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A Formalism for Argument Over Rules of Inference

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A Formalism for Argument Over Rules of Inference Tenacious Tortoises: A Formalism for Argument over Rules of Inference Peter McBurney and Simon Parsons Department of Computer Science University of Liverpool Liverpool L69 7ZF U.K. g fP.J.McBurney,S.D.Parsons @csc.liv.ac.uk May 31, 2000 Abstract Eighty years later, philosopher Susan Haack [9] took up the question of how one justifies the use of MP as a As multi-agent systems proliferate and employ different deductive rule of inference. If one does so by means of and more sophisticated formal logics, it is increasingly examples of its valid application, then this is in essence likely that agents will be reasoning with different rules a form of induction, which (as she remarks) seems too of inference. Hence, an agent seeking to convince an- weak a means of justification for a rule of deduction. If, other of some proposition may first have to convincethe on the other hand, one uses a deductive means of justi- latter to use a rule of inference which it has not thus far fication, such as demonstrating the preservation of truth adopted. We define a formalism to represent degrees across the inference step in a truth-table, one risks us- of acceptability or validity of rules of inference, to en- ing the very rule being justified. So how can one person able autonomous agents to undertake dialogue concern- convince another of the validity of a rule of deductive ing inference rules. Even when they disagree over the inference? acceptability of a rule, two agents may still use the pro- That rules of inference may be the subject of fierce posed formalism to reason collaboratively. argument is shown by the debate over Constructivism in pure mathematics in the twentieth century [21]: here the rule of inference being contested was double negation 1 Introduction elimination in a Reductio Ad Absurdum (RAA) proof: :È ! ɵ ´:È ! :ɵ FROM ´ and È In 1895, the logician Charles Dodgson (aka Lewis Car- INFER :: È roll) famously imagined a dialogue between Achilles FROM :: and a tortoise, in which the application of Modus Po- INFER È nens (MP) was contested as a valid rule of inference Although the choice of inference rules in purely for- mal mathematics may be arbitrary,1 the question of ac- É [4]. Given arbitrary propositions È and , and the two ceptability of rules of inference is important for Arti- ´È ! ɵ É premises È and , one can only conclude from these premises if one accepts that Modus Ponens ficial Intelligence for a number of reasons. Firstly, it is a valid rule of inference. This the tortoise refuses is relevant to modeling scientific reasoning. Construc- to do, much to the exasperation of Achilles. Instead, tivism, for example, has been proposed as a formalism the tortoise insists that a new premise be added to the for modern physics [3], as have other, non-standard log- ics. In the propositional calculus proposed for quantum È ^ ´È ! ɵµ ! É argument, namely: ´ . When Achilles does this, the tortoise still refuses to accept mechanics by Birkhoff and von Neumann [2], for ex- ample, the distributive laws did not hold: É as the conclusion, insisting on yet another premise: È ^ ´È ! ɵ ^ ´´È ^ ´È ! ɵµ ! ɵµ ! É ´ . The 1Goguen [8], for example, argues that standards of mathematical tortoise continues in this vein, ad infinitum. proof are socially constructed. 1 A _ B µ ^ ´A _ C µ ` A _ ´B ^ C µ Ê ´ does not include. For example, may be the use of the A _ B µ ^ C ` ´A ^ C µ _ ´B ^ C µ ´ contrapositive or RAA. There are three ways in which Indeed, it is possible to view scientific debates over al- the dialogue between A and B could then proceed. ternative causal theories as concerned with the validity First, A could attempt to demonstrate that Ê can be of particularmodes of inference, as we have shown with derived from the rules of inference which are contained regard to claims of carcinogenicity of chemicals based in B’s logic. Similarly, A could attempt to demonstrate Ê on animal evidence[13]. Intelligent systems which seek that Ê is admissible in B’s logic [20], i.e. that is to formally model such domains will need to represent an element of that set of inference rules under which these arguments [14]. the theorems of B’s logic remain unchanged.2 In ei- Secondly, it is not obvious that one logical formal- ther of these two cases, it would then be rational for ism is appropriate for all human reasoning, a subject B to accept , being a proposition whose proof com- of much past debate in philosophy (e.g. see [10]). A mences from agreed assumptions and which uses infer- many-valued logic proposed for quantum physics, for ence rules equivalent (in the sense of derivability or ad- instance, has also been suggested to describe religious missibility) to those B has adopted. In such a case, the reasoning in Azande and Nuer societies, reasoning difference of opinion is resolved, to the satisfaction of which appeared to contravene Modus Ponens [5]. In- both agents. deed, some anthropologists have argued that formal hu- Suppose then that A is unable to prove that Ê is man reasoning processes are culturally-dependent and derivable from or admissible in B’s logic. The second hence different across cultures [18]. To the extent that approach which A may pursue is to attempt to give non- this is the case, systems of autonomous software agents deductive reasons for B to adopt Ê. Examples of such acting on behalf of humans will need to reflect the di- reasons could include: scientific evidence for the causal versity of formal processes. In such circumstances, it mechanism possibly represented by Ê, where the rea- is possible that interacting agents may be using logics soning is in a scientific domain; instances of its valid ap- with different rules of inference, as is possible in the plication (e.g. the use of precedents in legal arguments); agent negotiation system of [15]. If one agent seeks the (possibly non-deductive) positive consequences for to convince another of a particular proposition then that B of adopting Ê (e.g. that doing so will improve the first agent may have to demonstratethe validity of a rule welfare of B, of A and/or of third parties); the (possi- of inference used to prove the proposition. Our objec- bly non-deductive) negative consequences for B of not tive in this work is to develop a formalism in which such adopting Ê (e.g. that not doing so will be to the detri- a debate between agents could be conducted. ment of B, of A and/or of third parties); or empirical evidence which would impact the choice of a particular logic.3 The precise nature of such arguments will de- 2 Arguments over rules of infer- pend upon the domain represented by the multi-agent system, and the nature of the proposition . Moreover, ence for A to successfully convince B using such arguments, B would require some formal means of assessing them, We begin by noting that a dialogue between two agents perhaps using a logic of values as outlined in [7]. Al- in which one only asserts, and the other only denies, a though currently being explored, these ideas are not rule of inferencewill not likely lead very far. A dialogue pursued further here. between agents concerning a rule of inference will need Suppose, however, that A exhausts all such argu- to express more than simply their respective positions ments, and still fails to convince B to adopt either Ê if either agent is to be persuaded to change its position. or . Then, a third approach which A could pursue is What more may be expressed? to represent B’s misgivings over the use of Ê in an ap- Suppose we have two agents, denoted A and B, and propriate formalism and use this to seek compromise that A seeks to convince B of a proposition . For ex- 2 ample, this may be a joint intention which A desires Note that Ê could be admissible in B’s logic yet not derivable from the axioms and inference rules of that logic. All derivable rules both agents to adopt. B asks for a proof of . Suppose that A provides a proof which commences from axioms are admissible, however [20]. 3Theory change in logic on the basis of empirical evidence has which are all accepted by B. Assume, however, that this been much discussed in philosophy, typically in a context of holist proof uses a rule of inference Ê which B says its logic epistemology [17]. 2 between the two of them. We term such a formalism for the probability that the application of the inference an Acceptability Formalism (AF) and see it as akin to rule is invalid. Thus, we cannot say that the application formalisms for representing uncertainty regarding the of the inference rule is valid in any one case, but we truth of propositions. Note that while B’s misgivings can say that, if applied to repeated samples drawn from concerning rule Ê may arise from uncertainty as to its the same population, it will be invalidly applied (say) at validity, they need not: B may be quite certain in reject- most 5% of the time. Thus, the calculation of Ô-values ing the rule. for statistical hypothesis tests, which is common prac- What would be an appropriate formalism for repre- tice in the biological and medical sciences [19], effec- senting degrees of acceptability of a rule of inference? tively associates each inference with a value from the fÔ : Ô ¾ ´¼; ½µg ½¼¼´½ Ôµ± At this point, A has adopted Ê and B has not, so that, in set .
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