Supplementary Material: Appendices
Appendices A-C derive causal model theory predictions for arguments in Goldvarg
and Johnson-Laird (2001) and Barbey and Wolff (2007). Derivations use the simpler
form of prevents, Equation 4a. Using Equation 4b instead would make no difference
except where noted. In the following  is intended to mean “according to causal model
theory, translates into the following structural equation.”
* following a conclusion indicates that this was not the modal response in Goldvarg
and Johnson-Laird (2001). However, as noted in the text, it might have been the modal
response in Barbey and Wolff’s (2007) replication.
Appendix A
Derivations of predictions for 15 two-premise causal arguments from Goldvarg and
Johnson-Laird (2001). Sixteenth argument is derived in the text.
A causes B  B := A
B allows C  C := B and X
Substituting the first relation in the second, we get C := A and X. This is the
representation of “B allows C.”
A causes B  B := A
B prevents C  C := ~B
Implies C := ~A. This is the representation of “A prevents C” (Equation 4a).
A causes B  B := A
Not B causes C  C := ~B
Implies C := ~A. This is the representation of “A prevents C.”
A allows B  B := A and X
B causes C  C := B
Implies C := A and X. This is the representation of “A allows C.”
A allows B  B := A and X
B allows C  C := B and Y
Implies C := A and X and Y. Combining X and Y into a single accessory variable, this is
the representation of “A allows C.”
A allows B  B := A and X
B prevents C  C := ~B
Implies C := ~(A and X). This is one representation of “A prevents C” (Equation 4c). *
A allows B  B := A and X
Not B causes C  C := ~B
Implies C := ~(A and X). Again, this implies “A prevents C”. *
A prevents B  B := ~A
B causes C  C := B
Implies C := ~A. This is the representation of “A prevents C.”
A prevents B  B := ~A
B allows C  C := B and X
Implies C := ~A and X. This is the representation of “A prevents C.”
A prevents B  B := ~A
B prevents C  C := ~B
Implies C := ~(~A). This is the representation of “A causes C.” *
If we represented “A prevents B” using Equation 4b, we would obtain “A allows C.”
A prevents B  B := ~A
Not B causes C  C := ~B
Implies C := ~(~A). This is the representation of “A causes C.” If we represented “A
prevents B” using Equation 4b, we would obtain “A allows C.”
Not A causes B  B := ~A
B causes C  C := B
Implies C := ~A. This is the representation of “Not A causes C” (by the second
processing assumption).
Not A causes B  B := ~A
B allows C  C := B and X
Implies C := ~A and X. This is the representation of “Not A allows C.”
Not A causes B  B := ~A
B prevents C  C := ~B
Implies C := ~(~A). This is the representation of “Not A prevents C.”
Not A causes B  B := ~A
Not B causes C  C := ~B
Implies C := ~(~A). This is the representation of “Not A prevents C” (by the second
processing assumption).
Appendix B
Derivations of predictions for 20 of 32 two-premise causal arguments from Barbey
and Wolff (2007, Experiment 1). Remaining 12 arguments derived in text and Appendix
A.
A causes Not B  ~B := A
B causes C  C := B
Implies C := ~A. This implies “A prevents C” (Equation 4a).
A causes Not B  ~B := A
B allows C  C := B and X
Implies C := ~A and X. This is the representation of “A prevents C” (Equation 4b).
A causes Not B  ~B := A
B prevents C  C := ~B
Implies C := A. This is the representation of “A causes C.”
A causes Not B  ~B := A
Not B causes C  C := ~B
Implies C := A. This is the representation of “A causes C.”
Not A allows B  B := ~A and X
B causes C  C := B
Implies C := ~A and X. This implies “Not A allows C.” (by the second processing
assumption). *
Not A allows B  B := ~A and X
B allows C  C := B and Y
Implies C := ~A and X and Y. This is the representation of “Not A allows C.”
Not A allows B  B := ~A and X
B prevents C  C := ~B
Implies C := ~(~A and X). This is the representation of “Not A prevents C” (Equation
4c).
Not A allows B  B := ~A and X
Not B causes C  C := ~B
Implies C := ~(~A and X). This is the representation of “Not A prevents C.”
A allows Not B  ~B := A and X
B causes C  C := B
Implies C := ~(A and X). This implies “A prevents C.” *
A allows Not B  ~B := A and X
B allows C  C := B and Y
Implies C := ~(A and X) and Y. This is the representation of “A prevents C” because A
makes C less likely.
A allows Not B  ~B := A and X
B prevents C  C := ~B
Implies C := A and X. This is the representation of “A allows C.”
A allows Not B  ~B := A and X
Not B causes C  C := ~B
Implies C := A and X. This is the representation of “A allows C.”
A prevents Not B  ~B := ~A
B causes C  C := B
Implies C := A. This is the representation of “A causes C.”
A prevent Not B  ~B := ~A
B allow C  C := B and X
Implies C := A and X. This is the representation of “A allows C.”
A prevents Not B  ~B := ~A
B prevents C  C := ~B
Implies C := ~A. This is the representation of “A prevents C.”
A prevents Not B  ~B := ~A
Not B causes C  C := ~B
Implies C := ~A. This is the representation of “A prevents C.”
Not A causes Not B  ~B := ~A
B causes C  C := B
Implies C := ~(~A). This is the representation of “Not A prevents C.”
Not A causes Not B  ~B := ~A
B allows C  C := B and X
Implies C := ~(~A) and X. This is the representation of “Not A prevents C.”
Not A causes Not B  ~B := ~A
B prevents C  C := ~B
Implies C := ~A. This is the representation of “Not A causes C.” * Allowing an
accessory cause for prevents would have given the modal conclusion “Not A allows C.”
Not A causes Not B  ~B := ~A
Not B causes C  C := ~B
Implies C := ~A. This is the representation of “Not A causes C.”
Appendix C
Derivations of causal model theory predictions for 14 three-premise causal arguments
from Barbey and Wolff (2007, Experiment 2). The 15th conclusion is derived in the text.
Conclusions are read from derived structural equation by relating the A term with a
positive valence to the D term with a positive valence.
A causes B  B := A
B causes C  C := B
C causes D  D := C
Substituting the first relation into the second and the resultant into the third, we get D :=
A. This is the representation of “A causes D.”
A allows B  B := A and X
B causes Not C  ~C := B
C causes Not D  ~D := C
Composing the three functions, we get ~D := ~(A and X) which is equivalent to D := A
and X. This is the representation of “A allows D.”
A allows B  B := A and X
B causes Not C  ~C := B
C causes D  D := C
Composing the three functions, we get D := ~(A and X). This is the representation of “A
prevents D.” *
A causes Not B  ~B := A
B allows C  C := B and X
C causes Not D  ~D := C
Composing the three functions, we get D := ~(~A and X) which should be read as “Not A
prevents D” and does not have a reading that gives both A and D a positive valence. *
A causes Not B  ~B := A
B causes C  C := B
C causes D  D := C
Composing the three functions, we get D := ~A. This is the representation of “A prevents
D.”
A allows B  B := A and X
B causes C  C := B
C causes D  D := C
Composing the three functions, we get D := A and X. This is the representation of “A
allows D.”
A causes Not B  ~B := A
B causes Not C  ~C := B
C allows D  D := C and X
Composing the three functions, we get D := A and X. This is the representation of “A
allows D.”
A causes B  B := A
B prevent C  C := ~B
C causes D  D := C
Composing the three functions, we get D := ~A. This is the representation of “A prevents
D.”
A causes B  B := A
B allows C  C := B and X
C causes D  D := C
Composing the three functions, we get D := A and X. This is the representation of “A
allows D.”
A causes Not B  ~B := A
B causes Not C  ~C := B
C causes D  D := C
Composing the three functions, we get D := A. This is the representation of “A causes
D.”
A causes B  B := A
B causes C  C := B
C causes Not D  ~D := C
Composing the three functions, we get D := ~A. This is the representation of “A prevents
D.”
A causes B  B := A
B causes C  C := B
C allows D  D := C and X
Composing the three functions, we get D := A and X. This is the representation of “A
allows D.”
A causes Not B  ~B := A
B causes C  C := B
C causes Not D  ~D := C
Composing the three functions, we get D := A. This is the representation of “A causes
D.”
A causes B  B := A
B causes Not C  ~C := B
C allows D  D := C and X
Composing the three functions, we get D := ~A and X. This is the representation of “A
prevents D.”