# Nonmonotonic Logic

### APPLICATIONS OF CIRCUMSCRIPTION TO FORMALIZING COMMON SENSE ( 1-Dec-2003)

In particular this paper properly treats variables that are not minimized.

### Prospects for a Kantian Machine

Rule-based ethical theories like Kant's appear to be promising for machine ethics because of the computational structure of their judgments. The ethical judgment is then an outcome of the consistency test, in light of the supplied rules. While this kind of test can generate nontrivial results, it might do no more than reflect the prejudices of the builder of the declarative set; the machine will "reason" straightforwardly, but not intelligently. This article is part of a special issue on Machine Ethics.

### 1 Partial Models and Nonmonotonic Inference

The advantages of this approach are a natural and intuitively satisfying formalization of diverse types of nonmonotonic reasoning, among them domain closure, the unique names hypothesis, and default reasoning. The purpose of this paper is to introduce a form of nonmonotonic inference based on the notion of a partial model of the world. If there are no partial models for a given theory, then for every conjecture a the operator Fa, becomes logical deduction, and no nonmonotonic inference takes place. If the partial models of a theory fully cover the intended models (that is, every intended model is an extension of some partial model), then a conjecture on the partial models takes into account all of the interpretations of the theory.

### 1 Partial Models and Nonmonotonic Inference

The advantages of this approach are a natural and intuitively satisfying formalization of diverse types of nonmonotonic reasoning, among them domain closure, the unique names hypothesis, and default reasoning. The purpose of this paper is to introduce a form of nonmonotonic inference based on the notion of a partial model of the world. If there are no partial models for a given theory, then for every conjecture a the operator Fa, becomes logical deduction, and no nonmonotonic inference takes place. If the partial models of a theory fully cover the intended models (that is, every intended model is an extension of some partial model), then a conjecture on the partial models takes into account all of the interpretations of the theory.

### Counterfactuals

A counterfactual is a statement such as, "if p, then Typical examples are, "If the electricity hadn't failed, dinner would have been ready on time," or "If the bedroom door were open, I could get the widget I left in there." The original problem reduces to proving the counterfactual (in some suitable sense) and to arranging for thusand-so to be true. If J. Edgar Hoover had been born Glymour and Thomason [5] seem to infer from this last observation that the study of nonmonotonic inference generally can be subsumed to some extent under an investigation of counterfactuals, but in light of the breadth of the nonmonotonic nature of commonsense reasoning (the fine problem, indicative conditionals, etc. Lewis points out that this may not be the case by examining the counterfactuals, "If Bizet and Verdi had been compatriots, Bizet would have been Italian," and, "If Bizet and Verdi had been compatriots, Bizet would not have been Italian."

### Counterfactuals

A counterfactual is a statement such as, "if p, then Typical examples are, "If the electricity hadn't failed, dinner would have been ready on time," or "If the bedroom door were open, I could get the widget I left in there." The original problem reduces to proving the counterfactual (in some suitable sense) and to arranging for thusand-so to be true. If J. Edgar Hoover had been born Glymour and Thomason [5] seem to infer from this last observation that the study of nonmonotonic inference generally can be subsumed to some extent under an investigation of counterfactuals, but in light of the breadth of the nonmonotonic nature of commonsense reasoning (the fine problem, indicative conditionals, etc. Lewis points out that this may not be the case by examining the counterfactuals, "If Bizet and Verdi had been compatriots, Bizet would have been Italian," and, "If Bizet and Verdi had been compatriots, Bizet would not have been Italian."

### 4Ip

Does Probability Have a Place in Nonmonotonic Reasoning? COMPUTER SCIENCE DEPARTMENT Stanford University Stanford, California 94305 Does Probability Have a Place in Nonmonotonic Reasoning? Should probabilities be used to implement nonmonotonic reasoning systems? Perhaps probabilities can be used effectively here; in any event, the general nature of auto-epistemic reasoning is sufficiently unclear that more fundamental questions about its nature need to be answered before any specific nonmonotonic implementation can be applied to it.

### 4Ip

Does Probability Have a Place in Nonmonotonic Reasoning? COMPUTER SCIENCE DEPARTMENT Stanford University Stanford, California 94305 Does Probability Have a Place in Nonmonotonic Reasoning? Should probabilities be used to implement nonmonotonic reasoning systems? Perhaps probabilities can be used effectively here; in any event, the general nature of auto-epistemic reasoning is sufficiently unclear that more fundamental questions about its nature need to be answered before any specific nonmonotonic implementation can be applied to it.

### Preferences and Nonmonotonic Reasoning

Selecting extended logic programming with the answer-set semantics as a "generic" nonmonotonic logic, we show how that logic defines preferred belief sets and how preferred belief sets allow us to represent and interpret normative statements. Conflicts among program rules (more generally, defaults) give rise to alternative preferred belief sets. Finally, we comment on formalisms which explicitly represent preferences on properties of belief sets. Such formalisms either build preference information directly into rules and modify the semantics of the logic appropriately, or specify preferences on belief sets independently of the mechanism to define them.