Abstract
Rudolf Carnap’s logical pluralism is often held to be one in which corresponding connectives in different logics have different meanings. This paper presents an alternative view of Carnap’s position, in which connectives can and do share their meaning in some (though not all) contexts. This re-interpretation depends crucially on extending Carnap’s linguistic framework system to include meta-linguistic frameworks, those frameworks which we use to talk about linguistic frameworks. I provide an example that shows how this is possible, and give some textual evidence that Carnap would agree with this interpretation. Additionally, I show how this interpretation puts the Carnapian position much more in line with the position given in Shapiro (Varieties of Logic. Oxford University Press, Oxford, 2014) than had been thought before.
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Notes
These syntactic rules may have later been expanded to include model-theoretic rules. Early quotes from Carnap suggest a purely proof-theoretic view of logic. Though he held that proof theory was the best way to “do” logic early in his career, his position about logic being a purely proof-theoretic endeavor changed after he met and spoke with Tarski, who convinced him that model-theory was a legitimate enterprise [see, for example, Carnap (1947)].
In particular, they will be given by the L-rules of the framework, which are the rules that govern transformations of logically true sentences into logically true sentences. We will not here concern ourselves with P-rules, which govern transformations of descriptively true sentences into descriptively true sentences. For more information, see (Carnap 1937, p. 133–135).
Pragmatic questions are in principle answerable. There is some question as to whether they are actually external, though. See Steinberger (2015) for an interesting discussion about how pragmatically to select the appropriate linguistic framework.
Thanks to Roy Cook for suggesting this example.
There are important relations between synonymy and analyticity on Carnap’s view. In effect, if Quine (1951) and subsequent authors are right, then Carnap cannot get his notion of analyticity, or his project, off the ground. As this paper is only meant to present an interpretation of Carnap’s views, and not assess whether they are viable, I will not address this further here.
There is some question as to whether Carnap would accept this type of suggestion. It seems that Carnap would have thought that structural rules are meaning determining as well. This type of holism seems to be part of what is expressed in the quote above, when he claims that the postulates and rules of inference determine the meaning of the “fundamental logical symbols”. The point I am trying to make still stands, though. It would be a mistake to think that the connectives in question shared a meaning, since even on the traditional view, they were never candidates for having the same meaning in the first place. Thanks to a referee for suggesting this possibility.
There is another option for interpretation here: it is possible to claim that the mathematicians are simply talking past each other. If we assume this, though, we would have to assume that any dispute they had would be a merely verbal dispute. However, it seems more charitable to assume that sometimes the mathematicians in question can have substantive disagreements, as in the case, for example, where they discuss the status of the intermediate value theorem, which the classicist is provable in the classical system, but not in the intuitionist system. See Shapiro (2014) for more details. Thanks to a referee for pushing me on this issue.
Thanks to Neil Tennant for bringing it to my attention and to a referee for doing so as well and suggesting much of the literature discussed here.
There is much literature on the topic of meta-language on Carnap’s view. See, for example, Tennant (2007) on a different response to Friedman’s PRA proposal. There, Tennant argues that “Friedman is demanding too much in thinking that the resources of such combinatorial analysis as Carnap requires should not exceed those of primitive recursive arithmetic.” (p. 103).
Thanks to a referee for pointing me to these passages.
Here, a transformance is a map between two languages such that “the consequence relation in [the first] is transformed into the consequence relation in [the second].” A reversible transformance is a transformance such that the reverse relation is also a transformance. Being equipollent in respect of a language amounts to the requirement that any sentences which are mapped to each other are consequences of each other in the language we are respecting. The details here are not as critical to the view as the fact that these are different requirements.
There are serious questions about whether the intuitionists Carnap took himself to be addressing would agree with these criticisms. See Koellner (forthcoming) for more details.
Smooth infinitesimal analysis (SIA) is an intuitionistic analysis system, in which all functions are smooth. Importantly, it is such that 0 is not the only nilsquare (elements whose square is zero, i.e. elements x such that \(x^2=0\)). This is because every function is linear on the nilsquares. From this, it follows that 0 is not the only nilsquare even though there are no nilsquares distinct from 0. This would be inconsistent in classical logic (because of the validity of LEM), and so intuitionistic logic is required. More formally, in a classical system, the sentence \(\lnot \forall x(x=0\vee \lnot (x=0))\) is a contradiction. In an intuitionistic logic, since the law of excluded middle is not valid, the sentence cannot be true. Importantly for us, the SIA system has a very simple and straightforward proof of the fundamental theorem of the calculus (that the area under a curve corresponds to its derivative). Rather than, as usual, taking approximations of the rectangles under a curve as they approach a width of 0, we take a rectangle under the curve which has the width of a nilsquare. No approximations are necessary, and we do not need the concept of “approaching zero”. See Bell (1998) for more details.
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Acknowledgments
Thanks to Roy Cook, Geoffrey Hellman, Tristram McPherson, Andrew Parisi, Marcus Rossberg, Kevin Scharp, Stewart Shapiro, Neil Tennant, and a referee for helpful comments on previous drafts. Thanks also to helpful audiences at the 2016 North American Meeting of the Association for Symbolic Logic, the 2016 Society for Exact Philosophy, the 2016 Ohio Philosophical Association, the College of Wooster Philosophy Round Table and the Winter 2016 Dissetration Seminar at Ohio State.
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Kouri, T. A New Interpretation of Carnap’s Logical Pluralism. Topoi 38, 305–314 (2019). https://doi.org/10.1007/s11245-016-9423-y
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DOI: https://doi.org/10.1007/s11245-016-9423-y