A=B The logic DL, based on Lf ; g, expands positive logic with the introduction of the following axioms for the additional vocabulary: A10) A11) A12) A13) A14) A15) A16) A17) (A ^ B) $ ( A_ B) (A _ B) $ ( A^ B) A ^ B ! ( (A ! B) ^ (A ^ B) ^ (A _ B) ^ ( A ^ B ! ((A ! B) ! ((A ! B) ! A)) A ! ( A ! A)
A $ A A ! ((A_ A) ^ ((A ! B) _ ( A ! B))) ( A) ! (((A_ A) ! B) _ (A^ A))
A))
One can de ne a strong negation using !, , ^ and in this terms: :A =def A ! ( B ^ B^ B); in using : one obtains all the theorems of classical logic. Consequently, one can recover classical logic within DL, thus making classical logic a sublogic of DL.
DL*, based on Lfln;kn; ; g, in its turn expands DL by adding the following axioms for the two families of propositional constants fk1; k2; :::kng and fl1; l2; :::; lng (ki 6= lj for any i; j 2 !, where ! is the lowest trans nite ordinal number): A18) :(ki_ A19) lj ^ A20) ki+1 lj
ki)
Let v : P ROP ! f1; 0g be a set of valuation functions from the set of propositional variables to the set f1; 0g. v is extended to F ORMfln;kn; ; g by the following interpretation functions i: i(p) = v(p), for every p 2 P ROP i(A ! B) = 1 i i(A) = 0 or i(B) = 1 i(A ^ B) = 1 i i(A) = i(B) = 1 i(A _ B) = 1 i i(A) = 1 or i(B) = 1 i( (A ^ B)) = 1 i i( A) = 1 or i( B) = 1 i( (A _ B)) = 1 i i( A) = i( B) = 1
If i( A) = i( B) = 1 then i( (A ! B)) = i( (A ^ B)) = i( (A _ B)) = i( ( B)) = 1 i( A) = 1 i i( A) = 1 If i( A) = 1 then i( A ! A) = 1 If i(A) = i( A) then i( A) = 0 If i(A) 6= i( A) then i( ( A)) = 0 i(ki) = 0 and i( ki) = 0 i(lj ) = 1 and i( lj ) = 1 A formula A is valid if for every interpretation function i, i(A) = 1. An interpretation i is a model for a set of formulas if for every A in , i(A) = 1. If for every model of , i(A) = 1, then A is a semantic consequence of , written as ` j= DL* A'. Let `j= DL* A' represent that A is a semantic consequence of any set of formulas, i.e., that it is valid. Using the previous de nitions, da Costa and Wolf prove the soundness and completeness of DL* with respect to the interpretations i.
Furthermore, the features of the interpretations have as a byproduct a
decision method similar to the decision method for the hierarchy of logics Cn
(1 n<!) (see [
A remarkable property of DL* is that it is dialetheic |or \negation-inconsistent"|, that is, it validates both at least one formula and its negation |lj and lj . Besides, the following schemas are not theorems of DL*: (A_ A)
(A^ (A_ 2
The logic HEGFL, introduced in [
Of special interest in a Hegelian logic are the principles concerning double negation. Estrada-Gonzalez and Ram rez-Camara observe that A j=HEGFL A does not hold, but A j=HEGFL A does. Also, they say that one could venture the hypothesis that this is so because while a rmation (A) alone still says nothing about its sublimation ( A) |for negation ( A) is needed for that|, A j= HEGFL A should hold because a rmation is somehow maintained in its sublimation. If this were along the right lines, A j=HEGFL A should be valid in HEGFL. And it is. Moreover, in HEGFL one has that both
A j=HEGFL A and A j=HEGFL A are valid. Why A j=HEGFL A fails and A j=HEGFL A holds? Because in the latter A is not missing, at least not in the same way as A is missing in the former: in having A, one somehow has A too, as it is the very proposition being negated, so the logical step to the sublimation can be done.2 Finally, Estrada-Gonzalez and Ram rez-Camara analyze Ficara's \Dialectical law of double negation": A j=HEGFL A^ A. This should hold in a Hegelian logic because \we capture the true nature of concepts only when, by negating their negation, we gain them in their completeness, which is contradictory." It can be easily veri ed that such a Dialectical law of double negation holds in HEGFL. 3
(Other) Contra-classical features of HEGFL
A contra-L logic of a given logic L is a logic that, over the same underlying
language, validates arguments that are not valid according to L. In particular, a
contra-classical logic is a logic that validates arguments that, over the same
language as that of classical logic, are not classically valid. Examples of such logics
are well known (see, for instance [
Another special kind of contra-classical logics are connexive logics.3 A connexive logic is characterized by the validity of the following schemas: A1) (A !
A) 2 To see why A j=HEGFL A and A j=HEGFL A are invalid, just assign v(A) = 1.
From this assignation it follows that v( A) = c and v( A) = c.
3 For an introductory survey, see [
A1) is known as Aristotle's Thesis and A2) as Variant of Aristotle's Thesis. B1) is known as Boethius' Thesis and B2) Variant of Boethius' Thesis.
Neither DL* nor HEGFL validate Aristotle's or Boethius' theses, they only invalidate the last schema. However, in an alike spirit to the logic DL*, one can de ne classical negation in HEGFL, and not only that, but also a classical conditional, that in certain combinations with the primitive negation and conditional in the formulas of HEGFL can validate or invalidate some connexive schemas.
The key to obtain the classical negation and conditional is by drawing upon the partition between designated and antidesignated set of values as follows. In HEGFL one can de ne the constants 1, c, c* as 1=def A^ A, c=def (A^ A) and c*=def (A^ A)|let Cx denote any constant. Considering that in (anti)designatedness jargon, classical negation exchanges designated values for antidesignated values and vice versa, classical negation can be de ned as :A =def ((A $ 1)^ c*), because one obtains the following table: A ((A $ 1)^ c*)4 1 c c 1 c 1
By the same token, considering that classical conditional is antidesignated if and only if the antecedent is designated and the consequent antidesignated, one can de ne classical conditional as (A B) =def ((::A ! (Cx^::B)) ! ((Cx^::A) ! ::A))^ c*), because one has the following table A B ((::A ! (Cx^::B)) ! 1 1 1 c 1 c c 1 c c c c c 1 c c c c ((Cx^::A) ! ::A))^ c*) 1 c c 1 1 1 1 1 1 4 Note, however, that classical negation so de ned con ates c and c. Therefore, once one adds it to HEGFL, one cannot obtain the values of the original formula A by double negation elimination.
Given the well-known fact that in the model theory structure of classical logic the set f:; g is a functionally complete set of connectives, and that in HEGFL one can model that structure by appealing to the division of designated and antidesignated values, one can de ne all classical logic within HEGFL, what makes classical logic a sublogic of HEGFL, as in the case of DL*.
With the new connectives thus introduced, and combining them with the basic connectives of HEGFL in the characteristic schemas of connexive logic, the following schemas are valid: :(A ! A) :(A ! :A) (A ! B) (A ! B) (A ! B) (A ! B) (A ! B) (A ! B) (A ! :B) (A ! :B) (A ! B) (A ! B) (A ! B) (A ! :B) (A B) (A :B) :(A ! B) (A ! B) (A ! B) (A B) :(A ! B) :(A B) :(:A ! A) :( A ! A) (A ! B) (A ! B) (A ! B) (A ! B) (A ! B) (A ! :B) (A ! B) (A ! :B) (A B) (A :B) :(A ! :B) :(A B) :(A :B) :(A ! B) :(A ! :B) :(A B) :(A ! B) :(A ! B) :(A ! B) :(A ! B) To prove the validity of all these schemas it is enough to note that the schemas with the form A ! B are always antidesignated, so the schemas with the form :(A ! B) are always designated. By the same reason, when a schema with the form A ! B appears as antecedent of a classical conditional, the classical conditional is designated. The proofs of the rest of the schemas can be made by simply making its tables and checking if they turn out to be true under every assignement of values to its propositional variables.
But one also obtains the unfortunate validation of the following schemas, which are versions of symmetry: and the invalidation of the following schemas (to avoid tedious repetition of several formulas, when possible I use the following notation (A!B)!: (A!: B) to say that from the connectives that appear between formulas one can choose any of them): (A
B) (A!: B) !: (A!B) (A B) (B ! A) (A ! B) ! (B A) (A ! B) ! (B ! A)
I can make a case for these results. In [
As I have remarked, the HEGFL conditional ! is not a connexive conditional, as it, together with HEGFL's primitive negation , does not validate any connexive schema. However, in using the classical conditional and negation, not only some variants of Aristotle's thesis, but also the weak Boethius' thesis is validated, an outcome that probably would have pleased Pizzi.
Additionally, the only connexive schemas that turn out to be valid are the ones that have the connexive conditional as second main connective (as in Aristotle's theses) or in the antecedent (as in Boethius' theses). The invalid schemas are those with the classical conditional in the antecedent. Moreover, the validation of the schemas that are versions of symmetry are not so bad at it could seem prima facie. The schema (A ! B) (B A) suggests that if there is a connection between the antecedent and the consequent, then the antecedent is necessary for the consequent. Similarly, (A ! B) (B ! A) can suggest that if there is a connection between the antecedent and the consequent, then the same can be said conversely: there is a connection between the consequent and the antecedent in any connexive conditional, a connection that holds because in the rst place the connection between the antecedent and the consequent already holds.
Until now, I have not touched upon the meaning of basic connectives of
HEGFL. As it could be seen, the connectives have very strange tables, as they
show that the conjunction of two designated formulas results in an antidesignated
5 In [
A A 1 c c c c c are the following:
A is not true i A is either true or false
A is false i A is not true
What such interpretation amounts to is to the falsity conditions of a
deminegation connective (see [
In this paper I made a quick study of two dialectical logics, DL* and HEGFL. I presented DL* in proof- and model-theoretic terms, and I mentioned its more distinguished properties. Then I presented the logic HEGFL as another dialectical logic, very di erent from DL*. Also, I discussed one form to de ne classical logic within HEGFL and I showed some unexpected connections of HEGFL to connexive logics and two di erent negative connectives.
It is worth mentioning that the decision method of HEGFL is truth-functional, and this characteristic lets to take few steps to establish whether an argument is valid. On the other hand, the decision method of DL* is not truth-functional, and this characteristic makes necessary to consider more steps to establish the validity of any argument. As a bonus, in HEGFL one can recover all the properties that DL* has with less linguistic resources.
As a future work, it remains to spell out in detail the meaning of all the connectives in HEGFL using a bivalent semantics. Also, HEGFL still lacks any proof-theoretic presentation, which could also shed light on some of its features. Finally, I deliberately ignored the discussion about to what extent DL* and HEGFL recover Hegel's philosophy. I do not aim at doing this in future work, but perhaps others could try to do it.