Abstract
At the end of the last chapter I invoked the idea that a formal system of logic, such as LP, is used metaphorically by the pluralist. It is essential to the pluralist position, and possibly to many other positions, that we should be able to make sense of this, and say something quite definite about it. Otherwise, our claims about appealing to formal systems of logic are empty. I look at three ways in which the pluralist makes use of a formal logical system. The first is in direct appeal to a rule or axiom to justify a move in an argument. The second is when the pluralist uses a formal theory in order to reconstruct another theory. This is done to understand the theory from another perspective. The third use is dialectical. In invoking or developing a formal theory to represent a form of reasoning, we bring some features of that reasoning into relief, and we obscure other features. We can evaluate the fit between the formal theory and the informal one. In the evaluation, we might well consider alternative formal representations. Thus, we enter a dialectic. Lastly, in order to remind us that pluralists are not the only ones who use formal logic informally, I look at how it is that mathematicians use formal logical theories.
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Notes
- 1.
Some of us tell our students that we do, but this is usually a cheap ruse. I have tried writing out the formal version of an philosophical argument, but found that either my argument in English sounded too trivial, or that I could not represent the ideas properly in a formal language. Of course small inference moves are easy to represent, but more involved arguments are much harder. Nevertheless, we have something like the logical structure of the argument in mind when we argue. So part of what this chapter is about is the relationship between formal logic and philosophical reasoning.
- 2.
There do exist some formal systems which combine several modal operators, (which are not duals of each other), but there are not many, and they are often not satisfactory (in their axioms or rules of inference). Occasionally, a philosophical argument can be elegantly expressed in a formal language and the inferences can be accounted for. It is very satisfying when we can do this.
- 3.
What I mean by ‘valid modulo induction’ is that the non-inductive part of the argument is valid, provided the induction move is carried out correctly, according to some axiom or rule of induction.
- 4.
The point is made in Sundholm (2012), but there, he makes the point for mathematical and logical arguments, not for philosophical arguments. However, I think that the point applies to philosophical arguments just as well. I shall return to this issue.
- 5.
There are formal logical theories where modus ponens does not hold. These are some of the relevant logics.
- 6.
We might think that this is a ‘border-line case’, that is, that in the case of appealing to formal rules in mid-argument is hardly metaphorical. It is just direct use of a logic. However, it is not quite, since we should distinguish between the formal system of logic and the informal use made of it. The informal use is, arguably, a metaphorical use of the formal logic. I do not much mind where we fall on this. Without compromising the thrust of the argument, we can ignore this stretch in the term ‘metaphorical’ without great loss.
- 7.
This might even turn into an opportunity to revise one’s monism, but I leave this sort of possibility aside, for the sake of simplicity.
- 8.
The possibility is an epistemic possibility – “for all I know”.
- 9.
Batens appeal to something he calls a ‘zero logic’. This is a degenerate logic where there are no rules of inference. As a result, no contradiction could be derived, since no derivations can be made. This is a degenerate logic, and extreme enough that we might be carried to suppose that it is not a logic at all. This is not important for the point here. If we can develop such a ‘logic’ then we can develop formal logics with very few rules of inference, and we could develop one where a contradiction in the form of a conjunction of two opposite formulas, could never be derived (just because the formal theory lacks a rule for forming conjunctions, even though there is a symbol for conjunction in the language (but then it is a ‘dummy’ connective)).
- 10.
In fact, in journal articles we have a lot of proofs, and this is because we are meant to be on the forefront of knowledge. The finding and conclusion should be surprising, curious or dubious (otherwise we would not publish the finding). It is for this reason that we see so much formal proof in advanced texts and journals. In less advanced material proofs are less necessary. They are given in textbooks in order to convince the student to whom the material is new, and to accustom the student to the language of proof.
- 11.
Circumstances are individuated by the people receiving the information, and their knowledge and background. The people might include people in the distant future, as when historians of mathematics interact with an historical text, or when a mathematician revisits an old text.
References
Moore, G. (1988). The emergence of first-order logic. In W. Aspray & P. Kitcher (Eds.), History and philosophy of modern mathematics (Minnesota studies in the philosophy of science, Vol. XI, pp. 95–135). Minneapolis: University of Minnesota Press.
Sundholm, G. (2012). Error. Topoi, An International Review of Philosophy, 31(1), 87–92.
Acknowledgements
I should like to thank Batens and Primiero for each separately suggesting to me that I wanted to use adaptive, or another paraconsistent logic, metaphorically rather than directly.
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Friend, M. (2014). Using a Formal Theory of Logic Metaphorically. 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_7
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