Concerning Peter Vickers's Recent Treatment of ‘Paraconsistencitis’

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Acknowledgements

The research for this article was supported by the Special Research Fund (BOF), Ghent University, and the Research Foundation—Flanders (FWO). We are greatly indebted to Joke Meheus for inspiring discussions on this topic. We are also grateful to Ofer Arieli for providing valuable references and to Peter Vickers for valuable comments on an earlier draft of this paper. Finally, we wish to thank the members of the Ghent reading group on inconsistencies in science, in particular Wim Christiaens, Jesse Heyninck, María Martínez, Jan Potters, and Rafal Urbaniak.

Notes

[1] Ex contradictione quodlibet, sometimes also called Ex falso quodlibet or the ‘principle of explosion’, is an inference rule valid in CL, according to which from a contradiction follows anything: A, ∼A⊢B.

[2] The view that scientific rationality does not always conform to the ideal of consistency fits into the more general philosophical leaning towards bounded rationality, according to which rational agents reason in view of limited cognitive resources and abilities (see e.g. Doyle 1992).

[3] Dialethism is a view that there are true contradictions, i.e. sentences that are both true and false.

[4] The term ‘paraconsistencitis’ stems from Meheus (2003), where it is used to indicate an epistemic bias: scholars suffering from it have a tendency to see inconsistencies everywhere.

[5] See also Harman (1986), who introduces various constraints on rational belief revision, among them minimality of change. He also discusses a ‘logical inconsistency principle’, according to which ‘logical inconsistency is to be avoided’ (Harman 1986, 11). Note though that according to Harman this principle is not without exceptions: he mentions the liar paradox as an example.

[6] PLs have been defended also on other grounds (see, e.g. Priest, Tanaka, and Weber 2013 for an overview), for instance, on the basis of arguments that deny that modus ponens or disjunctive syllogism are truth preservational. To start off our discussion, though, it is sufficient to focus on the argument above. Similarly, the adequacy of CL has been criticized as a suitable logic for the modelling of scientific reasoning for different reasons beside its explosive character (Weingartner 1994).

[7] Various complications arise if we try to give a more refined account of the exact nature of the commitment in question since the reasoning capacities of agents are restricted and they may not be aware of certain consequences of a given premise set and/or they may not be able to memorize and process all the consequences. For instance, Field (2009) characterizes the commitment in terms of degrees of beliefs and ‘obvious’ implications: ‘If it's obvious that A₁, … , A_n together entail B, then one ought to impose the constraint that P(B) is to be at least P(A₁)+ · · · +P(A_n)−(n−1), in any circumstance where A₁, … , A_n and B are in question’ (Field 2009, 260). Beall (2013) phrases the commitment in terms of a constraint according to which one is not to reject consequences: ‘If X⊢A, then it's irrational to accept X and reject A’ (Beall 2013, 4).

[8] Meheus (2002) offers a finer-grained analysis distinguishing between sensible and acceptable consequences. The latter are those statements that also hold according to the intended consistent replacement of a theory, while the former are the ones provided by the consequence relation of a logic. She argues that sometimes inferences are sensible though not acceptable (e.g. they may serve a heuristic value when searching for a consistent replacement). Moreover, for acceptance, derivability is not sufficient but sometimes extra-logical criteria have to be considered. When modelling the reasoning of scientists a focus on acceptability is fruitless since the intended consistent replacement may not be yet available but rather be the outcome of both analysing the given propositions and of extra-logical considerations.

[9] Often paraconsistent logicians argue that in view of such examples and the fact that sometimes it is rational to accept contradictions, modus ponens and similar rules are not truth preserving. A case in point where it is rational to accept a contradiction is the liar paradox (even non-paraconsistent logicians argue for acceptance of the contradiction, see e.g. Harman 1986).

[10] More generally, it can be shown that if we pair LP with a principle that rejects any contradictory disjuncts from disjunctive consequences, we get the full derivative strength of CL for consistent premise sets (see Priest 2006, chapter 8, and for a multi-conclusion variant Beall 2013). Inconsistency-adaptive logics (Batens 1999) integrate this behaviour in a dynamic proof theory with the effect that most of them are equivalent to CL for consistent premise sets.

[11] For example, Harman (1986) takes the radical position that logic has no significant role in reasoning.

[12] For an in-depth analysis of empirical studies such as the selection task and the suppression task, the reader is referred to Stenning and van Lambalgen (2008).

[13] A difference is that in Vickers's analysis these extra-logical considerations play a role only when revising beliefs, while in our illustration above they also play an ‘accumulative role’ when making choices among the disjuncts the logic offers. We have two more remarks. First, also when using LP the extra-logical consideration to avoid contradictions can become decisive for revising beliefs. In the example above we can derive (p∧∼p)∨(q∧∼q) from Γ via LP. Thus, we have the choice between accepting p∧∼p, accepting q∧∼q, or rejecting our premise set. The latter choice is given by an extra-logical consideration, which advises the rejection of contradictions in favour of revising beliefs. Second, it is likely that also in a reasoning process governed by CL extra-logical considerations are used by rational reasoners to make choices. Suppose we can derive (via CL) p∨q from some premise set, but neither p nor q is derivable. For instance, if p is significantly better supported by our evidence (or we have other reasons to reject q such as ethical or political reasons) we may base our further reasoning process on p rather than q.

[14] One may want to defend Vickers from this criticism by conjecturing that—in principle—he leaves open the possibility of a (rational) scientist applying ECQ. However, given his insistence of the non-existent danger of ECQ this seems extremely unlikely. For instance, Vickers argues that inferring anything via ECQ is ‘quite incredible’, concluding that ‘any claim that one should be wary of one's inferences because of ECQ is just a mistake’ (54).

[15] Take, for example, the characterizations of what it means that an agent employs a logic in Field (2009, 263):

I The way to characterize what it is for a person to employ a logic is in terms of norms the person follows, norms that govern the person's degrees of belief by directing that those degrees of belief accord with the rules licensed by that logic.

More concretely in terms of belief functions this means:

E Employing a logic L involves it being one's practice that when simple inferences A₁, … , A_n⊢B licensed by the logic are brought to one's attention, one will normally impose the constraint that P(B) is to be at least P(A₁)+ · · · +P(A_n)−(n−1).

Clearly, in Vickers's analysis ECQ is in no way a norm that ‘governs’ a person's beliefs as in I or E.

[16] While so-called non-monotonic logics usually only deal with the latter, ‘external’ dynamics (retraction in view of new external inputs), logics such as adaptive logics (Batens 2007; Straßer 2014) and Pollock's OSCAR system (Pollock 1995) deal also with the former, ‘internal’ dynamics.

[17] Meheus (2003) distinguishes between reasoning from inconsistent information and reasoning ampliatively from incomplete information (e.g. by means of abduction, induction, default reasoning, etc.). She argues that also in the latter case scientists sometimes arrive at inconsistent hypotheses and continue to reason with them in a heuristically fruitful way. A PL for abductive reasoning can, for instance, be found in Provijn (2012).

[18] The argument holds also if one objects to the notion of non-monotonic logics by distinguishing sharply between implication/deduction/argument and belief revision where only the former is governed by logics. According to this view, logics constrain a reasoner by providing rational choices, while for example evidence weighing determines that choice (Knorpp 1997; Beall 2013).

[19] The same holds for the consequence relations proposed in Rescher and Manor (1970). Another approach that falls in the grey zone between CL and non-classical approaches is the idea to ‘filter’ CL which has been developed, for example by Schurz and Weingartner (Weingartner 1994).