Adverbs

Author(s):  
James Higginbotham

Adverbs are so named from their role in modifying verbs and other non-nominal expressions. For example, in ‘John ran slowly’, the adverb ‘slowly’ modifies ‘ran’ by characterizing the manner of John’s running. The debate on the semantic contribution of adverbs centres on two approaches. On the first approach, adverbs are understood as predicate operators: for example, in ‘John ran slowly’, ‘ran’ would be taken to be a predicate and ‘slowly’ an operator affecting its meaning. Working this out in detail requires the resources of higher-order logic. On the second approach, adverbs are understood as predicates of ‘objects’ such as events and states, reference to which is revealed in logical form. For example, ‘John ran slowly’ would be construed along the lines of ‘there was a running by John and it was slow’, in which the adverb ‘slowly’ has become a predicate ‘slow’ applied to the event that was John’s running. Since adverbs are exclusively modifiers, they are classed among the syncategorematic words of terminist logic, the investigation of which carried the subject forward from Aristotle in the thirteenth century. (The contrasting ‘categoremata’ – grammatical subjects and predicates – are those words which have meaning independently.) They are of contemporary interest for philosophical logic and semantic theory, because particular accounts of them carry implications for the nature of combinatorial semantics and language understanding, and for ontology.

Author(s):  
Peter Fritz ◽  
Harvey Lederman ◽  
Gabriel Uzquiano

AbstractAccording to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise.


2008 ◽  
Vol 21 (4) ◽  
pp. 377-409 ◽  
Author(s):  
Scott Owens ◽  
Konrad Slind

Author(s):  
Crispin Wright

The paper explores the alleged connection between indefinite extensibility and the classic paradoxes of Russell, Burali-Forti, and Cantor. It is argued that while indefinite extensibility is not per se a source of paradox, there is a degenerate subspecies—reflexive indefinite extensibility—which is. The result is a threefold distinction in the roles played by indefinite extensibility in generating paradoxes for the notions of ordinal number, cardinal number, and set respectively. Ordinal number, intuitively understood, is a reflexively indefinitely extensible concept. Cardinal number is not. And Set becomes so only in the setting of impredicative higher-order logic—so that Frege’s Basic Law V is guilty at worst of partnership in crime, rather than the sole offender.


10.29007/n6j7 ◽  
2018 ◽  
Author(s):  
Simon Cruanes

We argue that automatic theorem provers should become more versatile and should be able to tackle problems expressed in richer input formats. Salient research directions include (i) developing tight combinations of SMT solvers and first-order provers; (ii) adding better handling of theories in first-order provers; (iii) adding support for inductive proving; (iv) adding support for user-defined theories and functions; and (v) bringing to the provers some basic abilities to deal with logics beyond first-order, such as higher-order logic.


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