Abstract
In this chapter, the pluralist arbitrates between two philosophical positions: the extensionalist and the constructivist. Both are anti-realists of a sort. The extensionalist position is that of Quine, and is represented by Bar-Am. The constructivist position is that of Sunholm and Martin-Löf. The two merit comparison because they both give a sensitive account of the history of logic, moreover, they give much the same account. The two positions differ on their final judgment of the modern trend. The extensionalist sees progress where the constructivist sees emptiness. To draw out the differences, we shall also meet the formalist and the realist. We shall see the accord and disaccord between these positions, especially in respect of their attitude towards logic.
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Notes
- 1.
The exception to this is what Batens has called a ‘zero logic’. This is limit case logic where there are no rules of inference or manipulation of symbols. In such a logic, no conclusions can be drawn. It is still a logic in virtue of the other characteristics: a formal language, a grammar, maybe some axioms. We ignore the case of zero logics for the rest of this chapter.
- 2.
We are assuming only single conclusion logic. For multiple conclusion logics, substitute conjunction for the comma between the conclusions. Provided the conjunction introduction rule is classical, the proof for the single conclusion will be longer, but the consequence relation will remain, essentially, unaffected.
- 3.
Intensional notions are those whose meaning is not merely the referent of ‘states of affaires’, terms or names. That is, getting the referent is insufficient for the meaning. And the meaning in a logical context will bear upon what can be deduced from a wff. The meaning of intensional notions is partly captured by Fregean sense, context or implicit understanding. It is discerned through the mode of presentation; how we express the notion, or the context in which we embed the notion.
- 4.
There are many non-equivalent ways of distinguishing and defining extension, intension and intention. I shelve discussion of these for another occasion. What is relevant here, is Bar-Am’s definitions and use of the notions, and how they relate to Sundholm’s concerns about logic.
- 5.
Arguably, Quine is quite extreme, and in places, thinks that philosophy should rid itself of intensional notions too (Quine 2008). A less extreme reading would interpret Quine to be asking philosophers to successively address intensional notions, in order to make them clearer, only when and if they can, so it is quite possible that some will remain in the discourse. The moderate extensionalist would accept that total purging of intensionality is not the goal. Rather, the goal is to rid ourselves of intensional notions in order to promote clarity in communication (Bar-Am 2010). Henceforth, we shall assume that the extensionalist occupies the more moderate position.
- 6.
There are some obvious weaknesses in the argument, as presented here since I have not given any support for the steps. There is support, and the strength or weakness of the argument is not our immediate concern.
- 7.
An extensional system is not just a formal system with an axiom of extensionality, or any formal system with no intensional operators, or some such. Extensionality, intensionality and intensions bear careful treatment. We saw a hint of this in Chap. 6. Here we are interested in extensional formal systems in the sense of formal system where epistemology and metaphysics is no longer a part of the logic or the language.
- 8.
One of the main differences in their historical treatment, is that Bar-Am lends close attention to Boole whereas Sundholm places more emphasis on Bolzano than Bar-Am.
- 9.
There might be a metaphysical or semantic justification. Today we would separate these from scientific justification, but such separation is foreign to Aristotle. Hence the embarrassment concerning the immediate judgment: man is a rational animal. Today we do not consider man to be essentially rational, since we consider other creatures to be rational, and some of us think that man is often irrational.
- 10.
Lull was one of the first to try to make the syllogism mechanical. He designed a physical method of constructing syllogisms by rotating concentric disks. This was in the fourteenth century.
- 11.
A proof, seen as an object, is simply a blueprint for making inferences, for coming to judgments (Sundholm 1998, 180).
- 12.
I am being quite careful, even in my oversimplification. Hilbert and Brouwer did not start their respective views ex nihilo. These ideas were usually confounded in the same system. However, they were starkly separated by both Hilbert and Brouwer.
- 13.
Martin-Löf shares with Brouwer the emphasis on epistemology and judgment in logic. However, he is not a Brouwerian intuitionist in that he distances himself from Brouwer’s idea that mathematical ontology is only mental construction. For this reason, he prefers to be called a constructivist rather than an intuitionist.
- 14.
We can be in error as to whether or not we are really justified in our judgment. The notion of validity is still in terms of a conditional: if we know the premises, then we know the conclusion. Moreover, the conditional is intuitionistic: a proof of the premises can be extended into a proof of the conclusion.
- 15.
We need not worry here about Gettier-type counter-examples since the justifications are always in the form of constructive proofs, and they can be put in normal form. As a result, under a constructivist conception of proof, there is no such thing as getting the truth in a justified way, but with the wrong sort of justification.
- 16.
Of course, since the deduction theorem holds for the type theory, we can turn a conclusion resting on something other than immediate judgments into a bona fide judgment by proposing the premises as assumptions that are discarded at the end of the proof by the familiar conditional introduction rule. The form of our conclusion is then that of a conditional statement.
- 17.
This is meant in the following sense: in a second-order language we can quantify over a predicate or relation, and when we do this we treat the predicate or relation as an object.
- 18.
While, arguably, Tarski was a realist, he was sensitive to anti-realist concerns.
- 19.
The origins of this view reach back to Plato, so are more ancient than Aristotle. Nevertheless, we see a resurgence of realist vocabulary in presentations of logic. We shall see that these are interspersed with formalist elements as well. What is common to the realist and the formalist is that logic loses its epistemic role.
- 20.
This view of formalism is sometimes referred to as game-theoretic formalism, as opposed to a more philosophical Hilbertian formalism. As a formal logician, when the formalist builds his formal systems, content is disregarded or thought to be an empirical matter, and therefore, is separated from the technical aspects of logic. Henceforth, we shall reserve the term ‘formalist’ for the formal logician who develops formal systems, which today we call ‘logical systems’.
- 21.
There is some discrepancy between how the term ‘realist’ is used in the USA and how it is used in the U.K. I take Wright’s definition (Wright 1986, 1). “Realism is a mixture of modesty and presumption. It modestly allow that humankind confronts an objective world, something almost entirely not of our making…. However, it presumes that we are, by and large and in favourable circumstances, capable of acquiring knowledge of the world and of understanding it.” I follow Wright in then thinking that there are two sorts of anti-realist, the sceptic and the Kantian idealist. The constructivist, in the chapter, is modeled after Martin-Löf and Sundholm, but also shares features with the Dummettian intuitionist. He cannot share all features, since some aspects of Martin-Löf’s constructivism are in conflict with Dummett’s intuitionism. I do not think that the differences matter for the purposes of this chapter.
- 22.
We do not have to go this far to begin the reverse process. We can skip the computers and computer languages. The divorce claim rests on two uncontroversial ideas. One is that computers (or computer programs) do not have knowledge, do not deal with the objects of a domain of interpretation, they perform no intentional action (of demonstrating). The second idea is that computer programs can be thought of as formal logical systems.
- 23.
The constructivist could block the argument here and say that such a proposed logic does not count as a logic. This might convince the constructivist, but such a block will not convince the pluralist. Therefore, the proposed block begs the question against the pluralist. There will be several points of departure like this in what follows. Grosso modo, the strategy is to declare that the conditions of the argument are unacceptable – since they are not constructively acceptable. This is legitimate but limiting. It is limiting because it will not convince an opponent.
- 24.
Priest (2002) uses the term model in the model theory sense. So I am deviating from his use here.
References
Bar-Am, N. (2008). Extensionalism; the revolution in logic. Berlin: Springer.
Bar-Am, N. (2010). Extensionalism in context. Philosophy of the Social Sciences, XX(X), 1–18.
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Priest, G. (2002). Beyond the limits of thought. Oxford: Clarendon.
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Quine, W. V. (2008). Confessions of a confirmed extensionalist. Cambridge, MA: Harvard University Press.
Sundholm, G. (1997). Implicit epistemic aspects of constructive logic. Journal of Logic, Language, and Information, 6, 191–212. Dordrecht: Kluwer.
Sundholm, G. (1998). Inference, consequence, implication: A constructivist’s approach. Philosophia Mathematica, VI, 178–194.
Sundholm, G. (2007). Semantic values of natural deduction derivations. Synthese, 148(3), 623–638.
Wright, C. (1986). Realism, meaning & truth. Oxford: Basil Blackwell.
Acknowledgments
I should like to thank both Bar-Am and Sundholm for detailed and useful comments on an early version of this chapter. I should also like to thank the Arché research group: Foundations of Logical Consequence, for useful comments during a presentation of an earlier version.
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Friend, M. (2014). Pluralism and Together Incompatible Philosophies of Mathematics. In: Pluralism in Mathematics: A New Position in Philosophy of Mathematics. Logic, Epistemology, and the Unity of Science, vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7058-4_13
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