Theory of Semi-Instantiation in Abstract Argumentation

D. M. Gabbay
2015 Logica Universalis  
We study instantiated abstract argumentation frames of the form (S , R, I), where (S , R) is an abstract argumentation frame and where the arguments x of S are instantiated by I(x) as well formed formulas of a well known logic, for example as Boolean formulas or as predicate logic formulas or as modal logic formulas. We use the method of conceptual analysis to derive the properties of our proposed system. We seek to define the notion of complete extensions for such systems and provide
more » ... for finding such extensions. We further develop a theory of instantiation in the abstract, using the framework of Boolean attack formations and of conjunctive and disjunctive attacks. We discuss applications and compare critically with the existing related literature. * Research supported by the Israel Science Foundation Project 1321/10: Integrating Logic and Networks. For example, let ∆ = {A, A → ⊥} (with → being implication, and ⊥ being falsity) and the attack relation be from A to A → ⊥ but not from A → ⊥ to A. We are not explaining why this attack relation is defined so. There are logics, like the Lambek calculus where modus ponens works from the left but not from right. Option 2. Such instantiated systems can also arise from general methodolgoical considerations. Let us ask ourselves a very simple question: Question. What is the added value of abstract argumentation networks over, say, classical propositional logic? Obviously they have the same expressive power. Many papers by various authors have demonstrated such equivalence. My favourite is my own paper [37], showing that the attack relation is really the Peirce-Quine dagger connective (x ↓ y = def. ¬x ∧ ¬y) of classical logic. So, we ask, what is the added value of such networks? My answer to this is that in these networks we bring some meta-level features into the object level Dung argumentation networks, expand classical propositional logic with the meta-predicate "x is false". When we write z ։ x (i.e., z attacks x) we are saying z = "x is false", or z ↔ "x is false" So the liar paradox becomes x ։ x, "I am false". So the added value of abstract argumentation networks to classical propositional logic is the meta-predicate False(x). So the language now has (¬x, x ∧ y, x ∨ y, x → y) and the additional connective False (x). Now the minute we accept this view we must also allow and address expressions like x ։ A where A is a wff, i.e., x = "A is false" and we then must also allow meaning that x and y together say that A is false, and now, of course, once we go this far we must also address B ։ A. The latter is nothing but the equivalence If you think about it, once we add to any logic a new connective "C(x)", we must be able to address y ↔ C(x), it being just another wff. Having established some interest in semi-instantiated argumentation networks, let us now get to business and describe the machinery and problems involved. Let (S , R) be an abstract argumentation frame. This means that S is a non-empty set and R ⊆ S × S . Let L be a logic, with a set of well formed formulas WFF(L) and let µ be either semantics or proof theory for this logic. Assume that we have models for this logic which we denote by {m}, and/or theories for this logic which we denote by {∆}. We assume that a notion L for this logic is available such that for each m or ∆ and for each Φ ∈ WFF(L) the relation ∆ L Φ or m L Φ can get 3 answers. Yes (= 1), no (= 0) or undecided (= 1 2 ). As an example of a logic let us take intuitionisitc propositional logic H, with consequence ⊢ H we can have: holds Another example is 3-valued classical propositional logic with the Kleene truth table for {0, 1 2 , 1}. Call it K. See [19] and Definition C.2. Given a model assignment m to the atoms, we have m K Φ is the value that m gives to Φ, denoted by m(Φ). It is a value in {0, 1, 1 2 }. Let (S , R) be a network and let L be a logic with L . Consider the instantiation function, I : S → WFF(L). Consider (S , R, I). This is an instantiated argumentation network. We seek to define the notion of complete extensions for (S , R, I) and give algoirthms for finding such extensions. After performing a conceptual analysis of this problem, we reached the following definition. A model m (or a theory ∆) of L generates an extension for (S , R, I) if the function λ m (or λ ∆ ) defined on S below is a legitimate Caminada labelling giving rise to a complete extension on (S , R). The function λ is: The problem is how to identify/compute, using purely argumentation methods, such extensions for (S , R, I). This is the task of this paper. 1 Note that the emphasis is on 1 The reader should note that we are not defining, as a stipulated technical definition, the complete extensions of (S , R, I) as those legitimate Caminada labellings arising from models or theories of the logic. We are deriving this definition from conceptual analysis of the idea of instantiation. To make the point absolutely clear, suppose we instantiate the elements of S by names of Chinese restaurants in London. We can define by stipulation extensions for such Chinese systems as those legitimate Caminada extensions λ such that if • λ(x) = in then the Chinese restaurant associated with x made a profit in 2014 • λ(x) = out, then the Chinese restaurant associated with x made a loss in 2014 • λ(x) = undecided, then the Chinese restaurant associated with x came out even in 2014 The above stipulation has nothing to do with a Chinese restaurant attacking another, and is nothing more than means of restricting the Caminada labellings on (S , R). using geometric syntactical argumentation methods to find the extensions of (S , R, I). What we can do and we do not want to do is to systematically generate all models m of the logic or all theories ∆ of the logic and check whether λ m or λ ∆ generate a legitimate Caminada extension. We want to syntactically transform (S , R, I) into an argumentation network. Put differently, we want to identify and use the argumentation network meaning of the logic. We consider three main logics. 1. Classical propositional logic based on 3 valued Kleene truth table. 2. Monadic predicate logic without equality based on Kleene table. 3. Modal logic S5 based on Kleene table. This paper solves the problem. However, many of the results are technical and are done in the Appendices. The methodological schema is simple: Given (S , R, I) with I being an instantiation into WFF(L) we follow the steps below: Step 1. Rewrite any wff Φ of L into an equivalent formula (in L) which is argumentation friendly and convenient form. Finding the right friendly form is not immediate and requires some analysis and trial and error. Once we find a convenient form for any Φ ∈ WFF(L) we need to prove the equivalence. This may involve some technical manipulations and is therefore done in an appendix. Step 2. The instantiation I(x) = Φ x can now be assumed to be in this special form. When x attacks y in (S , R), we get after instantiation that Φ x attacks Φ y . This attack between formulas of L, needs to be given a meaning. The two formulas may be consistent in L, so what does it mean that one attacks the other? So we transform any Φ of L into an argumentation template called a Boolean attack formation "equivalent" to Φ which we denote by BF(Φ). Such formations can attack each other because they are defined as argumentation systems with input output nodes. Note that we do not want to say something like "Φ attacks Ψ if {Φ, Ψ} is not consistent in L" because we do not want to use L. We turn Φ and Ψ syntactically into argumentation networks and remain solidly within the argumentation framework world. We now have BF(Φ x ) attacking BF(Φ y ) where each BF is an argumentation network with input and output nodes. The attack formation associated with Φ, encodes the logical meaning of Φ. This has to be defined and proved. Because of the technical complexity of the transformation from Φ to BF(Φ), this is also done in an appendix. Step 3. We now have the original (S , R) network instantiated by Boolean attack formations. So this is a system of network of networks. Thus steps 1 and 2 reduced the problem of instantiating a network (S , R) by wffs of a logic L, into the problem of instantiating (S , R) by some special argumentation networks called BF (Boolean attack
doi:10.1007/s11787-015-0133-9 fatcat:5jajxyv4zfeohieuekjmdaawgi