Philosophers may wonder about the relationship between invariance and other similar sounding ideas in the literature- lawlikeness, projectability and so on. As I explained in CHF and elsewhere I see laws of nature as, so to speak, limiting cases on the invariance spectrum. Roughly, laws are highly invariant generalizations (or generalizations that describe highly invariant relationships). However, a generalization can be relatively invariant without looking like anything that we would naturally describe as a law– invariance is thus (in this respect)  a more general notion. Relatedly it is a notion that is better suited to characterize aspects of ordinary causal cognition than views according to which lay causal cognition involves representations of laws of nature and the like. As for projectability this is a notion that is far too vague and indiscriminating to be useful– among other limitations, it fails to distinguish generalizations that will continue to hold under changing circumstances (which has to do with invariance) from generalizations that will continue to hold only if background and other circumstances do not change .  
In the previous post I claimed that assumptions about invariance play a substantial role in causal judgment and reasoning. In this post I focus on some experimental  and theoretical work by psychologists illustrating this. In that literature it is a very common practice to ask subjects to make graded judgments about the “causal strength” of a relationship rather than a dichotomous judgment yes/no about whether the relationship is causal.  One standard version of such a probe  asks, regarding some scenario, “on a seven point scale,  how appropriate would it be to describe  this relationship as causal?” (In fact as discussed in CHF a number of different strength probes have been employed and there are important methodological questions about which is most appropriate for different purposes.) Because they are graded rather than dichotomous, causal strength judgments have the virtue that they can be used to track whether subjects are influenced in their judgments by graded factors like extent of invariance– other things being equal,  one would expect causal strength judgments to be higher for more invariant causal relationships. Summarizing  some very complex empirical research,  this is what is found in experiments conducted by Tania Lombrozo and colleagues that focus various forms of background invariance– see, e.g.,  Vasilyeva et al. 2018.
Vasilyeva, N.], Blanchard, T.   & Lombrozo, T. “Stable Causal Relationships Are Better Causal Relationships.” Cognitive Science, 42(4): 1265-98 
Experimental results from Patricia Cheng’s lab also  that are described in CHF show how invariance considerations (albeit of a somewhat different sort than those just described) influence causal strength judgments. I mentioned above that in addition to invariance under background conditions there are also other varieties of invariance that seem to influence causal judgment. Suppose C causes E or is a candidate for this and that C and E are both dichotomous random variables. It is natural to expect that if this causal relationship is genuine,  C will continue to cause E as the probability distribution of C changes, at least for some range of changes. This is a kind of invariance condition– invariance of the C–> E relationship under changes in the distribution of the values of  C — but a different  condition from the invariance under changes in background conditions discussed previously. Yet another invariance condition takes the following form: Suppose E  can be caused by either of two distinct factors, C or  A.  Suppose that the effect of C on E does not depend on the frequency with which A occurs or the details of the A–> E relationship and similarly for the effect of C on the A–>E relationship. This is also a kind of invariance condition– it has to do with whether the C–>E relationship is invariant under changes in the frequency of A or other features of A–>E relationship. Intuitively, when such an invariance condition holds completely, C and A do not “interact” with respect to E. Cheng shows that when the relationship between C and E satisfies the  two invariance conditions just described and certain other conditions ae met, it is possible to estimate a quantity that she calls “causal power” from contingency information involving C and E, with A also assumed to be present but unobserved. She also shows that at least in a range of cases,  causal power tracks people’s causal strength judgments better than alternative candidates such as the “associationist” measure  Dp mentioned  previously. Given Cheng’s assumptions, causal power is also the “rational” or normatively correct measure to use in estimating causal strength in certain contexts. For example, as she shows, in contrast to Dp,  causal power is a measure that correctly guides the extrapolation of causal claims to certain kinds of new situations– a consequence of the fact that Cheng’s measure incorporates invariance considerations.

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