Abstract
From Leibniz to Krauss philosophers and scientists have raised the question as to why there is something rather than nothing (henceforth, the Question). Why-questions request a type of explanation and this is often thought to include a deductive component. With classical logic in the background only trivial answers are forthcoming. With free logics in the background, be they of the negative, positive or neutral variety, only question-begging answers are to be expected. The same conclusion is reached for the modal version of the Question, namely ‘Why is there something contingent rather than nothing contingent?’ (except that possibility of answers with neutral free logic in the background is not explored). The categorial version of the Question, namely ‘Why is there something concrete rather than nothing concrete?’, is also discussed. The conclusion is reached that deductive explanations are question-begging, whether one works with classical logic or positive or negative free logic. I also look skeptically at the prospects of giving causal-counterfactual or probabilistic answers to the Question, although the discussion of the options is less comprehensive and the conclusions are more tentative. The meta-question, viz. ‘Should we not stop asking the Question’, is accordingly tentatively answered affirmatively.
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Notes
Lehmann (2002) did not provide rules for vacuously quantified sentences.
There is one important difference between \(N/PFL_{=,\Box }\) on the one hand and \(LPCE + \mathbf {S5}\), the system in (Hughes and Cresswell 1996), on the other hand: \(\phi \leftrightarrow \forall x \phi \) (provided that x is not free in \(\phi \)) is an axiom scheme of \(LPCE + \mathbf {S5}\), whereas only the left-to-right direction is an axiom scheme of \(N/PFL_{=,\Box }\). Semantically, the difference is that in \(N/PFL_{=,\Box }\) the world-relative domains of quantification can be empty, whereas they cannot in \(LPCE + \mathbf {S5}\). The formal relevance of \(\mathbf {S5}\) consists in the fact that one does not need to assume a certain primitive rule called \(UGL\forall ^{n}\). The material relevance of \(\mathbf {S5}\) is due to the fact that it is generally taken to the correct logic for metaphysical or counterfactual necessity—see (Williamson 2013) for an argument. The dialectical relevance of \(\mathbf {S5}\) is that it gives very strong modal resources to those who attempt a deductive explanation.
Note that, even if \(\phi \leftrightarrow \forall x \phi \) (with x not free in \(\phi \)) were one of the axiom schemes (as in \(LPCE + \mathbf {S5}\)—see footnote 4), this would still hold.
Hughes and Cresswell (1996, p. 293) mention three other inference rules. For one of these, see note 4. The two other rules are the rule of necessitation (if \(\vdash _{N/PFL_{=, \Box }} \phi \), then \(\vdash _{N/PFL_{=, \Box }} \Box \phi \)) and the rule of universal generalisation (if \(\vdash _{N/PFL_{=, \Box }} \phi \), then \(\vdash _{N/PFL_{=, \Box }} \forall x \phi \)). For the two others, note the following two things. First, they can be made redundant, e.g. one can stipulate that all the axioms are necessary. (Necessity is closed under modus ponens.) Second, in neither case is the conclusion of the inference of the right syntactic form.
Note that the proved equivalences in Case 3 are unaffected even if one were to add \(\phi \leftrightarrow \forall x \phi \) (with x not free in \(\phi \)) as an axiom scheme.
I say ‘broadly speaking’, because Hintikka (1962) points to some problems that I will not elaborate on.
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Previous versions of this paper have been presented at the Fifth Graduate Student Conference (27 March 2015, Leuven), the SePPhia Seminar (1 April, 2015), the Congress for Logic, Methodology and Philosophy of Science (7 August 2015, Helsinki), and the CEFISES Seminar in Louvain-la-Neuve (13 January 2016). I would like to thank the audiences for their comments and questions. Furthermore, I would like to thank the anonymous reviewers for their useful reports.
Appendix: Tree Proofs for Lemma 3.2
Appendix: Tree Proofs for Lemma 3.2
Before giving the tree proofs, let me remind the reader that \(\left( \alpha \wedge \beta \right) \) is definitionally equivalent to \(\lnot \left( \alpha \rightarrow \lnot \beta \right) \) and, consequently, \(\lnot \left( \alpha \wedge \beta \right) \) is definitionally equivalent to \(\lnot \lnot \left( \alpha \rightarrow \lnot \beta \right) \), which logically entails \(\left( \alpha \rightarrow \lnot \beta \right) \) (cf. the tree rules for double negation). So, one can use the rules for material implications.
For convenience, whenever the reductio assumption of a tree proof is a quantified sentence or the negation thereof the marker \(^{*}\) will be added directly.
Proof
(Case 1-i) For convenience and without loss of generality, let us consider only \(P\left( t\right) ^{*}\) and \(\exists x \left( x = t \wedge P\left( x \right) \right) ^{*}\).
Proof
(Case 1-ii) For convenience and without loss of generality, let us consider only \(P\left( t\right) ^{*}\) and \(\exists x \left( x = t \wedge P\left( x \right) \right) ^{*}\).
Proof
(Case 3.1)
Proof
(Case 3.2)
Proof
(Case 3.3)
Proof
(Case 3.4)
Proof
(Case 3.1-ii)
Proof
(Case 3.2-ii)
Proof
(Case 3.3-ii)
Proof
(Case 3.4-ii)
Proof
(Case 3.4-ii Cont.)
Proof
(Case 4-ii)
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Heylen, J. Why is There Something Rather Than Nothing? A Logical Investigation. Erkenn 82, 531–559 (2017). https://doi.org/10.1007/s10670-016-9831-9
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DOI: https://doi.org/10.1007/s10670-016-9831-9