Abstract
We can distinguish two non-equivalent ways in which a natural language argument can be valid: it can be interpretationally or representationally valid. However, there is just one notion of classical first-order validity for formal languages: truth-preservation in all classical first-order models. To ease the tension, Baumgartner (Synthese 191:1349–1373, 2014) suggests that we should understand interpretational and representational validity as imposing different adequacy conditions on formalizations of natural language arguments. I argue against this proposal. To that end, I first show that Baumgartner’s definition of representational validity is extensionally inadequate. I present a number of natural language arguments that we pre-theoretically hold to be representationally valid, but are not representationally valid according to Baumgartner’s definition. I then point to two further untenable features of Baumgartner’s definitions: (i) according to Baumgartner’s definition of a representationally correct formalization, we cannot arrive at formalizations in a recursive way, and (ii) Baumgartner’s definition of representational validity is non-monotonic. I conclude that interpretational and representational validity cannot be understood as merely imposing different adequacy conditions on formalizations. If we want to capture our interpretational and representational intuitions, we need two different formal definitions of validity.
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Notes
Etchemendy (1990) introduced the distinction between interpretational and representational semantics in the context of his famous critique of Tarski’s definition of logical consequence (cf. Tarski 1956). For discussions of Etchemendy’s critique of Tarski, see for example Priest (1995), Ray (1996), Sher (1996), Gómez-Torrente (1996), Gómez-Torrente (1999), Gómez-Torrente (2009), Hanson (1997), Patterson (2008), but also Etchemendy (2008).
From here on, all page numbers refer to Baumgartner 2014.
Etchemendy (1990, Chap. 8) famously argued that there are also arguments that are interpretationally, but not representationally valid. The most prominent examples are sentences about the size of the world, as, e.g. “\(\exists \hbox {x}\exists \hbox {y} (\hbox {x}~\ne ~\hbox {y}\))”. Etchemendy claims that—if true—this sentence is trivially true under every interpretation of the non-logical constants, as it does not contain any. Thus, every argument with this sentence as its conclusion will be declared interpretationally valid. (If the sentence is false, every argument with the conclusion “\(\lnot [\exists \hbox {x}\exists \hbox {y}~(\hbox {x}~\ne ~\hbox {y}\))]” is said to be interpretationally valid.) I will not discuss these examples of Etchemendy here.
To be precise, the interpretational definition of validity demands truth-preservation under all interpretations, rather than truth-preservation under all substitutions. The difference between the interpretational and the so-called substitutional definition of validity becomes important once we consider models with domains with unnamed objects. To avoid cumbersome formulations I will, however, follow Baumgartner in ignoring the difference between these two definitions whenever I can.
I use Greek upper case letters as parameters for natural language sentences (or sets thereof), and Greek lower case letters as parameters for formal language sentences (or sets of thereof).
A notable exception is Halbach 2010.
My formulation of (\(r-\mathrm{COR}\)) slightly differs from Baumgartner’s original definition as I make the reference to a correspondence scheme explicit (see the next paragraph for an explanation). For reasons of simplicity, I omit indexing models to \(\varphi \). (A model \({\mathfrak {M}}^{\varphi }\) provides an interpretation of the categorematic terms occurring in \(\varphi \).).
More precisely, a model need not stand for a whole world, but only for a part of it, i.e. a model stands for a situation.
“\(t^{{\mathfrak {M}}}\)” stands for the interpretation of the constant \(t\) in the model \({\mathfrak {M}}\). To state this more precisely, let \({\mathfrak {M}}\) consist of the interpretation function \({\mathfrak {I}}\) and the non-empty domain D. We then have \(t^{{\mathfrak {M}}}\,= {\mathfrak {I}}(t)\) with \({\mathfrak {I}}(t)~\in \) D iff \(t\) is an individual constant, and \({\mathfrak {I}}(t)~\subseteq \hbox {D}^{n}\) iff \(t\) is an \(n\)-place relation constant.
See p. 1361 for Baumgartner’s exact definition of \(i\)-correctness.
More precisely, Baumgartner’s definition of \(i\)-correctness does not demand that the extensions be identical, but only that the actual extension of the natural language term corresponds to the extension of the term of the formal language interpreted in the model under consideration. For Baumgartner’s notion of correspondence, see p. 1360.
Baumgartner stresses that the permissible reinterpretations need not only involve “English names and predicates that are listed in standard English dictionaries” as “the space of possible reinterpretations provided by the model-theoretic machinery far exceeds the expressive power of natural English” (p. 1361).
To apply the definition of \(r\)-correctness to whole arguments, arguments must be understood as complex statements: the argument \(<\!\!\gamma , \psi \!\!>\) is understood as the material conditional \({\gamma }_{1}\,\wedge \,{\ldots }\wedge \, {\gamma }_{n}\,\rightarrow \psi (\hbox {with}\,\gamma = \{{\gamma }_{1,} {\ldots }, {\gamma }_{n}\)}). (See p. 1363)
If your intuitions differ here, you will not accept (fa1) as an \(i\)-correct formalization of (a). As (fa1) is not FOL-valid anyway, this will have no consequences for the question whether or not argument (a) is interpretationally valid (see the definition of interpretational validity below). In the end, we are interested in the (interpretational) (in)validity of (a) and thus any discussion about the “correct intuitions” concerning the above reinterpretation is irrelevant.
Argument (a) was introduced as a paradigm example of an argument that is intuitively judged to be representationally, though not interpretationally valid. As (fa2) is FOL-valid, argument (a) would be declared interpretationally valid by Baumgartner’s account in case one judges (fa2) to be \(i\)-correct (see the definition of interpretational validity below). We should therefore concede that the above reinterpretation is intuitively inadmissible.
See Baumgartner (2014), p. 1371, for more on the notion of a minimally complex formalization.
See Baumgartner (2014), p. 1369, for the exact definition of rr-correctness.
Nevertheless, in a different context, Baumgartner & Lampert demand that “the expressions assigned to the categorematic parts of a formula by its realization [i.e. its correspondence scheme – the author] are logically independent and neither tautologous not contradictory.” (Baumgartner and Lampert 2008, p. 110).
More precisely, proponents of the view that there are two informal notions of validity, interpretational and representational validity, will hold that there are two correlated notions of logical possibility and, therefore, of logical independence.
The distinction between determinables and determinates is due to Johnson (1921), Chap. XI.
If you do not think that humans are humans by necessity, choose your favorite example of a necessary property. Of course, this type of example only works under the assumption that there are objects that have certain properties by necessity (or, with obvious modifications, under the assumption that there are objects lacking certain properties by necessity).
Baumgartner and Lampert (2008), p. 105, explicitly consider whether there is a more suitable formalism than first-order predicate logic to deal with statements involving color predicates. This would directly be relevant for argument (b).
I here only refer to sentences which are Boolean connections of atomic sentences. I do not want to commit myself to the claim that we can formalize, e.g. molecular sentences involving atomic sentences in the scope of intensional operators in a recursive way (as e.g. in “Ben believes that Ann likes Cen.”).
Of course, this assumes that the necessary adjustments concerning the identities and non-identities of the constants and variables are made. Although “Fa” might be an adequate formalization of “Daryl is a woman” and “Ga” might be an adequate formalization of “Russell is a man” considered individually, “\({Fa} \wedge {Ga}\)” is not an adequate formalization of “Daryl is a woman and Russell is a man.” Obviously, the case discussed in the main text is not of this kind.
See especially Sainsbury (2001), Chap. 6, for a discussion of the claim that formalizations provide systematic representations of the truth conditions of natural language sentences. See also Epstein (1990) and Brun (2004) for more on systematic formalization. Halbach (2010), Chap. 3 and Chap. 7, also assumes that formalizations of molecular sentences are build up from formalizations of their atomic parts.
See Peregrin and Svoboda 2013, pp. 2906–2907, for a similar example. They also use their example to show that IN does not allow us to arrive at formalizations in a recursive way.
Needless to say, (g) seems to constitute a further counter-example to Baumgartner’s definition of representational validity: the natural language argument (g) is intuitively representationally valid, though there seems to be no \(r\)-correct and FOL-valid formalization of (g).
Similarly, Peregrin and Svoboda 2013, p. 2909, claim that “[s]uch an exclusion [of inadmissible models – the author] brings us beyond the boundaries of logic [...].”
For the sake of simplicity, I use a possibilist semantics where we have one domain of objects for all worlds.
For the sake of simplicity, I spell out the semantics only for a language without variables.
Of course, the role of the correspondence scheme has to be spelled out more carefully, but here is not the place to do so.
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Acknowledgments
I am indepted to Michael Baumgartner, Wolfgang Freitag, Wolfgang Spohn and Holger Sturm for discussions and comments on an earlier draft of this paper.
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Zinke, A. On exhibiting representational validity. Synthese 192, 1157–1171 (2015). https://doi.org/10.1007/s11229-014-0607-y
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DOI: https://doi.org/10.1007/s11229-014-0607-y