Gemma Robles

The basic constructive logic for a weak sense of consistency

In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic BKc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending BKc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent logics.

A General Characterization of the Variable-Sharing Property by Means of Logical Matrices

As is well known, the variable-sharing property (vsp) is, according to Anderson and Belnap, a necessary property of any relevant logic. In this paper, we shall consider two versions of the vsp, what we label the “weak vsp” (wvsp) and the “strong vsp” (svsp). In addition, the “no loose pieces property,” a property related to the wvsp and the svsp, will be defined. Each one of these properties shall generally be characterized by means of a class of logical matrices. In this way, any logic verified by an actual matrix in one of these classes has the property the class generally represents. Particular matrices (and so, logics) in each class are provided.

A Routley–Meyer Semantics for Gödel 3-Valued Logic and Its Paraconsistent Counterpart

Routley–Meyer semantics (RM-semantics) is defined for Gödel 3-valued logic G3 and some logics related to it among which a paraconsistent one differing only from G3 in the interpretation of negation is to be remarked. The logics are defined in the Hilbert-style way and also by means of proof-theoretical and semantical consequence relations. The RM-semantics is defined upon the models for Routley and Meyer’s basic positive logic B+, the weakest positive RM-semantics. In this way, it is to be expected that the models defined can be adapted to other related many-valued logics.

A Routley-Meyer Type Semantics for Relevant Logics Including B r Plus the Disjunctive Syllogism

Routley-Meyer type ternary relational semantics are defined for relevant logics including Routley and Meyer’s basic logic B plus the reductio rule

Strong Paraconsistency and the Basic Constructive Logic for an Even Weaker Sense of Consistency

\vdash A\rightarrow \lnot A\Rightarrow \vdash \lnot A

The Basic Constructive Logic for Negation-Consistency

and the disjunctive syllogism. Standard relevant logics such as E and R (plus γ) and Ackermann’s logics of ‘strenge Implikation’ Π and Π′ are among the logics considered.

An Interpretation of Łukasiewicz’s 4-Valued Modal Logic

In a standard sense, consistency and paraconsistency are understood as the absence of any contradiction and as the absence of the ECQ (‘E contradictione quodlibet’) rule, respectively. The concepts of weak consistency (in two different senses) as well as that of F-consistency have been defined by the authors. The aim of this paper is (a) to define alternative (to the standard one) concepts of paraconsistency in respect of the aforementioned notions of weak consistency and F-consistency; (b) to define the concept of strong paraconsistency; (c) to build up a series of strongly paraconsistent logics; (d) to define the basic constructive logic adequate to a rather weak sense of consistency. All logics treated in this paper are strongly paraconsistent. All of them are sound and complete in respect a modification of Routley and Meyer’s ternary relational semantics for relevant logics (no logic in this paper is relevant).

Dual Equivalent Two-valued Under-determined and Over-determined Interpretations for Łukasiewicz’s 3-valued Logic Ł3

In this paper, consistency is understood in the standard way, i.e. as the absence of a contradiction. The basic constructive logic BKc4, which is adequate to this sense of consistency in the ternary relational semantics without a set of designated points, is defined. Then, it is shown how to define a series of logics by extending BKc4 up to minimal intuitionistic logic. All logics defined in this paper are paraconsistent logics.

The Basic Constructive Logic for a Weak Sense of Consistency defined with a Propositional Falsity Constant

A simple, bivalent semantics is defined for Łukasiewicz’s 4-valued modal logic Łm4. It is shown that according to this semantics, the essential presupposition underlying Łm4 is the following: A is a theorem iff A is true conforming to both the reductionist (rt) and possibilist (pt) theses defined as follows: rt: the value (in a bivalent sense) of modal formulas is equivalent to the value of their respective argument (that is, ‘ A is necessary’ is true (false) iff A is true (false), etc.); pt: everything is possible. This presupposition highlights and explains all oddities arising in Łm4.

A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes

AbstractŁukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension of Routley and Meyer’s basic positive logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” Łukasiewicz logics are paraconsistent.