Let \(L\) denote the language of Peano arithmetic. A (counterfactual) world \(W\) is any subset of \(L\). These worlds need not be consistent. Let \(\mathcal{W}\) denote the set of all worlds. The actual world \(W_\mathbb{N}\in\mathcal{W}\) is the world consisting of all sentences that are true about \(\mathbb{N}\).
Consider the function \(C:L\rightarrow\mathcal{W}\) which sends the sentence \(\phi\) to the world we get by “correctly” counterfactually assuming \(\phi\). The function \(C\) is not formally defined, because we do not yet have a satisfactory theory of logical counterfactuals.
Hopefully we all agree that \(\phi\in C(\phi)\) and \(\phi\in W_\mathbb{N}\Rightarrow C(\phi)=W_\mathbb{N}\).
Given an (infinite) directed acyclic graph \(G\), and a map \(v\) from sentences to vertices of \(G\), we say that \(C\) is consistent with \(G\) and \(v\) if \(C(\phi)=C(\psi)\) for all \(v(\phi)=v(\psi)\), and whenever \(W_{\mathbb{N}}\) and \(C(\phi)\) disagree on a sentence \(\psi\) there must exist some causal chain \(\psi_1,\ldots \psi_n\) such that:
\(v(\psi_1)=v(\phi)\),
\(\psi_n=\psi\),
\(W_{\mathbb{N}}\) and \(C(\phi)\) disagree on every \(\psi_i\), and
\(v(\psi_i)\) is a parent of \(v(\psi_{i+1})\).
These conditions give a kind of causal structure such that changes from \(W_{\mathbb{N}}\) and \(C(\phi)\) must propagate through the graph \(G\).
Given a function \(f:\mathcal{W}\rightarrow\mathbb{R}\), we say that \(C\) optimizes \(f\) if for all \(\phi\in W\) and \(W\neq C(\phi)\) we have \(f(C(\phi))>f(W)\).
Many approaches to logical counterfactuals can be described either as choosing the optimal world (under some function) in which \(\phi\) is true or observing the causal consequences of setting \(\phi\) to be true. The purpose of this post is to prove that these frameworks are actually equivalent, and to provide a strategy for possibly showing that no attempt at logical counterfactuals which could be described within either framework could ever be what we mean by “correct” logical counterfactuals.
A nontrivial cycle in \(C\) is a list of sentences \(\phi_1,\ldots,\phi_n\), such that \(\phi_{i}\in C(\phi_{i+1})\), \(\phi_n\in C(\phi_1)\), and the worlds \(C(\phi_i)\) are not all the same for all \(i\).
Given a partial order \(\succ\) , we say that \(C\) optimizes \(\succ\) if for all \(\phi\in W\) and \(W\neq C(\phi)\) we have \(C(\phi)\succ W\).
Our main result is that the following are equivalent:
\(C\) optimizes \(f\) for some function \(f\).
\(C\) is consistent with \(G\) and \(v\) for some DAG \(G\) and map \(v\).
\(C\) has no nontrivial cycles.
\(C\) optimizes \(\succ\) for some partial order \(\succ\).
Proof:
1 \(\Rightarrow\) 2: Construct the graph \(G\) with a vertex for every world in the image of \(C\). The map \(v\) sends \(\phi\) to the vertex associated with \(C(\phi)\). Insert an edge to \(W_{\mathbb{N}}\) from every other vertex. Insert an edge from the vertex associated with \(W_1\neq W_{\mathbb{N}}\) to the vertex associated with \(W_2\neq W_{\mathbb{N}}\) whenever \(f(W_1)<f(W_2)\). Clearly \(C(\phi)=C(\psi)\) for all \(v(\phi)=v(\psi)\).
Assume \(W_{\mathbb{N}}\) and \(C(\phi)\) disagree on a sentence \(\psi\). If \(\psi\in W_{\mathbb{N}}\) then \(v(\phi)\rightarrow v(\psi)\), since \(v(\psi)\) is the vertex associated with \(W_\mathbb{N}\) which is a child of every vertex. Therefore you get a length 1 path from \(v(\phi)\) to \(v(\psi)\). Otherwise, \(\psi\notin W_{\mathbb{N}}\), so \(\psi\in C(\phi)\). In this case, note that since \(\psi\) is in both \(C(\phi)\) and \(C(\psi)\), and these worlds are not the same, it must be that \(f(C(\psi))>f(C(\phi))\). Again, this means that you get a length 1 path from \(v(\phi)\) to \(v(\psi)\). Therefore \(C\) is consistent with \(G\) and \(v\).
2 \(\Rightarrow\) 3: Consider a nontrivial cycle \(\phi_1,\ldots,\phi_n\). If any of these sentences were in \(W_\mathbb{N}\), then they would all be true in \(W_\mathbb{N}\), since \(C(\phi)=W_\mathbb{N}\) whenever \(\phi\in W_\mathbb{N}\). This would contradict the fact that \(\phi_1,\ldots,\phi_n\) is a nontrivial cycle.
Otherwise, since \(\phi_i\in C(\phi_{i+1})\), there must be a path from \(v(\phi_{i+1})\) to \(v(\phi_i)\) in \(G\), since \(C(\phi_{i+1})\) and \(W_\mathbb{N}\) differ on \(\phi_i\). Concatenating these paths together would give a cycle in \(G\) unless the \(v(\phi_i)\) are all the same vertex. However, if all of the \(v(\phi_i)\) are the same vertex, then all of the \(C(\phi_i)\) would be the same world, which would contradict the fact that \(\phi_1,\ldots,\phi_n\) is a nontrivial cycle.
3 \(\Rightarrow\) 4: Consider the partial order on the image of \(C\) constructed by saying that \(W_1\succ W_2\) if \(W_1=C(\phi)\) for some \(\phi\in W_2\), and taking the transitive closure of these rules. If this were not a partial order, it would have to be because we created a cycle of worlds \(W_1,\ldots,W_n\), such that each \(W_i=C(\phi_i)\) with \(\phi_i\in W_{i+1}\) and \(\phi_n\in W_{1}.\) This is would be a nontrivial cycle.
Extend this partial order to all of \(\mathcal{W}\) by saying that if \(W_1\) is in the image of \(C\) and \(W_2\) is not, then \(W_1\succ W_2\). Note that this \(C\) optimizes this partial order by definition.
4 \(\Rightarrow\) 1: Let \(C\) optimize the partial order \(\succ\). Consider the restriction of \(\succ\) to the image of \(C\). This is a partial order on a countable set. Order the worlds in this partial order \(W_1,W_2,\ldots.\) Embed the partial order into \(\mathbb{R}\) by repeatedly defining \(f(W_n)\) such that:
\(f(W_n)>0\),
\(f(W_n)>f(W_i)\) for all \(i<n\) and \(W_n\succ W_i\), and
\(f(W_n)<f(W_i)\) for all \(i<n\) and \(W_i\succ W_n\).
Define \(f(W)=0\) if \(W\) is not in the image of \(C\). Clearly this function \(f\) is constructed such that C optimizes \(f\).
\(\square\)
This result is useful not just for establishing the equivalence of (a certain type of) optimization and acyclic causal networks, but also for providing a strategy for showing that “correct” logical counterfactuals cannot arise as an optimization process or through an acyclic causal network. To show this, one only has to exhibit a single nontrivial cycle. In the simplest case, this can be done by exhibiting a pair of sentences \(\phi\) and \(\psi\) such that \(\phi\) and \(\psi\) are clearly counterfactual consequences of each other, but do not correspond to identical counterfactual worlds.
Do the “correct” logical counterfactuals exhibit a nontrivial cycle?
