When Sellars wrote his article on causal modalities and counterfactual conditionals in 1958, he was reacting to the arguments of Chisholm and Goodman. Sellars sets out to disarm those arguments. In doing so he develops a theory of conditionals that accommodates the troubling and puzzling examples. It is quite a decent theory of conditionals, when it is put in modern form, and could have held its ground in debates in the years before Stalnaker and Lewis changed the game.
- The core of the problem page 1
- Sellars’ representation of nature: the thing-kind framework page 2
- Conditionals in this framework page 2
- What conditionals are in this framework page 3
- Sellars’ theory in modern form page 4
- Principles that hold for Sellars’ conditionals page 5
- Handling the examples about counterfactuals page 7
- Nesting and iteration of conditionals page 8
1. The core of the problem
One point about conditionals was central and crucial. It had been assumed that the logic of conditionals must mimic the logic of valid arguments. That would mean that Weakening would hold: an argument will remain valid if extra premises are added. If A implies C then (A & B) implies C, and by that mimicry, “if A then C” must imply “If (A & B) then C”. But “this match would light if struck” does not imply “this match would light if moistened and struck”.
The crucial point about the failure of Weakening is that reasoning under supposition will fail for conditionals. For Weakening can be ‘proved’ as follows:
- if A then C (Given)
- A & B Supposition
- A 2 conjunction elimination
- C 1, 3 modus ponens
- if A & B then C 2-4, conditional introduction
The rule of conditional introduction is the principle for discharging a supposition, which works fine for valid arguments. It fails when mimicked for conditionals.[1] Given this, there quickly follow other examples of reasoning with conditionals which are not in accord with earlier logics, such as C. I. Lewis’ modal logic of ‘strict implication’.
2. Sellars’ representation of nature: the thing-kind framework
To make his point, Sellars introduces what he takes to be the canonical form of the language of science, namely the language of the thing-kind framework. That is “the conceptual framework in terms of which we speak of what things do when acted upon in certain ways in certain kinds of circumstance” (1958: 225).
The basic pattern he describes as follows:
Suppose we have reason to believe that
𝜙-ing Ks (in circumstances C) causes them to 𝜓
(where K is a kind of thing – e .g., match). Then we have reason to believe of a particular thing of kind K, call it x, which is in C, that
x, would 𝜓, if it were 𝜙-ed.
And if it were 𝜙-ed and did 𝜓, and we were asked “Why did it 𝜓?” we would answer, “Because it was 𝜙-ed”; and if we were then asked, “Why did it 𝜓 when 𝜙-ed?” we would answer “Because it is a K.” If it were then pointed out that Ks don’t always 𝜓 when 𝜙-ed, we should counter with “They do if they are in C, as this one was.” (Sellars 1958: 248.)
The point is that the antecedent is an input (action or interaction) statement, the consequent an output statement, and neither input nor output statements describe circumstances (standing conditions). As an example I would take: “if this vase be thrown against the wall it will break”.
Sellars is not easy to read, but this article combined with the earlier “Concepts as involving laws and inconceivable without them” (1948), yields a clear and simple account of just what proposition is expressed by a conditional, on Sellars’ view.[2]
A given thing x of kind K will have a history, which is a trajectory in its state-space. What is characteristic of kind K is not only that specific state-space, but a selection from the set of logically possible histories: the family of histories such a thing can really have, its nomologically possible histories.
“In speaking of a family of possible worlds, what are we to understand by a “world”? Let us begin with the following: A world is a spatio-temporal structure of atomic states of affairs which exhibits uniformities of the sort we have in mind when we speak of the laws of nature.” (1948: 293)
This passage he immediately follows with the admonition to abandon the term “world”, and to speak of possible histories instead:
“Our basic framework is thus a family of possible histories, one of which is the actual and possible history.” (ibid.)
In that framework, kind K is characterized by a restricted family of possible histories: the histories which alone are possible for things of kind K.
3. Conditionals in this framework
Starting with the basic pattern in the quoted passage from Sellars, it is clear that “x, would 𝜓, if it were 𝜙-ed” is true in a situation where x’s kind and the circumstances are ‘just right’. It seems also that in the same conditions “x will 𝜓 , if it is 𝜙-ed” is true, and that this is in addition quite independent of whether the antecedent is true or false. But in a situation where the kind and circumstances are not ‘just right’ we need to think about other conditions. It may be that the conditional is just false, but it is also possible that the antecedent is impossible in those conditions, or the antecedent is possible but not in combination with the consequent. (I will give several examples below.) In that case, the conditional is not just a counterfactual, but a counterlegal.
Rather than ‘proof-texting’ with quotations, here is my summary for the special case in the 1958 passage quoted above:
Explication. To assert “x, would 𝜓, if it were 𝜙-ed” is to assert that the situation is felicitous for this conditional to be true.
The kind and circumstances are felicitous for that conditional exactly if there is a kind K to which x belongs, and there are circumstances C that x is presently in, which are together compossible with x being 𝜙-ed and 𝜓-ing, and it is the case that in all histories possible for any things of kind K in circumstances C, that they 𝜓 when 𝜙-ed.
I italicized “any” to show that on this understanding there is an underlying universal claim, overriding differences that may mark individuals of the same kind in the same circumstances.
The condition of compossibility is crucial to accommodate puzzling examples.
Example 1: There may be a kind of match which lights when moistened (with a special chemical that oxidizes rapidly when wet) and does not light if struck, for striking it removes the chemical. There is also the familiar kind of match which will light if struck only when dry. But there is no kind of match which will light when both moistened and struck. The antecedent is not impossible: take any match at all, and at will you can both moisten and strike it. But its combination with the consequent is impossible.
Example 2: The electric circuit in this room is such that the light will go on if either switch A or switch B, but not both, is in the down position. Currently both switches are in the up position. So both conditionals “if A is clicked the light will be on” and “If B is clicked the light will be on” are true, but the conditional “If A is clicked and B is clicked the light will be on” is false. Again, the antecedent is not impossible: you can click both switches. But the result will not be to turn on the light.
4. What conditionals are in this framework
In this Explication, the conditional in question is construed as an existential statement of fact: there is a kind K and circumstance C that characterize this situation such that …. An existential statement is in effect a disjunction, generalized to cover the case of infinitely many alternatives. So “If A then B” will amount to:
the case described in A amounts to a disjunction of combinations of kinds and circumstances {K(i) and C(i): i is in J}, each of which is compossible with A and B, and for which all histories that satisfy A also satisfy B
The implication relation signified by ‘all histories that satisfy … also satisfy …’ encodes the usual valid argument relationship.
So, on this view, a conditional is a disjunction involving a modality. Weakening does not hold for conditionals, due to the compossibility requirement.
We note one oddity, a difference from more familiar theories of conditionals, that has to do with modality. Suppose A logically implies B. In familiar theories of conditionals the conditional ‘if A then B’ is then automatically true. But there may be kinds K and circumstances C which are not compossible with A and B. In that case the conditional “if A then B” will not be logically valid. In fact, in that case this conditional is simply the disjunction of all kinds K(i) and circumstances C(i) compossible with A and B.
For example if something is scratched with a human nail then it is scratched with something human. But a steel vase cannot be scratched with a human nail. So then, on this Sellarsian account, it is not correct to assert, when looking at a steel vase in normal circumstances, that it would be scratched by something human if it were scratched by a human nail.
Yes, it is odd. But if someone were to assert that, while we are looking at a steel vase in normal circumstances, that would be very odd too! In this case, the counterfactual is a counterlegal conditional, and counterlegals pose a host of further difficulties for our intuitions (see e.g. Fisher 2017).
5. Sellars’ theory in modern form
Sellars 1948 and 1958 together spell out the thing-kind framework, with the elaboration into the possible histories of things of certain kinds in certain circumstances, but not in the precise form that we now require. Yet it is clear enough to provide a fairly straightforward way to put this in modern form.
If the main concepts are to be generalized, intuitively, then we can read “kind” and “circumstance” to stand for “whatever grounds the laws of nature” and “boundary conditions”. But I will keep the symbols K and C and the mnemonic “kind” and “circumstance” throughout.
A model will be a triple <H, P, F>, each element a non-empty set: I will call H the set of situations, P is a partition of H, and F, the family of propositions, is a set of subsets of H.
The elements of H, which correspond to Sellars’ possible histories, are also triples <K, C, 𝜋>. We think of possible situations as involving a single particular (whether a physical system or a whole world). This particular is classified as being of a certain kind K, as being in certain circumstances C, and as having a history 𝜋 (which could have various further attributes, not specifically spelled out here, such as being 𝜙-ed).
The family F may not be the family of all subsets of H. But we specify that it includes both H and the empty set Λ, and is closed under intersection, union, and relative complement. So F is a complete field (Boolean algebra) of sets. But moreover we specify that P is part of F, where we now define P as the family of all sets <K, C> with the definition:
<K, C> = {x in H: there is a factor 𝜋 such that x = <K, C, 𝜋>}.
In other words, a cell of this partition P is the set of all possible histories for an individual of kind K in circumstances C. How many cells does this partition have? How many kinds, and how many circumstances, are there that can go together in this way? We leave that open, so there are large models and small models, even models with just a single cell in that partition.
When are kind and circumstances in a given cell X of P felicitous for the conditional A →B? Compossibility is just being possible together. So what is required is that X ∩ A ∩B is not empty.
Abbreviation. “◊( X ∩ A ∩B)” for “( X ∩ A ∩B) ≠ Λ”
We can then define the operation that forms conditionals:
Definition. A → B = ∪{X in P: ◊( X ∩ A ∩B) and X ∩ A ⊆ B}
Abbreviation. To facilitate reading I will abbreviate “◊( X ∩ A ∩B)” to “◊” whenever the context allows that to be clear.
By this definition, every such conditional is a proposition, for if A and B are in F then A → B is a union of cells in partition P, all of which belong to F, which is closed under union. So F is closed under the operation →.
A conditional proposition is in general quite a large set of situations. It comprises all the situations in which the kind and circumstances are just right to make B follow from A. That is just what we saw above, in Sellars’ exposition, in terms of disjunctions.
Truth. If x is a situation and A is a proposition then A is true in x exactly if x is a member of A.
Notice that if x and y are both situations belonging to a given cell X of P then any conditional true in x is true in y. The special factor 
What principles hold or fail in the logic of Sellars’ theory of conditionals?
6. Principles that hold for Sellars’ conditionals
Modus Ponens. A ∩ (A → B) ⊆ B
If A and A →B are both true in x then B is true in x.
For if the premises are true in situation x, then x is in A and x is in ∪{X in P: ◊ & X ∩ A ⊆ B}. But if x is in the union of the family {X in P: ◊ & X ∩ A ⊆ B} then x is in one of the cells, Xo, in that family. So x is in A and in Xo and X ∩ A ⊆ B}. there x is in B.
Impossibility. (A → ~ A) = Λ
For there is no cell X such that ◊( X ∩ A ∩ ~A) .
Definition. ◊(A, B) = ∪{X in P: ◊( X ∩ A ∩B)}
Entailment. If A ⊆ B then (A → B) = ◊(A, B)
For if A ⊆ B then, for any cell X in P it is the case that ◊( X ∩ A ∩B) and X ∩ A ⊆ B if and only if ◊( X ∩ A ∩B).
This has the corollary that (A → A) = ◊(A, A).
The principle that A → B implies A → (A & B) does not seem to have a standard name; I’ll call it Carry-over. This has the corollaries that A → ~ A implies A → (A & ~A), and that ~A → A implies ~A → (A & ~A), which mark the cases in which A is impossible or necessary.
In terms of propositions:
Carry-over. (A → B) ⊆ [A →(A ∩ B)]
For suppose situation y is in the union of the cells in the family {X in P: ◊ & X ∩ A ⊆ B), and hence in a specific cell Xo in that family. Since Xo, A, B are compossible then so are Xo, A, A ∩ B. And since Xo ∩ A ⊆ B it is the case that Xo ∩ A ⊆ A ∩ B. So y is in [A → (A ∩ B)].
In normal modal logic the principle that if B1, … , Bn implies C then □B1, … , □Bn A implies □C, is sometimes called Goedel’s rule. So I’ll choose that to name the analogous principle for conditionals: If A, B1, … , Bn implies C then A → B1, … , A → Bn implies A → C. Notice that this is much weaker than the Classic or Intuitionist principle that if A, B ├ C then B ├ A → C.
In terms of propositions rather than sentences it becomes:
Goedel. If (B1 ∩ … ∩ Bn) ⊆ C then(A → B1) ∩ … ∩ (A → Bn) ⊆ (A → C)
Without real loss of generality I will provide the argument just for the case n = 2.
Suppose B1 ∩ B2 ⊆ C. Suppose also that (A → B1) ∩ (A → B2) is true in y. It follows that y is in a cell X such that ◊(X ∩ A ∩B1) and X ∩ A ⊆ B1, and that y is in a cell Y such that ◊( Y ∩ A ∩ B2) and Y ∩ A ⊆ B2.
Since y can only be in one cell, it must be a cell Xo such that ◊(Xo ∩ A ∩B1) and ◊( Xo ∩ A ∩ B2) and Xo ∩ A ⊆ B1and Xo ∩ A ⊆ B2. Therefore Xo ∩ A ⊆ C.
In addition, because ( Xo ∩ A ∩B1) is not empty it follows that ( Xo ∩ A) is not empty, and since that is part of C, it follows that ◊( Xo ∩ A ∩ C).
Therefore Xo, which contains y, is part of the union of the cells X such that ◊( Xo ∩ A ∩ C) and X ∩ A ⊆ C, which is A → C. Therefore A → C is true in y.
The Principle that “and” distributes over “if … then”, A → (B & C) ├ (A → B) & (A → C), and conversely, has its formulation in terms of propositions:
∩-Distribution 1. A → (B ∩C) ⊆ [(A → B) ∩ (A → C)]
If cell X is such that X ∩ A ⊆ B ∩ C then it is certainly such that X ∩ A ⊆ B and X ∩ A ⊆ C.
Moreover, if ◊( X ∩ A ∩ B ∩ C) then ◊( X ∩ A ∩ C).
∩-Distribution 2. [(A → B) ∩ (A → C)] ⊆ [A → (B ∩C)]
For suppose that situation y is in both [(A → B) and (A → C). Since y cannot be in more than one cell, y is in some cell Xo such that ◊( X ∩ A ∩ B) and ◊( X ∩ A ∩ C), and moreover X ∩ A ⊆ B and X ∩ A ⊆ C. The latter two facts entail that (X ∩ A) ⊆ (B ∩ C). The former two facts each entail that X ∩ A is not empty. But together with (X ∩ A) ⊆ (B ∩ C) that implies that ◊( X ∩ A ∩ B ∩ C).
Of the corresponding principle, that “or” distributes over “if .. then”, generally called Conditional Excluded Middle or CEX, one part holds but the other part fails, as we will see below.
CEX 1. (A → B) ∪ (A → C) ⊆ [A → (B ∪ C)]
For if X ∩ A ⊆ B then X ∩ A ⊆ (B∪ C). Note that ◊( X ∩ A ∩B) implies that ◊[ X ∩ A ∩ (B ∪ C)] . We may also note that CEX 1 follows from the simpler principle (again I don’t have a standard name, but it is a special case of Goedel):
Weakening on the right. If B ⊆ C then (A → B) ⊆ (A → C).
For if ◊( X ∩ A ∩B) and B ⊆ C then ◊( X ∩ A ∩ C).
7. Handling the examples about counterfactuals
The examples I gave above with matches and light switches can be represented by little models of this sort. So we know, for example, that Weakening fails for conditionals, in general.
Weakening. If A → C ⊆ [(A ∩ B) → C]. FAILS.
The simple reason is that ◊( X ∩ A ∩ C) does not guarantee that ◊( X ∩ A ∩ B ∩ C).
CEX 2. [A → (B ∪ C)] ⊆ (A → B) ∪ (A → C). FAILS.
For example, C = ~B, then the part of A in any cell will be included in (B ∪ ~B) but there will in general be cells in which A is not included either in B or in ~B, but overlaps both.
Centering. A ∩ B ⊆ (A → B). FAILS.
Suppose that y is in A ∩ B. Then there is a cell Xo in P such that y is in Xo ∩ A ∩ B. Since P is a partition, the cells do not overlap, so y can be in the union of the cells {X in P: ◊ & X ∩ A ⊆ B} only if it is in one of those cells. But Xomay not be one of those, it may be a cell where A overlaps both B and ~B.
There is a traditional principle that is much discussed in this subject area, Import-Export (A & B) → C ├ A → (B → C), and its converse A → (B → C) ├ [(A & B) → C].
Import-Export 1. [A → (B → C)] ⊆ [(A ∩ B) → C]. FAILS
Counterexample. There is only one cell, X. A is non-empty, but A and B do not overlap, B is a non-empty part of C. So (B → C) = X. X is compossible with A and A is part of X, hence of (B → C). So it is also the case that X = [A → (B → C)].
Since B is part of C, (A ∩ B) is also part of C, but (A ∩ B) is empty, So (A ∩ B ∩ C) is empty. So X is not [(A ∩ B) → C], which is in fact the empty set, and does not have X as a part.
Import-Export 2. (A ∩ B) → C ⊆ [A → (B → C)]. FAILS.
Counterexample (see Diagram below). In this model there are exactly two cells, X0 and X1.
◊( Xo ∩ A ∩B ∩ C) and Xo ∩ A ∩ B ⊆ C, but Xo ∩ B is not part of C.
X1 ∩ A ∩ B is not empty but it is not part of C, and hence also X1 ∩ B is not part of C.
From these data we derive the following:
- [(A ∩ B) → C] = X0
- (B → C) = Λ
- Neither (Xo ∩ A) nor (X1 ∩ A) is empty
- [A → (B → C)] = Λ
Therefore (A ∩ B) → C is not included in [A → (B → C)].
Let’s also show how we can model the example of the two switches.
Example. The light will go on if either switch A or switch B, but not both, is in the down position. Currently both switches are in the up position. So both conditionals “if A is clicked the light will be on” and “If B is clicked the light will be on” are true, but the conditional “If A is clicked and B is clicked the light will be on” is false. Again, the antecedent is not impossible: you can click both switches, and the result will be that the light is still off.
The model is very simple. There is only one cell X. A and B are both non-empty, and each is part of C. So X = (A → C) = (B → C). But A and B do not overlap, so (A ∩ B) → C] = Λ.
8. Nesting and iteration of conditionals
Equating conditionals with suitably chosen disjunctions (finite or infinite), if accepted in general for all conditionals, has therefore yielded a specific family of logical principles to govern reasoning with conditionals.
The failure of Import-Export, in both directions, shows that nesting of conditionals in Sellars’ theory is not trivial. The nested conditional [A → (B → C)] is not in general equivalent to a statement in which there is no nesting of conditionals.
But this flexibility does not go very far. For Sellars, counterfactual conditionals are grounded on underlying lawlike strict conditionals, due to what is the case in all possible histories for things of a given kind. That is a differentiating feature, it is unlike anything along the lines of ‘nearest possible world’ relations, on which conditionals would be seen as grounded by Stalnaker and Lewis.
As a putatively difficult example, take
(*) If this vase would break if thrown against the wall then it would break if dropped on the floor
This has the form (A → C) → (B → C).
To begin, let us see how the antecedent can imply the consequent. There is a certain list of kinds of vases (glass, ceramic, steel) and circumstances (wall of brick, of wool, …). For some of these combinations of kind and circumstance it is the case that those vases will always, necessarily, break if thrown against the wall. Consider the antecedent conditional “If this vase were to be thrown against the wall then it would break”, and two cases:
- the conditions are normal and this is a steel vase
- the conditions are normal and this is a porcelain vase
In case 1. the antecedent conditional is a counterlegal. In the second case it is lawlike and true. But in that second case, the consequent “If this vase were to be dropped on the floor then it would break”, is also lawlike and true. We might put it this way: in neither case is this matter very iffy.
Reduction of a certain nesting. (A → C) → (B → C) = (A → C) ∩ (B → C)
For suppose first that y is in cell Xo and is in (A → C) → (B → C). Then ◊( X ∩ (A → C) ∩ (B→C)) and Xo ∩ (A → C) ⊆ (B → C).
But Xo ∩ (A → C) is non-empty only if Xo is a member of the family {X in P: ◊( X ∩ A ∩C) & X ∩ A ⊆ C}, for distinct cells do not overlap. Therefore (A → C) is true in y. Similarly, Xo ∩ (B → C) is non-empty only if Xo is a member of the family {X in P: ◊( X ∩ B ∩ C)& X ∩ B ⊆ C}, and so (B → C) is true in y as well.
Secondly, suppose y is in cell Xo and is in (A → C) ∩ (B → C). Then Xo belongs to both families {X in P: ◊( X ∩ A ∩ C) & X ∩ A ⊆ C} and {X in P: ◊( X ∩ B ∩ C) & X ∩ B ⊆ C}. Therefore the intersections of Xo with (A → C) and with (B → C) are not empty but both equal Xo itself. Hence also Xo ∩ (A → C) is part of (B → C), trivially, so y is in (A → C) → (B → C).
REFERENCES
Fisher, Tyrus (2017) “Counterlegal dependence and causation’s arrows: causal models for backtrackers and counterlegals”. Synthese 194: 4983-5003.
Fitch, Frederick (1952) Symbolic Logic. New York: Ronald Press.
Sellars, Wilfrid (1948) “Concepts as involving laws and inconceivable without them”. Philosophy of Science 15: 287-315.
Sellars, Wilfrid (1958) “Counterfactuals, Dispositions, and the Causal Modalities”. Minnesota Studies in the Philosophy of Science II: 225-308. Open Access at https://cla.umn.edu/mcps/publications/minnesota-studies-philosophy-science
Thomason, Richmond H. (1970) “A Fitch-style formulation of conditional logic”. Logique et Analyse Nouvelle Série 13: 397-412.
NOTES
[1] Fitch had modified reasoning under supposition for modal logic in 1952, and in 1970 Richmond Thomason would similarly modify reasoning under supposition for the logic of conditionals.
[2] It is a drawback for this account that it does not distinguish between present tense subjunctive and present tense indicative conditionals. For Sellars’ story will be the same, in this summary, if “would” and “were” are replaced by “will” and “is”. We may for now just note that the typical examples to distinguish subjunctive and indicative conditionals are not present tense, however, but of the “what would have been if” type.
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