The Relational Syllogistic Revisited

The Relational Syllogistic Revisited Ian Pratt-Hartmann1 The relational syllogistic is an extension of the language of Classical syllogisms in which predicates are allowed to feature transitive verbs with quantified objects. It is known that the relational syllogistic does not admit a finite set of syllogism-like rules whose associated (direct) derivation relation is sound and complete. We present a modest ex- tension of this language which does. 1 Introduction By the Classical syllogistic, we mean the language featuring the four sentence-forms of standard Aristotelian syllogisms: “Every p is a q”, “Some p is a q”, “No p is a q” and “Some p is not a q”, where p and q are substituted by common (count) nouns. By the relational syllogistic, we mean the extension of the Classical syllogistic in which predicates are additionally allowed to feature transitive verbs with quantified objects. For example, the following is a valid argument in this language. Some artist admires no beekeeper Every beekeeper admires some artist Some artist is not a beekeeper Indeed, consider any artist, a, who admires no beekeeper. If he is not a beekeeper himself, the conclusion is certainly true. On the other hand, if he is a beekeeper, the second premise guarantees the existence of an artist b, whom he admires, and who cannot therefore be a beekeeper (since otherwise a would not admire b), whence the conclusion is again true. Thus, the relational syllogistic allows us to formulate arguments that—subjectively at least—are more difficult than those we encounter 1University of Manchester 195 LiLT Volume 9 Perspectives on Semantic Representations for Textual Inference. Copyright c ⃝ 2014, CSLI Publications. 196 / Ian Pratt-Hartmann in the Classical syllogistic. For the Classical syllogistic, the familiar collection of syllogisms con- stitutes, together with one or two ancillary rules, a (sound and) com- plete inference system: if an argument in this language is valid, in the sense that there is no possible world making its premises true and its conclusion false, then that validity can be demonstrated using syllogis- tic inference steps alone. This was first shown, in a slightly weakened form, by Corcoran Corcoran (1972) and Smiley Smiley (1973). Specifi- cally, these authors proved certain systems of syllogism-like rules to be (sound and) refutation-complete: if an argument is valid, then an absur- dity can be derived from its premises together with the negation of its conclusion. Subsequently, Pratt-Hartmann and Moss Pratt-Hartmann and Moss (2009) obtained the following results: (i) there is a finite set of syllogism-like rules in the Classical syllogistic which is sound and complete (not just refutation-complete); (ii) there is a finite set of syllogism-like rules in the relational syllogistic which is sound and refutation complete; (iii) there is no finite set of syllogism-like rules in the relational syllogistic which is sound and complete. In other words, for the relational syllogistic, indirect reasoning is essential. This paper presents a modest extension of the relational syllogistic, which additionally features sentences of the form: “If there are p’s, there are q’s.” The following …

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