Do Ceteris Paribus Laws Exist? A Regularity-Based Best System Analysis

Abstract

This paper argues that ceteris paribus (cp) laws exist based on a Lewisian best system analysis of lawhood (BSA). Furthermore, it shows that a BSA faces a second trivialization problem besides the one identified by Lewis. The first point concerns an argument against cp laws by Earman and Roberts. The second point aims to help making some assumptions of the BSA explicit. To address the second trivialization problem, a restriction in terms of natural logical constants is proposed that allows one to describe regularities, as specified by basic generics (e.g. ‘birds can fly’) and universals (e.g. ‘all birds can fly’). It is argued that cp laws rather than strict laws might be a part of the the best system of such a regularity-based BSA, since sets of cp laws can be both (a) simpler and (b) stronger when reconstructed as generic non-material conditionals. Yet, if sets of cp laws might be a part of the best system of a BSA and thus qualify as proper laws of nature, it seems reasonable to conclude that at least some cp laws qualify as proper laws of nature.

Appendix: The Default Modus Ponens

This appendix gives a definition of a default modus ponens (DMP; see Sect. 5). To this end, two languages $\mathfrak {L}_{{\mathbf{GNC}}}$ and $\mathfrak {L}_{\mathrm{m }}$ are employed, where $\mathfrak {L}_{{\mathbf{GNC}}}$ and $\mathfrak {L}_{\mathrm{m }}$ refer to sets of GNC formulæ and material formulæ, respectively. Both languages admit only monadic predicates $P,\,P_1,\,P_2$, ...  Open literals and closed literals are expressions of the form (1) $Px$ and $\lnot Px$ as well as (2) $Pa$ and $\lnot Pa$, respectively, for arbitrary predicates $P$, individual variables $x$, and individual constants $a$. Language $\mathfrak {L}_{{\mathbf{GNC}}}$ allows for formulæ of the form where $\alpha$ and $\beta$ are conjunctions of open literals and single open literals, respectively. In contrast, $\mathfrak {L}_{\mathrm{m }}$ contains arbitrary Boolean combinations of closed literals. Formula $\alpha [a/x]$ results from uniformly substituting $x$ in $\alpha$ with $a$.

For sets of formulæ $\Gamma$ and formulæ $\alpha$ of language $\mathfrak {L}_{{\mathbf{GNC}}}$ [$\mathfrak {L}_{\mathrm{m }}$], $\Gamma \vdash _{{\mathbf{GNC}}} \alpha$ [$\Gamma \vdash _{{\mathbf{FOL}}} \alpha$] indicates derivability in a logic ${\mathbf{GNC}}$ for GNCs [first-order logic]. System ${{\mathbf{GNC}}}$ corresponds semantically to Lewis’s (1973) sphere semantics without centering. System ${{\mathbf{GNC}}}$ is also required to make and ${\mathbf{GNC}}$-inconsistent given $\alpha [a/x]$ is ${\mathbf{FOL}}$-consistent. For open formulæ $\alpha$ and sets of open formulæ $\Gamma$ the expression $\Gamma \vdash _{{\mathbf{FOL}}} \alpha$ abbreviates $\{\gamma [a/x]$: $\gamma \in \Gamma \} \vdash _{{\mathbf{FOL}}} \alpha [a/x]$, where $a$ is a fixed individual constant. Let ∆ be a set of of formulæ of language $\mathfrak {L}_{\mathrm{m }}$ with the following property: ∆ contains a finite number of formulæ in which any given individual constant a occurs. The DMP can now be defined as follows:²⁶

Definition 1

(Default Modus Ponens) Suppose that α[a/x] is the conjunction of all formulæ in ∆ in which a occurs. Infer $\gamma [a/x]$ from $\alpha [a/x]$ based on $\Gamma$ and ∆ if there is a formula $\beta$ such that

• (1)

• (2) $\{ \alpha \} \vdash _{{\mathbf{FOL}}} \beta$, and

• (3) for all $\gamma '$ such that (i) (ii) $\{\alpha \} \vdash _{{\mathbf{FOL}}} \beta '$, and (iii) it holds that $\{\beta \} \vdash _{{\mathbf{FOL}}} \beta '$.

Observe that conditions (2) and (3ii) ensure that DMP inferences pertain only to GNCs with instantiated antecedents as specified by α[a/x].

Let me finally give an example, based on System CP (see Sect. 6). Suppose that $\Gamma =\{$(A1’), (A2’), (A3’), (A4’)$\}$ and that one aims to infer $P_4a$ for a given individual $a$ from $\Gamma$ and $P_1a \& \lnot P_2a \& \lnot P_3a$, based on (A4’) For this inference, conditions (1) and (2) are trivially satisfied. Furthermore, $P_1a \& \lnot P_2a \& \lnot P_3a$ implies (i) $P_1a$, (ii) $P_1a \& \lnot P_2a$, and (iii) $P_1a \& \lnot P_2a$, which correspond to the antecedent formulæ of (A1’)–(A3’), respectively. Only the consequent formulæ of (A2’) and (A3’) contradict the consequent formulæ of (A4’). Thus, merely (ii) and (iii) are required to be implied by $P_1a \& \lnot P_2a \& \lnot P_3a$, which is easily verified.