#### Supplement to Connexive Logic

## Supplement on Terminology

Kapsner (2019) introduces the following restricted versions of Aristotle’s theses, Boethius’ theses, Unsat1, and Unsat2:

Plain humble AT and AT′: For any satisfiable ~A, ~(~A→A) is valid. For any satisfiableA, ~(A→~A) is valid.

Plain humble BT and BT′: For any satisfiableAand satisfiableA→B, (A→B)→~(A →~B) is valid. For any satisfiableAand satisfiableA→ ~B, (A→ ~B) → ~(A→B) is valid.

Plain humble Unsat1: In no model,A→ ~Ais satisfiable (for any satisfiableA), and in no model,~A→Ais satisfiable (for any satisfiableA).

Plain humble Unsat2: In no model,A→BandA→ ~Bare simultaneously satisfiable (for any satisfiableA).

Logics that satisfy the restricted versions of Aristotle’s and
Boethius’ theses are then *plainly weakly humbly
connexive*. According to Lenzen (2019a) the idea of restricting
Aristotle’s and Boethius’ theses to self-consistent
propositions can be found in Leibniz already, see section 2. However,
if a formula *A* is satisfiable in classical logic, then
⊬ *A* → ~* A*, so that in this weak sense of connexivity, restricting AT′ to satisfiable antecedents makes the whole enterprise of connexive logic pointless from the standpoint
of classical logic. Restrictions of connexive principles in terms of
consistency or possibility requirements have been discussed also in
Unterhuber 2016, whereas Vidal (2017) considers a restriction to
contingent formulas *A* and * B* in Aristotle’s
and Boethius’ theses. Kapsner 2019 discusses even more notions
of connexivity, namely modal humble, glutty humble, and paraconsistent
humble connexivity. Combinations of Kapsner’s defining
properties gives rise to even further notions of connexivity such as
the notion of a logic being *Kapsner strong* from
Estrada-González & Ramirez-Cámara 2016.

Still another conception
of connexive logic that also imposes further conditions in addition to
validating Aristotle’s and Boethius’ theses was used by
Routley (1978, Routley *et al*. 1982) and Routley and Routley
(1985), see sections 3.1 and 3.3.

One may assume that the non-symmetry of implication is tacitly assumed in many works on connexive logic, but since this condition is used to draw a terminological distinction in Jarmużek and Malinowski 2019a, it is taken into account in the following table that surveys the terminology and notions of connexivity used in the literature as well as some (but not all) combinations of certain defining properties.

notionprinciples and conditionsreferencesconnexiveAT, AT′, BT, BT′, non-symmetry of implication this entry and numerous other publications minimally connexive ditto Estrada-González and Ramirez-Cámara 2016 properly connexive ditto Jarmużek and Malinowski 2019a diverging and additional notionsprinciples and conditionsreferencesconnexive at least one of AT, AT′, BT, BT′ Mares and Paoli 2019 connexive AT, AT′, BT, BT′ Jarmużek and Malinowski 2019a connexive AT, AT′, BTw, BTw′ Weiss 2019 half-connexive BTw, BTw′ Weiss 2019 subminimally connexive some but not all of AT, AT′, BT, BT′, non-symmetry of implication Estrada-González and Ramirez-Cámara 2016 properly quasi-connexive ditto Jarmużek and Malinowski 2019a demi-connexive some but not all of AT, AT′, BT, BT′ Wansing, Omori, Ferguson 2016 quasi-connexive ditto Jarmużek and Malinowski 2019a weakly connexive AT, AT′, but BT, BT′ only in rule form Wansing and Unterhuber 2019 weakly connexive AT, AT′, BT, BT′ Kapsner 2012 strongly connexive AT, AT′, BT, BT′ + UnSat1, UnSat2 Kapsner 2012 superconnexive strongly connexive + ( A→ ~A) →BKapsner 2012 plainly humbly connexive plain humble AT, AT′, BT, BT′ + plain humble UnSat1, UnSat2 Kapsner 2019 plainly weakly humbly connexive plain humble AT, AT′, BT, BT′ Kapsner 2019 modally humbly connexive modal humble AT, AT′, BT, BT′ + modal humble UnSat1, UnSat2 Kapsner 2019 modally weakly humbly connexive modal humble AT, AT′, BT, BT′ Kapsner 2019 Gluttily humbly connexive glutty humble AT, AT′, BT, BT′ + glutty humble UnSat1, UnSat2 Kapsner 2019 Gluttily weakly humbly connexive glutty humble AT, AT′, BT, BT′ Kapsner 2019 paraconsistently humbly connexive paracons. humble AT, AT′, BT, BT′ + paracons. humble UnSat1, UnSat2 Kapsner 2019 paraconsistently weakly humbly connexive paracons. humble AT, AT′, BT, BT′ Kapsner 2019 Kapsner strong UnSat1, UnSat2 Estrada-González and Ramirez-Cámara 2016 subminimally connexive and Kapsner strong subminimally connexive + UnSat1, UnSat2 Estrada-González and Ramirez-Cámara 2016 hyperconnexive AT, AT′, BTe, BTe′, non-symm. of implication + failure of Conj. Simplif. Sylvan 1989 (in combination with Routley 1978)

In the above table, the conditions defining Kapsner’s notion of modal humble connexivity are as follows, cf. also the modalized versions of Aristotle’s and Boethius’ theses as entailment statements in Vidal 2017:

Modal humble AT, AT′: ◊

~A⊨ ~(~A→A), ◊A⊨ ~(A→ ~A).

Modal humble BT, BT′: ◊A∧ ◊ (A→B) ⊨ (A→B) → ~(A→ ~B), ◊A∧ ◊ (A→ ~B) ⊨ (A→ ~B) → ~(A→B).

Modal humble UnSat1: In no model, ◊~A∧ (~A→A) is satisfiable, and in no model, ◊A∧ (A→ ~A) is satisfiable.

Modal humble Unsat2: In no model, ◊A∧ (A→B) ∧ (A→ ~B) is satisfiable.

For glutty humble connexivity, Kapsner’s assumes a many-valued
semantics with a designated glutty truth value **B**,
“understood as “both true and false”, and a negation
operator ~ such that ~*A* receives the value **B**
if *A* does. Such a semantics gives a paraconsistent logic,
where *ex contradcitione quodlibet*, *A* ∧
~*A* ⊢ *B*, does not hold. The defining conditions
are the following:

Glutty humble AT, AT′: For any satisfiable ~

A, resp.Athat does not take valueB, ~(~A→A), resp. ~(A→ ~A) takes a designated value.

Glutty humble BT, BT′: For any satisfiableA,B, andA→B, reps.A, ~B, andA→ ~Bthat do not take valueB, (A→B) → ~(A→ ~B), resp. (A→ ~B) → ~(A→B) takes a designated value.

Glutty humble UnSat1: In no model, ~A→A, resp.A → ~Ais satisfiable (for any satisfiable ~A, resp.Athat does not take valueB).

Glutty humble Unsat2: In no model,A→BandA→ ~Bare simultaneously satisfiable (for any satisfiableA,Bthat do not take valueB).

Kapsner’s conditions for paraconsistent humble connexivity make use of conjunction and disjunction in the object language:

Paraconsistent humble AT, AT′: For any satisfiable ~

A, resp.A, (A∧ ~A) ∨ ~(~A→A), reps. (A∧ ~A) ∨ ~(A→ ~A) is valid.

Paraconsistent humble BT, BT′: For any satisfiableAandA→B, reps.AandA→ ~B, (A∧ ~A) ∨ ((A→B) ∧ ~(A→B)) ∨ ((A→B) → ~(A→ ~B)), resp. (A∧ ~A) ∨ ((A→B) ∧ ~(A→B)) ∨ ((A→ ~B) → ~(A→B)) is valid.

Paraconsistent humble UnSat1: ~A→AandA →~Aare satisfiable only in valuations in whichA∧ ~Ais also satisfied.

Paraconsistent humble Unsat2:A→BandA→ ~Bare simultaneously satisfiable only in valuations in whichA∧ ~Ais satisfied or (A→B) ∧ (A→ ~B) is satisfied.