Abstract
We motivate and develop an extension of Nelson’s constructive logic N3 that adds a counterfactual conditional to the existing setup. After developing the semantics, we will outline how our account will be able to give a nice analysis of natural language counterfactuals. In particular, the account does justice to the intuitions and arguments that have lead Alan Hájek to claim that most conditionals are false, but assertable, without actually forcing us to endorse that rather uncomfortable claim.
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- 2.
In other words, it is like the Michael Dunn’s semantics (cf. [1]) for the so-called Belnap-Dunn logic or FDE.
- 3.
Note, however, that for the notation we follow Krister Segerberg’s notation in [13]. More specifically, we use \(\sqsupset \) and for would- and might-conditionals respectively. This has an intuitive appeal since these symbols are half-box and diamond which reflect the truth conditions for those conditionals.
- 4.
Note that we have \(w\Vdash _{0} A\) iff \(w\not \Vdash _{1} A\) here since we are reviewing the conditional logic based on classical logic.
- 5.
See [6, p.15] for the original discussion of these requirements and their motivation. If, instead of the last requirement, we add
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If \(x\in f_A(w)\) and \(y\in f_A(w)\) then \(x=y\),
we get the system preferred by Robert Stalnaker.
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- 6.
One reason why this alternative might seem tempting is that it leaves our original condition free to serve as the falsification clause for an added might-conditional. Might-conditionals, however, are yet another topic we don’t have the space to cover in this piece.
- 7.
This also works for the case with Wansing’s connexive logic C, a variant of N4.
- 8.
We emphasize again that \((A\wedge {-}A){\rightarrow }B\) is invalid in N4.
- 9.
Basically the same idea is applied by Priest in [11] in which he discusses the cancellation account of negation.
- 10.
One of the reasons why Lewis was not too concerned about the “right" choice between the two clauses for the truth of conditionals is this: He realized that, in the classical case he was considering, he could define either one of the two conditionals in terms of the other ([6, p. 26]). Here is how to define the old condition in terms of the new one: \(A\sqsupset _{old}B =_{def}\) \((A\sqsupset _{new} A)\supset (A\sqsupset _{new} B)\). Now, if this was possible in our setting as well, of course, then we would be faced with disaster again. Luckily, the equivalence does not hold in our system if we replace the material conditional with the constructive Nelson conditional. The same is true of \({-}(A\sqsupset _{new} A)\vee (A\sqsupset _{new} B)\).
- 11.
For more on connexive logics in general, see [19].
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- 13.
We would like to thank Massimiliano Carrara for directing our attention to this issue.
- 14.
See, for example, [16].
- 15.
A more intricate condition on the two relation was proposed by a reviewer, and we thank her or him for the inspiration:
If \(w \le x\) and \(x R_A x'\) then there is a \(w'\in W\) such that \(w R_A w'\) and \(w' \le x'\).
This condition is analogous to what in many-dimensional modal logics is called left-commutativity (see [2, p. 221]). Once parsed, this condition indeed seems eminently plausible. With the vocabulary we have introduced so far, however, it seems that no difference to the consequence relation is made by imposing the condition. This will change when, in later work, we will introduce a suitable might-conditional, a topic we have to leave out for reasons of space. When we will address this, we will be sure to come back to the reviewer’s condition.
- 16.
A quote from [14, p. 202].
- 17.
In this sense, the example serves to show that we are not overly and unnecessarily ambitious here.
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Acknowledgments
Hitoshi Omori is a Postdoctoral Research Fellow of Japan Society for the Promotion of Science (JSPS). We would like to thank the anonymous referees for their helpful comments that improved our paper. We would also like to thank Massimiliano Carrara, Roberto Ciuni, and the participants of Kyoto Philosophical Logic Workshop I for useful comments and discussions.
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Kapsner, A., Omori, H. (2017). Counterfactuals in Nelson Logic. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_34
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