In the paper by Oscar Estrada et al. it is introduced the concept of "Possibilistic Intuitionistic Logic (PIL)"; In that paper, the authors present results analogous to those of the well-known intuitionistic logic, such as a Deduction Theorem, a Generalized version of the Deduction Theorem, a Cut Rule, a weak version of a Refutation Theorem, a Substitution Theorem and Glivenko's Theorem. The concept of Safe Beliefs was introduced in the paper "Safe Beliefs for Propositional Theories" by Osorio et al. as an extension of the concept of Answer Set. In this paper, we present the analogous concept to Safe Beliefs: Possibilistic Safe Beliefs Candidate.
Uncertainty is an attribute of information. The pioneering work of Claude Shannon [18] on Information Theory led to the universal acceptance that information is statistical in nature; As a consequence, dealing with uncertainty was confined to the Theory of Probability.
Dubois et al. presented Possibilistic Logic on [3]; Possibilistic Logic enables to handle information with uncertainty. Their logic is based on the Theory of Possibility of Zadeh [20]; On the paper by Dubois et al. [3], it is included an axiomatization for Possibilistic Logic and an extended resolution-based method which is viable to be implemented on a computer.
According to Pascal Nicolas et al. [9], Possibilistic Logic provides a sound and complete machinery for handling qualitative uncertainty with respect to a semantics expressed by means of possibility distributions which rank order the possible interpretations. Pascal mentions that in possibilistic logic it is dealt with uncertainty by means of classical two-valued (true or false) interpretations that can be more or less certain, more or less possible. Possibilistic Logic is not concerned to deal with a vagueness representation in a multi-valued framework but instead, it stays in the framework of classical logic to which it adds a way to graduate the confidence one has in each proposed information [9].
On the other hand, Intuitionistic Logic has become important to the area of computer science since the publication of a paper by David Pearce [17], on which it is established a link between Intuitionistic Logic (and furthermore, all the class of intermediate logics) and Answer Set Programming (ASP) [6,8].
Considering the importance of the ASP formalism, in combination with its usefulness within the context of intermediate logics, and the need for a better way of handling uncertain information, it was proposed in [10] the Possibilistic Intuitionist Logic (PIL) formalism. It was presented only propositional calculus.
On this this paper, following the same flow of ideas, we present the concept of Possibilistic Safe Belief, which tries to establish a link between Safe Beliefs [16] and Possibilistic Intuitionistic Logic [10]. The concept of Safe Belief as presented in [16] is a generalization of the concept of Answer Set. With this in mind, one can argue in favor of developing Possibilistic Safe Beliefs in order to broaden the scope of applications; Hence, a Possibilistic Safe Belief would allow us to analyze situations involving uncertainty, for example, in decision making problems.
First, in Section 2, we present some background on Possibilistic and Intuitionistic Logic, and we present some theorems from [10] and show some examples. In Section 3, we introduce our main contribution for this paper, the concept of Possibilistic Safe Belief and a Lemma establishing a resemblance with some result of Safe Belief as presented on [16]. Finally, in Section 4 we give the conclusions and sketch some ideas for future work.
We present the necessary background for this paper. The propositional calculus is built as in the well-known book by Mendelson [7].
Recall that the following are valid meta-theorems of Intuitionism:
The following definitions can be found in [3], to which the reader is referred for further details.
A necessity-valued formula is a pair (ϕ α), where ϕ is a classical propositional formula and α ∈ (0, 1]. (ϕ α) expresses that ϕ is certain to the extent α, that is, N (ϕ) ≥ α, where N is a necessity measure which models our state of knowledge. The constant α is known as the valuation of the formula and is represented as val(ϕ).
On [3], it is proposed an axiom system for Possibilistic Logic:
with the inference rules, (GMP) (ϕ α), (ϕ
We now present the axioms for Possibilistic Intuitionistic Logic (PIL) [10].
, Int2 where all of them weighted by 1, that is:
Together with the following rules of inference:
A Deduction Theorem is given on [3] for Standard Possibilistic Logic, which is based on Classical Logic. However, the Deduction Theorem we present here is taken from [10], which is given in the context of PIL.
Theorem 2 (Deduction Theorem [10]). Let Γ ∪ {ϕ, ψ} be a set of formulas in PIL, and α ∈ (0, 1]. Then we have that
Furthermore, on [10] it is proven the following result.
). Let Γ ∪ {ϕ, ψ} be a set of PIL formulas and α, β ∈ (0, 1]. Then we have that
If, furthermore, β ≥ α, then we have the equivalence:
The proof given for the previous theorem on [10] uses theorems also valid on classical logic, so this theorem also holds in Standard Possibilistic Logic. Besides, observe that as a consequence of the previous result, if we fix β = 1 on Theorem 3(2), then we obtain Theorem 2 as a corollary.
We now define the concept of "dependence" on a formula [10]. Let (ϕ α) be a formula in a set Γ of formulas, and assume that we are given a deduction D 1 , D 2 , . . . , D n from Γ , together with justification for each step in the deduction. We shall say that D i depends on (ϕ α) in this proof if and only if:
1. D i is (ϕ α) and the justification for D i is that it belongs to Γ , or 2. D i is justified as a direct consequence by GMP or by the Rule (S) of some previous formulas of the sequence, where at least one of these preceding formulas depends on (ϕ α).
). Let Γ ∪ {ϕ, ψ} be PIL formulas and α, β ∈ (0, 1]. Assume that Γ ∪ {(ϕ β)} P IL (ψ α), and let Γ = {(φ 1 ρ 1 ), (φ 2 ρ 2 ), . . . , (φ p ρ p )} ⊆ Γ be the set of formulas on which the proof of (ψ α) from Γ ∪ {(ϕ β)} depends on; Then,
Observe that if in the previous result we let Γ = ∅, then the theorem states that if (ϕ β) P IL (ψ α), then P IL (ϕ → ψ min {∅}), that is, P IL (ϕ → ψ 1), which is what one would expect; For instance, is clear that (ϕ 0.3) P IL (ϕ 0.3), so when applying this version of the Generalized Deduction Theorem we obtain
As a consequence of Theorem 3 one obtains the following results. Consider the set of PIL formulas Γ = {(ϕ → ψ 0.2), (ϕ → ¬ψ 0.7), (σ → ϕ 0.3)}. It it easy to see that Γ P IL (¬ϕ 0.2), and also that Γ ∪ {(¬ϕ 0.2)} P IL (¬ψ 0.2). So, by the previous theorem, we have that Γ P IL (¬ψ 0.2). Now it is presented a version of the Substitution Theorem and a version of Glivenko's Theorem for Possibilistic Intuitionistic Logic; The proofs of this results can be found in [10].
Theorem 7 (Substitution Theorem [10]). Let ϕ be a classical propositional formula and let ψ be a subformula of ϕ; Let ϕ be the formula which results by substituting some, or none, occurrences of ψ in ϕ by a formula σ.
Example 2. Consider the set of PIL formulas Γ = {(¬¬ϕ → ϕ 0.3)}. Since P IL (ϕ → ¬¬ϕ 1), it is easy to show that Γ P IL (ϕ ↔ ¬¬ϕ 0.3). Now, by the previous theorem we have that Γ P IL ((ϕ → ψ) ↔ (¬¬ϕ → ψ) 0.3) Theorem 8 (PIL Glivenko's Theorem [10]). Let Γ ∪{(ϕ α)} be PIL formulas, and assume that Γ P os (ϕ α); Let Γ * = {(γ 1 α 1 ), (γ 2 α 2 ), . . . , (γ n α n )} ⊆ Γ be such that the proof of (ϕ α) from Γ depends on the formulas on Γ * , then we have that
, and P os (¬α ∨ α 1) it is easy to show that Γ P os (ϕ 0.7). Now, by Glivenko's Theorem we have that Γ P IL (¬¬ϕ 0.7) .
We now present our main results.
Proof. Since T ∪ (¬ M 1) P IL (M 1) then T ∪ (¬ M 1) P IL (¬¬M 1) (recall that P IL (ϕ −→ ¬¬ϕ 1)). By Monotony we have that T ∪ (¬ M 1) P IL T ∪ (¬ M 1) ∪ (¬¬M 1). Conversely, we obtain immediately that T ∪(¬ M 1)∪(¬¬M 1) P IL T ∪(¬ M 1).
Note that the Lemma 3 has resemblance with some result of Safe Belief as presented on [16].
On this brief note we presented Possibilistic Intuitionistic Logic in a purely axiomatic way. This allowed us to state some basic, but relevant results. These are partial results, but as a first attempt, they pave the way for results on reduction of theories. For future work, we leave the task of finding a suitable semantics. Also, we presented a definition for Possibilistic Safe Beliefs candidate. We have several questions, for example: Is there an equivalence between Safe Beliefs [16] and our Possibilistic Safe Beliefs Candidate?, that is, is it true that M is a possibilistic safe belief candidate of a theory T if and only if M * is a safe belief of the theory T * ?. Such result would allow us to establish a relation between Safe Beliefs as presented on [16] and Possibilistic Intuitionistic Logic (PIL) [10].
Here, we define the concepts of classical projection of a possibilistic theory, the Inconsistency Degree and Possibilistic Safe Beliefs Candidate.