Abstract
Antecedent-strengthening, a trivially valid inference of classical logic of the form: P → Q ⊨ (P & R) → Q, has a counterpart in everyday reasoning that often fails. A plausible solution to the problem involves assuming an implicit ceteris paribus (CP) qualifier that can be explicated as an additional conjunct in the antecedent of the premise. The qualifier can be explicated as ‘everything else relevant remains unchanged’ or alternatively as ‘nothing interferes’. The qualifier appears most prominently in the context of the discussion of laws in the sciences, where these laws are often expressed with a CP qualifier. From an analysis of the qualifier’s role in the problem of antecedent-strengthening, we can learn more about CP qualifiers in general and in their application to the laws used in the sciences.
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Notes
See ibid.
Assume that the operator ‘CP’ in ‘CP (∀x ∃y (Fx → Gy))’ can be explicated as an additional antecedent, i.e. as ‘CCP → ∀x ∃y (Fx → Gy)’. Suppose that ‘CCP’ depends on x; then either the sentence representing a CP law is not closed or the operator must be explicated as acting within the quantifier’s scope, i.e. the CP law is of the form ‘∀x (CCP → ∃y (Fx → Gy))’. Suppose that ‘CCP’ does not depend on x; then ‘CCP → ∀x (Fx → Gx)’ is equivalent to ‘∀x (CCP → ∃y (Fx → Gy))’. Similarly for y, such that finally we have ‘∀x ∃y (CCP → (Fx → Gy))’.
See Schurz (2002, 360) for a different construction with the same result. (Note that for Gy ↔ y = x & G′y the above end result reduces to ‘∀x (Fx & CCP → G′x))’).
A list of such forms can be obtained from Priest (2001, 10, 13, 74).
By qualifying the statement I mean qualifying the two occurrences of ‘kings’ differently, such that we get, e.g. ‘all kings of the present are no real kings (do not act like kings should)’.
We can represent ‘∀x | Kx (Bx)’ as ‘∃x Kx & ∀x (Kx → Bx)’. Hence, ‘all kings are not kings’, interpreted as a restricted quantification (‘∀x | Kx (¬ Kx)’), can be represented as ‘∃x Kx & ∀x (Kx → ¬ Kx)’, which is a contradiction.
Assuming that ‘In (x, ¬ P)’ entails ‘¬ In (x, P)’, we can write ‘∀x | In (x, ¬ P) (In (x, P))’ as ‘∃x ¬ In (x, P) & ∀x (¬ In (x, P) → In (x, P)’, which is again a contradiction.
For perspicuity, formalize ‘In (x, P)’ as ‘Px’ (similarly for Q) and ‘In (x, P & R)’ as ‘Px & Rx’. Then let ‘if P, then Q’ and ‘if P & R, then Q’ be formalized, respectively, as ‘∃x (Px) & ∀x (Px → Qx)’ and ‘∃x (Px & Rx) & ∀x (Px & Rx → Qx)’. The second does not follow from the first in classical logic, evidently because ∃x (Px) ⊭ ∃x (Px & Rx) (though, of course, we have: ∀x (Px → Qx) ⊨ ∀x (Px & Rx → Qx)).
See again Priest (2001, 76).
See the examples above at note 2.
Formalise again ‘In (x, P)’ as ‘Px’ and so on. Then ‘if P & ¬ Q, then Q’ is ‘∃x (Px & ¬ Qx) & ∀x (Px & ¬ Qx → Qx)’, again a contradiction.
Here I concede the possibility that an inference with false premises is valid. Recall that an inference is valid iff: if, given an arbitrary interpretation, the premises are true, then so is the conclusion. If this latter conditional is a material conditional, then, as is well-known, P & ¬ P ⊨ Q, i.e. the inference from a contradiction is valid. If that conditional is interpreted with a tacit restricted quantification, P & ¬ P ⊭ Q (since there is no case where the premise is true), i.e. the inference from a contradiction is invalid.
‘If P & Q, then Q’ is ‘∃x Px & Qx & ∀x (Px & Qx → Qx)’, the second conjunct of which is a tautology.
There are well-known examples for non-necessary factors of an explanandum being non-explanatory. Consider e.g. Salmon’s famous example (see Salmon 1989, 50), where that John takes Jane’s birth-control pills is taken to explain (but in fact is non-necessary for) his not getting pregnant. Still, you may doubt whether necessary factors explain. Does the fact that Jane is a woman really explain her pregnancy? I claim that it does (though it explains only badly or incompletely). How else should we distinguish non-explanatory (Jane’s illness after her pregnancy, John’s taking Jane’s pills) from explanatory (Jane’s being a woman, John’s being a man) factors?
That sufficient factors are not necessarily explanatory is illustrated below in the example of Mary and Bill.
One such example is: ‘The cactus flowers if and only if it rains. It rains if and only if the cactus flowers’ (Sanford 1989, 203).
This may be due to a subtle ambiguity in the word ‘condition’. It seems that in everyday contexts being a condition is identified with being a necessary condition (see e.g. the short discussion of ‘P only if Q’ vs. ‘P on the one condition that Q’ in Sanford (1989, 202)). If so, then a sufficient condition is always a necessary and sufficient condition. Moreover, either a sufficient but not necessary condition is a contradiction in terms or the everyday use of ‘condition’ is not consistent with the differentiation of conditions into only necessary, only sufficient and necessary and sufficient ones.
I here gloss over a technical problem. An explanandum might be a proposition or alternatively the entity whose existence that proposition asserts. To properly distinguish these two manners of speaking would require a full theory of explanation that I am unable to present here.
Could not Bill have come earlier without starting his icebreaking? No, for in that case not the mere presence of an icebreaker but something else (some specific action of Bill’s or Mary’s) would have been a necessary and sufficient condition of a good party, i.e. the necessary-and-sufficient condition claim we previously assumed would not be true.
To be exact, ‘that P is a sufficient condition of (Q)’ entails ‘in all situations where P, Q’ but is not equivalent to it. If it were, we could not account for conditions-of as opposed to conditions-that. I insist on the difference, yet I am unable to present a positive account of conditions-of here.
I assume that a contradiction is a logically false proposition. Hence, ‘∃x (Ax & Bx & ¬ Bx) & ∀x (Ax & Bx & ¬ Bx → Cx’, though not entailing a proposition of the form ‘P & ¬ P’, is a contradiction.
See Lycan (2001, 28).
The second example is from Earman and Roberts (1999, 453).
The second example is adapted from Woodward (2002, 310).
An earlier version of this paper, titled ‘Antecedent-Strengthening and Ceteris Paribus Laws’, was presented at the GWP 2016 in Düsseldorf.
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Held, C. Ceteris-Paribus Qualifiers. J Gen Philos Sci 48, 439–452 (2017). https://doi.org/10.1007/s10838-017-9369-9
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DOI: https://doi.org/10.1007/s10838-017-9369-9