Abstract
Most literature on vagueness deals with the phenomenon as applied to predicates. On the contrary, even the idea of vague connectives seems to be taken as an oxymoron. The goal of this article is to propose an understanding of vague logical connectives based on vague quantifiers. The main idea is that the phenomenon of vagueness translates to connectives in terms of the property of Abnormality. I also argue that Prior’s Tonk can, according to this approach, be considered a vague connective. In order to do so, I provide a sound and complete interpretation for it, based on Strict-Tolerant semantics, in which Tonk has a non-normal truth-table.
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Notes
Inasmuch as they are logical, quantifiers are not just any sort of predicate. In particular, their meaning is not as variable as regular predicate letters’ is: their extensions do vary from model to model, but not arbitrarily. A little more on this will be said in Sect. 3, though not a lot.
A brief commentary on notation: I will use as the name of the function which interprets the symbol , but when the distinction between the two generates no confusion, I will loosely use the word “quantifier” to refer to either the symbol or the function.
Counting the elements of a fuzzy set is a big problem in itself. The easiest option is to take what is known as the \(\Sigma \)-count, which takes the sum of the degree of membership of each element. Problems with \(\Sigma \)-count appear because very different sets turn out to have the same number of elements. For instance, a set with many things which are not very F and a set with few things which are very F. But when dealing with connectives, we will only work with three different degrees, instead of the whole real interval. Thus, such subtleties are of no concern.
Notice also that propositional quantifiers are considered to be second order quantifiers.
In a sense, connectives are used to assert a certain number of the subformulas to which they are applied. More specifically, sentences involving them can be paraphrased by sentences which employ numerical predicates applied to sentences: “A and B” can be paraphrased as “Both ‘A’ and ‘B’ are true”, “A or B” can be paraphrased as “At least one of ‘A’ and ‘B’ are true”, “Not A” can be more awkwardly paraphrased as “None of these are true: ‘A”’
Remember that a rule is admissible in a set A of inferences if and only if, for all its instances, either some of their premises are not in A or its conclusion is in A.
Some authors in this tradition, such as Ripley (2013), prefer a more pragmatic reading in terms of some sentences being neither assertible nor deniable, although that is not the path we follow here
The way the structural rules come into play will be slightly different in some cases, without altering the main issue.
Thanks to Alba Cuenca for the design!
Contingently, our tree does not branch, since all rules are one-premise, but of course this would not be so if we had richer vocabulary.
Now is time to remember the \(\Sigma \)-count. When we speak about the set of true arguments of
. we are not always referring to a crisp set, given that in the present framework, sentences can have intermediate values. For example, take a 1 constant \(\top \) and a 1/2 constant \(\leftthreetimes \). Thus, if we use \(\Sigma \)-count to determine the cardinality of the set of true arguments, we should for instance say that
has 1.5 true arguments, and
has one.
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Acknowledgements
I would like to thank the annonymous referee, for their helpful comments and kind words; to the Buenos Aires Logic Group, for being the absolute best; to all the audiences that have listened to and commented on different versions of this work, specially to Roy Cook and Dave Ripley. This paper could not have been written without the financial aid of the National Scientific and Technical Research Council (CONICET).
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Teijeiro, P. Vague connectives. Philos Stud 180, 1559–1578 (2023). https://doi.org/10.1007/s11098-022-01817-2
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DOI: https://doi.org/10.1007/s11098-022-01817-2