Abstract
Monotonicity-based inference is a fundamental notion in the logical semantics of natural language, and also in logic in general. Starting in generalized quantifier theory, we distinguish three senses of the notion, study their relations, and use these to connect monotonicity to logics of model change. At the end we return to natural language and consider monotonicity inference in linguistic settings with vocabulary for various forms of change. While we mostly raise issues in this paper, we do make a number of new observations backing up our distinctions.
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Notes
- 1.
An extensive overview of monotonicity inference with generalized quantifiers can be found in (Peters and Westerståhl 2006).
- 2.
There is also an intuitive temporal aspect to the whales example, where extensions change with the passage of time. Such more intensional aspects of monotonicity inference will be considered briefly at the end of this paper.
- 3.
In particular, with this definition, it is never true that D-relatively many A are D. We will not discuss other variants of relative “many” here.
- 4.
This inequality is equivalent to \(kx + x > mn\) which is implied by the earlier \(kx > mn\).
- 5.
A realistic concrete use of monotonicity in mathematics is the convergence test for improper integrals discussed in (Icard et al. 2017).
- 6.
In this combined calculus, monotonicity applies to both set inclusion for denotations and greater-than for numbers. The former is a type-lifting of the latter, and many more complex type-theoretic lifts support monotonicity reasoning (van Benthem 1991). However, beyond these, in natural language monotonicity can apply to many orderings that are sui generis: conceptual, temporal, spatial, and so on. Can the style of analysis in this paper be generalized to cover these?.
- 7.
Many inferences are intuitively about single occurrences of parts of expressions. But some require comparing coordinated occurrences, like in the logical rule of Contraction, where two identical premises can be contracted to just one.
- 8.
Extensions to richer type logics of relevance to natural language seem an open problem, cf. (van Benthem 1991) on the case of the Boolean Lambda Calculus.
- 9.
This device has not been studied yet in the literature, to the best of our knowledge, but as we shall see momentarily, it is close to second-order logic.
- 10.
This can be simplified to \(\lnot \exists x (Px \wedge Qx)\, \wedge \exists x (\lnot Px \wedge Qx) \wedge \exists x \lnot Qx\, \wedge \, Px \wedge \lnot Qx.\)
- 11.
Enlargement Monotonicity is then expressed by modal combinations like \(\langle \subseteq \rangle \langle \equiv _P^{+}\rangle \).
- 12.
It is easy to see with simple concrete examples of PAL update that downward monotonicity fails as well for announced formulas: \([!\varphi ]\psi \) does not imply \([!(\varphi \wedge \alpha )]\psi \).
- 13.
The exact information content of an announcement \(!\varphi \) is that \(\varphi \) was true before the announcement (the caveat is needed since announcing an epistemic statement \(\varphi \) might change its truth value), and if \(\psi \) subsequently adds no new information, this means that the \(!\psi \) update does not change the model. Thus, a way of taking dynamic entailment is as a valid implication \(Y\varphi \rightarrow \psi \), where Y is a one-step backward-looking temporal operator beyond the language of PAL, cf. (Sack 2007).
- 14.
To make the above questions fully precise, we need to define syntactic polarity of occurrences in PAL formulas, where occurrences inside announced formulas may lack polarity. Also, given the intensional setting for PAL of a universe of many epistemic models connected through updates, the earlier semantic notion of monotonicity can be phrased in a number of ways. Finally, we need not confine ourselves to syntactic properties of single occurrences of predicates. A proper notion of monotonic inference for formulas \([!\varphi ]\alpha \) might involve correlated simultaneous replacements of proposition letters in both \(\varphi \) and \(\alpha \). We leave these detailed issues for follow-up work. A first exploration of possible Lyndon-style theorems for PAL can be found in (Yin 2020).
- 15.
(Liu and Sun 2020) discuss such inference patterns in the ancient Chinese language.
- 16.
With this richer linguistic vocabulary in monotonicity reasoning, the more general orderings of Footnote 7 may also come to the fore. Thomas Icard (p.c.) gives the nice example of “The tree is tall. The tree grows. Therefore, the tree is still tall.”.
- 17.
The difference between inclusions locally true in the actual world and inclusions true also in other worlds remains somewhat hidden in common phrasings of upward monotonicity inference as a pattern “from \(\varphi (P)\) to \(\varphi (P \vee Q)\)”. The inclusion from P to \(P \vee Q\) is universally valid, so usable anywhere.
- 18.
There are many further intensional aspect to monotonicity inference that we cannot address here. For instance, such inferences seem sensitive to description. In the ancient Mohist example that “Your sister is a woman. But loving your sister is not loving a woman”, the issue may be under which description we are viewing the loving (‘as a relative’ vs. ‘romantically’). This distinction is widespread. Oedipus killed a man on the road, but did not realize that the man was his father. Did he kill his father? Under one description: yes, under another: no. For many further instances of the role of description in intensional contexts, see (Aloni 2001), (Holliday and Perry 2015). Should we consider a more refined notion of monotonicity inference where inference can take place at either the level of denotations, or that of descriptions?.
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Acknowledgments
We thank Thomas Icard, Zhiqiang Sun, Jouko Väänänen, Dag Westerståhl, Jialiang Yan, the audience of the DALI Workshop (Prague 2020), the LoLaCo audience at the ILLC Amsterdam, and the referees of successive versions of this paper for helpful comments and useful information. This research is supported by the Major Program of the National Social Science Foundation of China (NO. 17ZDA026).
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van Benthem, J., Liu, F. (2020). New Logical Perspectives on Monotonicity. In: Deng, D., Liu, F., Liu, M., Westerståhl, D. (eds) Monotonicity in Logic and Language. TLLM 2020. Lecture Notes in Computer Science(), vol 12564. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-62843-0_1
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