Abstract
The point of formalisation is to model various aspects of natural language. Perhaps the main use to which formalisation is put is to model and explain inferential relations between different sentences. Judged solely by this objective, a formalisation is successful in modelling the inferential network of natural language sentences to the extent that it mirrors this network. There is surprisingly little literature on the criteria of good formalisation, and even less on the question of what it is for a formalisation to mirror the inferential network of a natural language or some fragment of it. This paper takes some exploratory steps towards a quantitative account of the main ingredient in the goodness of a formalisation. We introduce and critically examine a mathematical model of how well a formalisation mirrors natural-language inferential relations.
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Notes
We assume throughout that arguments consist of sentences. This is not entirely uncontroversial; see Russell (2008).
As noted by commentators, e.g. in Peregrin & Svoboda (2013, p. 2899); for discussion, see that article as well as Sainsbury (2001, ch. 6), Epstein (2001, ch. V), Baumgartner and Lampert (2008), Blau (1977) and Brun (2004; 2008). The relatively scant literature on the topic tends to speak of the ‘adequacy’ of formalisation. I prefer the word ‘goodness’, since it is more suggestive of the idea that the match between natural and formal languages is a matter of degree.
Formal semanticists tend to be less concerned with inference than philosophical logicians; see Heim and Kratzer (1998) for a textbook illustration of the semanticists’ approach. For the relation between semantic form—the form attributed to a sentence by a compositional meaning theory—and that sentence’s formalisation in some logic, see e.g. Lepore and Ludwig (2002).
Here Quine is thinking of the formal language as interpreted, unlike us.
Quine (1960, p. 160); I have respectively italicised and de-italicised parts of the sentence.
As we shall have occasion to observe once more in Sect. 5, inferential-role preservation cries out for supplementation with a grammatical constraint.
Throughout this paper, sentences of a formal language are uninterpreted. We take a formal language to be individuated by its language (vocabulary and grammar) and semantics. Standard logics have their standard semantics. We allow empty domains for first-order logic.
Strictly speaking, the conclusion is either a singleton set or the empty set.
Strictly speaking, ‘\(B\) therefore \(A\)’ below is the quotation of the sentence with \(B\) and \(A\) replaced by ‘Something is fair’ and ‘Alberta is fair’. We shall not fuss over such matters.
See Williamson (2007, ch. 4) for discussion of this inference.
Peregrin & Svoboda (2013, p. 2916) hint at something like the idea to follow.
How to weight the various desiderata constraining formalisation is a separate question, not addressed here.
On whether logic should concern itself with formal consequence, see Read (1994).
If \(\chi _{i}^{\Phi }\) takes values \(s\) (for success) and \(f\) (for failure) respectively, where \(s > f\), a few manipulations (which go through because all the sums are bounded) show that the resulting measure \({\mathcal {M}}_{(s,f)}(\Phi ) \in [f,s]\) is given by \({\mathcal {M}}_{(s,f)}(\Phi )= (s-f){\mathcal {M}}_{(1,0)} (\Phi ) + f\).
Subtracting 1, which does not affect the value of the infinite cardinal.
We assume that the \(F\)-validity or \(F\)-invalidity of \(F\)-arguments for the values of \(F\) of interest is a determinate matter.
Who the ‘we’, presupposed by the word ‘our’, are is a flexible matter: the account can model individual subjects’ judgments, or those of the inferential community as a whole, or some subset of it.
See e.g. the early Machina (1976).
‘Pre-theoretically’ meaning prior to the enterprise of formalisation; naturally, the \(N\)-sentences themselves may be theoretical in some other sense (e.g. they may be sentences of mathematics or science).
This is an empirical claim, of course, and is compatible with the fact that sometimes we are in an opposite situation. In the relatively rare case in which all the inferences of interest fall into this category, the second reply does not apply.
A crucial question about how systematic formalisation is also here broached, namely whether and if so in what sense formalisations must be the result of some systematic procedure of formalising. See some of the cited works by Blau, Brun, Baumgartner & Lampert for discussion.
Proof. Assume without loss of generality that the index set \(I\) is an ordinal. For \(\alpha \) an ordinal smaller than \(I\), define \(\chi ^{\alpha } \in \{0,1\}^{I}\) as follows:
$$\begin{aligned} \chi ^{\alpha }(i) = \left\{ \begin{array}{rl} 1 &{} \text {if } i < \alpha \\ 0 &{} \text {if } i \ge \alpha \end{array}\right. \end{aligned}$$By Dominance, \(M(\chi ^{\alpha }) < M(\chi ^{\beta })\) iff \(\alpha < \beta \). If \(I\) is uncountable, the non-empty, disjoint, open, real-intervals (\(M(\chi ^{\alpha }), M(\chi ^{\alpha + 1})\)) where \(\alpha < I\) are uncountably many, which is a contradiction. The reason is that the topology of the reals is ccc; indeed the reals are separable, as any non-empty real interval contains a rational number, of which there are countably many. It is immediate that \(N\) has uncountably many valid arguments (e.g. let the conclusion be a member of the premiss set) and uncountably many invalid arguments (e.g. let the conclusion be any contradiction and the premiss set be any subset of the countably infinite set of logical truths).
An analogous project in the foundations of economics is to look for more general domains than the real numbers in order to accommodate utility functions for preference rankings without countable order-dense subsets (i.e. non-separable total orders, in the usual order topology). See Chipman et al. (1971) for more.
Adapting the argument in footnote 24.
That is, an ascending sequence whose length is \(2^{\aleph _0}\), construed as an ordinal—the least ordinal equinumerous to the set of functions from \(\aleph _0\) to \(2 = \{0, 1\}\).
Invalidity versions of these principles are also derivable, e.g.
$$\begin{aligned} Subset: \text {If } P \nRightarrow c \text { and } P^{-} \subseteq P \text { then } P^{-} \nRightarrow c, \end{aligned}$$is derivable from \(Superset\), which expresses the idea that deductive inference is monotonic.
More generally, a language with an infinity \(|N |\) of sentences and a compact entailment relation has a supervenience base of size no greater than \(|N |^{<\omega } = |N |\).
Proof. Suppose the countably many validity facts \(P_1 \Rightarrow c_1, P_2 \Rightarrow c_2, \dots , P_n \Rightarrow c_n, \dots \) are a Reflexivity-, Superset- and Cut-supervenience basis for the \(2^{\aleph _0}\) validity facts of the second type. Order the countable set \(\{\exists _{\ge n}: n \text { is a natural number}\}\) using the standard order on the indices. Then define \(Q_n\) recursively as follows: \(Q_0 = P_1\); \(Q_{n+1} = Q_n \cup \{\)the second least \(k\) such that: \(\exists _{\ge k} \in P_{n+1}\) and \(k >\) the greatest \(i\) such that \(\exists _{\ge i} \in Q_n\}\). Finally, define \(Q_\omega = \bigcup _{n \in {\mathbb {N}}} Q_n\). Note that \(|Q_n |= n\) and that \(Q_n\) omits at least one element from each \(P_{n}\). Thus \(Q_\omega \) is an infinite set which is not a superset of any \(P_{n}\). It follows that the countably many validity facts \(P_1 \Rightarrow \exists _{\infty }, P_2 \Rightarrow \exists _{\infty }, \dots , P_n \Rightarrow \exists _{\infty }, \dots ,\) do not determine the validity of \(Q_\omega \Rightarrow \exists _{\infty }\).
Compare Shapiro (1998, p. 137)
Thanks to Gila Sher and Otávio Bueno for the invitation to speak at the UNILOG workshop held in April 2013 in Rio de Janeiro, which prompted this paper, and for their continuing interest in the ideas herein presented. I received helpful comments at the Rio workshop, at a Colloquium in honour of Dan Isaacson in Oxford in June 2013, at my department’s Theoretical Work in Progress seminar in January 2014, at the Joint Session of the Aristotelian Society and Mind Association held in Cambridge in July 2014, and at the European Congress for Analytic Philosophy in Bucharest in September 2014. Thanks in particular to: Arnold Koslow, Catarina Dutilh Novaes, Chris Timpson, Daniel Rothschild, Hugo Dixon, Jonathan Payne, Martin Pickup, Ralf Bader, Susanne Bobzien, Tim Button and Tim Williamson, as well as to three Synthese referees. Special thanks finally to Owen Griffiths for generous discussion.
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Paseau, A.C. A measure of inferential-role preservation. Synthese 196, 2621–2642 (2019). https://doi.org/10.1007/s11229-015-0705-5
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DOI: https://doi.org/10.1007/s11229-015-0705-5