Abstract
I use some recent formal work on measuring causation to explore a suggestion by James Woodward: that the notion of causal specificity can clarify the distinction in biology between permissive and instructive causes. This distinction arises when a complex developmental process, such as the formation of an entire body part, can be triggered by a simple switch, such as the presence of particular protein. In such cases, the protein is said to merely induce or "permit" the developmental process, whilst the causal "instructions" for guiding that process are already prefigured within the cells. I construct a novel model that expresses in a simple and tractable way the relevant causal structure of biological development and then use a measure of causal specificity to analyse the model. I show that the permissiveinstructive distinction cannot be captured by simply contrasting the specificity of two causes as Woodward proposes, and instead introduce an alternative, hierarchical approach to analysing the interaction between two causes. The resulting analysis highlights the importance of focusing on gene regulation, rather than just the coding regions, when analysing the distinctive causal power of genes.
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Notes
 1.
A reading more consistent with the original thesis is that we should treat causes with “consistency with respect to a criterion” (Oyama 2003, p. 183)—an issue discussed further in Griffiths and Gray (2005). If this is right, then Woodward’s properties provide the very kind of criteria that Developmental Systems Theory would welcome, assuming they are consistently applied.
 2.
Here is a simple approach to computing causal specificity. A joint probability distribution over two discrete random variables \(X\) and \(Y\) can be represented using a table (such as those in Table 3), where each cell contains P(x, y)the probability of observing both \(X=x\) and \(Y=y\). To calculate the (noncausal) mutual information between the two variables, we first calculate the pointwise mutual information for each cell, \(\log\frac{P(x, \;y)}{P(x)P(y)}\), noting that numerator here is the value of the cell, and the denominator is simply the product of the sums of both the row and column in which the cell lies. By convention, if either \(P(x)\) or \(P(y)\) is zero then pointwise mutual information is 0. The mutual information is then the expectation of the pointwise mutual information computed in all cells, thus \(I(X; Y) = \sum P(x, y) \log \frac{P(x, \,y)}{P(x)P(y)}\). Now, if the joint distribution in the table was produced under intervention, then this same operation on the table computes a causal specificity measure. To see why, note that pointwise value we now calculate will have a numerator (the cell value) of \(P(\widehat x)P(y \widehat x)\) and a denominator of \(P(\widehat x)P(y)\). Cancelling \(P(\widehat x)\), we now compute \(\log \frac{P(y \widehat {x})}{P(y)}\) for each cell, and the expectation of this is \(I(\widehat X;Y) = \sum P(\widehat{x}) P(y\widehat{x}) \log\frac{P(x\widehat{y})}{P(y)}\), which is causal specificity as defined by Griffiths et al.
 3.
This is because the mutual information between two variables cannot exceed the lowest entropy of the two variables, and the maximum entropy of a variable with two states is 1 bit. Formally, \(I(X; Y) \le \min (H(X), H(Y))\).
 4.
A specificity of zero tells us that the dial is not a cause at all. So I could simply say that the dial is a cause when the switch is on, but not a cause when the switch is off, at least in this case. Having a graded measure of causation will, however, turn out to be important in more complex cases in the following sections.
 5.
Woodward’s discussion appears to shift between these two modes of analysis. He tells us that “[...] the dial gives one relatively fine grained control over which station is received, assuming that the switch is on.” (Woodward 2010, my italics), thus drawing attention to specificity against one background. When he talks of the switch, however, he appears to consider it against all backgrounds: “one can’t modulate or finetune which station is [received] by varying the state of the switch.”
 6.
My thanks to Paul Griffiths for drawing my attention to the Galton box.
 7.
We can state this equivalently by saying that the conditional entropy is zero: \(H(BS,P=p_4) = 0\). But the formulation in Eq. (1) will serve us better below.
 8.
To avoid any confusion: I’m now treating the on–off switch as a foreground cause and the tuning dial as a background cause. This is the opposite of what I did in “Tuning Woodward’s radio” section. This is no cause for alarm, I am making a different point here.
 9.
This is, in fact, how the simulations that generate the distributions actually work.
 10.
The quote is similar to that cited by Woodward (2010, 302)
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Acknowledgements
I am indebted to the Theory and Method in Biosciences group at the University of Sydney, in particular to Stefan Gawronski, Paul Griffiths, Arnaud Pocheville, and Karola Stotz, and for their feedback and assistance. Funding was provided by Swansea University Templeton World Charity Foundation.
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Appendix
Appendix
The appendices give further details for constructing the Waddington box and generating the distributions that were analysed in the paper. The code for producing this (and other) Waddington boxes, and the definitive source for how the simulations work, is freely available at https://github.com/brettc/waddingtonbox.
The layouts
There are three aims in limiting the way pins can appear in a layout. The first is to reduce the total number of layouts to something that can be analysed in a reasonable length of time. The second aim is ensure that all layouts permit the ball to reach the buckets (in combination with the physics rules defined below). The third aim is to ensure the layouts produce something interesting—a variety of different mappings.
I consider the layout of individual rows of pins first, as the full number of layouts will be constructed by combining these. The top row is a special case, which I will discuss last. The remaining four rows each contain 15 holes which could contain pins. The unrestricted number of possible pins layouts for such a row is thus \(2^{15} =32{,}768\). I now add two restrictions.

(L1)
Pins must have at least three holes between them, and

(L2)
there must be at least three pins in each row.
This reduces the number of possible pin configuration in a single row to 99.
Now for the top row. Why is it different? I could have just had five rows with 15 holes, but the placement of the slots means that many of the pins on the top row will never interact with the ball. Instead, the top row has two sets of three holes below the slots, and each set can contain zero or one pins. So the total layouts for the top row are \(4^2 = 16\). The total number of layouts with these five rows is \(16 \times 99^4 = 1{,}536{,}953{,}616\). To further reduce this, I add one last restriction:

(L3)
A pin can never lie directly above another pin on the next row.
This brings the total number of layouts down to \(3{,}306{,}664\), which is small enough for an exhaustive analysis.
The physics
To implement a simple physics that captures a ball falling through a series of rows with pins I make the following assumptions:

(R1)
A ball falls in one of 15 discrete channels that line up with the 15 pins.

(R2)
A ball interacts with each row independently, arriving at the row in one channel and exiting at one or two channels.
The way that a ball interacts with the pins on each row obeys the following rules:

(R3)
The ball glances off a pin If there is a pin in the channel to the right of the ball it shifts one channel left; if there is a pin to the left it shifts one channel to the right.

(R4)
The ball bounces on a pin; it could go either way If there is a pin in the same channel as the ball it takes two paths, moving left two channels with probability 0.5 and right two channels with probability 0.5.

(R5)
The ball hits the edge of the box If either of the previous two rules would cause the ball to leave the 15 channels, the ball goes to two channels to the opposite of the pin it encountered.
Rule (R4) means that the path a ball takes can split, and we must then track both balls as they descend. A path can split multiple times and carries with it a probabilities. These probabilities are then accumulated in the buckets. The four buckets, \(\{b_1 \ldots b_4\}\) collect the ball falling in the following channels:

(R6)
The four buckets collect from the 15 channels \(b_1 \leftarrow (1, 2, 3, 4)\), \(b_2 \leftarrow (4, 5, 6, 7, 8)\), \(b_3 \leftarrow (8, 9, 10, 11, 12)\), \(b_4 \leftarrow (12, 13, 14, 15)\)
Notice that the buckets overlap—some channels end up in both buckets. This works the same as if they bounced on a pin: with a probability of 0.5 that they go either way.
The splitting of pathways is the key to Galton’s box. A ball falling on top of a pin has an equal probability of going left or right. If the ball falls directly onto another pin, the pathways split again. In the Quincunx layout of Galton’s box, the pathways produce a binomial distribution across the buckets at the bottom (which approximates a normal distribution). In the Waddington box, this is what allows the layouts to generate a range of different probability distributions over the buckets.
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Calcott, B. Causal specificity and the instructive–permissive distinction. Biol Philos 32, 481–505 (2017). https://doi.org/10.1007/s1053901795680
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Keywords
 Causation
 Specificity
 Instructive
 Permissive
 Information
 Waddington