[Updated with corrections March 19, 2023]

• 1 Suppositions and selective belief suspension                                page 1
• 2 Rules for reasoning: what is allowed under supposition               page 3
• Appendix I. The logical principles engendered by the rules            page 5
• Appendix II. The literature about supposition                                  page 6

I am going for a week’s holiday in La Jolla in July, and so I am taking my lightest summer clothes.  But suppose it rains all week?  I’ll also take some extra books.

Suppositions are a common device in practical reasoning.  They are there also closely, if not straightforwardly, connected with conditional judgments.  From the above imagined monologue we may infer an implicit assertion of “If it rains, I will read”.

In the philosophical literature there are two guiding ideas about supposition (in its modern sense, illustrated above).  The first is that it is a move in conversation or argumentation, posing a statement and inviting exploration of its implications.  The second is that it is a ‘provisional updating”, closely related to the well-developed topic of how opinion is to change in response to actual new evidence or experience.  Both approaches bring supposition into close relation to the logic of conditionals.

In Appendix II I will sketch an overview of the literature. 

My approach here is of the first sort, which seems to me more fundamental.  I will also relate reasoning under supposition to the logic of conditionals.

1.   Suppositions and selective belief suspension

A supposition is not an assertion.  “Suppose that A” is an injunction or invitation to explore the implications, in context, of the statement that A.  While statements made in the course of a conversation or argument may have other functions as well, I will assume here for simplicity that they are all either assertions or suppositions (and come marked as such).

Practical deliberation.  The example of holiday planning that I gave would be misleading if we took it as a general guide to reasoning with suppositions.  It is quite special in that the speaker, who does not know whether or not it will rain, can rely on any and all knowledge and belief that s/he does have already.  

Even for practical deliberation that is not always the case.   We may momentarily suspend judgments on possibilities that we normally discount, reflecting that our opinions have not always been entirely accurate. 

However there are at least three other familiar uses of suppositions where drawing on what is previously taken as given is very much restricted.  The more extreme is reductio ad absurdum.  The less extreme, but more common, is imaginative exploration in pursuit of subtler practical goals. Somewhere in between is the reliance on some specific, common knowledge or belief among participants in a discussion or between speaker and audience, with a bracketing or suspending of certain other shared knowledge or belief.  While this is very common, it does not seem that we have any way of identifying what is kept, and what is bracketed, in general.

Reductio ad absurdum.  Suppose I am an atheist, certain that it is true that God does not exist, and wish to show that the belief that He does is absurd.  I could not very well proceed with “Suppose God exists.  In fact he does not exist.  Putting the two together we have a contradiction.  So the supposition that God exists leads to an absurdity.”  I ‘reiterated’ in my proof a line that preceded it, so to say, one of my beliefs given beforehand.  That is not admissible:  when making the supposition I must at the same time suspend or bracket a whole lot of the beliefs I have.  Indeed, if the reductio ad absurdum is to be a logical proof I must suspend all of my factual beliefs.

Imaginative exploration.  The less extreme case may arrive especially when there is an ostensible ulterior motive in the inquiry.  “I know your brother is here to help you with your tax return. But if he were not, what would you do?”  One response would be “Since we know that he is here the “But if” question does not arise”.  That would ignore what was really the purpose the questioner had, namely to find out how able or adept or well prepared or knowledgeable the addressee is with respect to financial affairs.  The “But if” requests a supposition, with a bracketing of a very specific, indeed specified, bit of knowledge, namely that the brother is there.  Specific, but of course bringing in train various other bracketings, such as that he is easily reachable by telephone.

Conversation. What happens here is most easily noticed when it goes wrong.  “If Peter had lit the fuse the bomb would have exploded.” “No, it wouldn’t!  For Peter is very prudent, he would have first disconnected the fuse.”  Here the first speaker keeps constant the knowledge that the fuse is connected to the bomb and leaves aside any special knowledge about Peter, while the second brackets the knowledge that the fuse is connected and keeps constant a certain belief about Peter.  

Linguistic pointers to the “ceteris paribus”.  Acknowledging the tacit selectivity in what is kept constant we may add “all things being equal”, “ceteris paribus”.  The content of this clause, certainly not explicit, must be a specification of what is meant to be bracketed or held constant.  The difference between indicative and subjunctive conditionals seems at first blush to point to a systematic difference that place.   In the famous example “Suppose Oswald didn’t shoot Kennedy/ Suppose Oswald hadn’t shot Kennedy”, it seems that in the former we hold constant that Kennedy was shot, and in the latter that Oswald was the only shooter present.  

There are two difficulties with this.  First, while we can often quite readily tell what the difference is meant to be (as in this example), we have no general guide to how to make the difference.  We only have metaphors, such as that of imagining what is near, nearer, or far among possible worlds.  These suggest a form for reflection, but the same point applies: though able to give quick answers in particular cases, we can’t say how it works.  The second difficulty is that even if there is a guide, it is very far from perfect.  For in the example above, about Peter and the bomb, both conditionals are in the subjunctive.  Yet the content of “all else being equal” is as different in these conditionals as it is in the Kennedy/ Oswald example.

All the efforts in the early history of this subject, by Chisholm, Goodman, Sellars, and others in the 40s, 50s, and early 60s, to find a general answer to the content of the “ceteris paribus” failed.  Stalnaker and Lewis changed the game by shifting the focus to a general form for the truth conditions, instead on what the truth conditions were or could be.  This ended that philosophical epoch.

Tentative conclusion, and project.  What exactly a supposition carries as a requirement to suspend or bracket cannot be answered in general, at least not non-trivially.  Whatever determines the content of the “ceteris paribus” we may assume to be determined by contextual factors not shown on the linguistic surface.

But we can still try to find some constraints to be met, if any principles for reasoning under supposition are to hold in general.  And we can see if there might be a systematic way to signal what is to be kept constant, in the ideal case where nothing is left tacit or implicit.

Project.  Let’s take as clue the point made at the beginning, that there appears to be an intimate connection between supposing and conditionals.  In the example some reasoning went on from the supposition “it rains during the holiday” to the conclusion “If it rains during the holiday I will read”.   Thomason (1970) presented natural deduction rules for what can be brought into that reasoning.  I will explain the ones that were not in dispute between different logics of conditionals, present their rationale, and identify the logical principles that they engender.

2.   Rules for reasoning: what is allowed in general under supposition

Let us imagine a person reasoning in monologue, like in the example at the start.  He reasons in time, so the statements come one by one, they can be numbered.  At a certain point the new line, say 17, begins with “Suppose that A”.  After a certain amount of reasoning, in the following lines, say, 18-29, there is a conclusion “So, if A then B” (line 29) which, to introduce a good terminology, discharges the supposition.  

What is allowed, in general, in those lines 18-28, between the supposition and the conclusion?

I will use symbol → for “if .. then”, but proceed informally here, and add an Appendix to summarize the results in formal fashion.

I will suppose, without argument, that Modus Ponens is valid for the conditional.  

First suggestion for an addition:  simple repetition is admissible here.  So if we suppose that A then we can infer A there.  Then we can discharge the supposition and conclude  A →A.

Second suggestion: any purely logical reasoning is allowed there.  So under supposition A, noting that X logically implies Y we can make that inference, and then decide to discharge either to A → X or to A → Y.  So if we are able to conclude that (A → X) we can also conclude that (A →Y).  In other words, reflecting on this, we see  that if X implies Y then (A → X) implies (A →Y). 

This also allows us to reason that if A implies B and A implies C then A implies (B & C), therefore A implies whatever (B & C) implies.  This yields the general principle

 if the argument from premises X to conclusion C is valid, and

if  (A → B) is the case for each premise B in set X, then (A → C) is the case. 

We can see this as showing that our discharging procedure is coherent.  That is, we can verify that the rule of Conditional Introduction holds for → (to which we have been appealing informally so far) is a way of stating what our supposition-discharging procedure does.  For now, when we are given that from a supposition A we can logically derive B, we can argue as follows:

A implies A, also A implies B; but A and B together imply B.  Therefore A →B.

Now we have to face the really serious question:  can any of the lines preceding line 17, let’s say line 13,  appear reiterated in that passage 18-29?  That means: can any of those lines be the same as line 13, with the justification precisely that the statement in question is there in the lines before the supposition (17) was entered?

A radical answer would just be: No!. And the justification for this No! would be that there is no general answer to what any given supposition requires us to bracket.

Third suggestion.  There is a less radical answer, offered as part of a larger theory, by Richmond Thomason (1970).  He proposes first of all the rule (which he calls “reit 1”) that after the supposition of A, in line 17 for example, we may insert any statement of form B if A → B has already appeared before that supposition.  His argument for it is this:

The rule of reit 1 is, I think, an uncontroversial feature of conditional reasoning; if a conditional statement has been posited, the supposition of its antecedent allows its consequent to be asserted. “If the bill were passed, it would be declared unconstitutional.” “Well, suppose it were passed. Then, according to what you say, it would be declared unconstitutional …”.  (Thomason 1970: 406)

That is not really enough for a perfect justification. The rule ‘reit 1’ ensures that previously all asserted conditionals must be part of what is kept constant, when new suppositions are entered.  That is a kind of ‘consistency over time’ requirement.

We have to take this rule very strictly.  For example the following argument is invalid:

1. A → C                                    (Given)
2. A & B                                     Supposition
3. A                                             2  conjunction elimination
4. A → C                                    1, reit
5. C                                             3, 4 modus ponens
6. (A & B) →C                           2-4, conditional introduction  

Here line 4. is illegitimate, for the reiterated conditional does not have the exact supposition (A & B) as its antecedent.  (A previous line would have had to be (A & B) → (A → B) to license line 4 in this derivation.) Rule reit 1 provides a limited way of importing information, when reasoning under a given supposition, from outside that supposition.

And it is good to see how this argument is classified as invalid.  For remember, the literature on counterfactual conditionals had from the beginning the seminal example that “This match would light, if struck” does not imply “This match would light, if wet and struck”.

Fourth suggestion.  Thomason has another proposal that is also of a truly general form.  With the conditional in hand we can introduce a defined notion of necessity:  □A means (~A → A).  

Thomason proposes the rule, which he calls “reit 2”, that if □A appears as a line before the supposition on line 17 then A can be in those lines 18-29.  

Here is Thomason’s argument for his reit 2:

To simultaneously assert something of the form (~ A → A) and deny something of the form (B → A) is to say that a situation in which A does not obtain cannot be conceived, and yet that A does not obtain in some situation in which B is posited. (Thomason 1970: 407, with my symbolism replacing his)

This yields the principle that if it is necessary that (either not-A or B) then A → B.  The effect of accepting this rule is that  □(A ⊃ B) logically implies (A → B).

For example if Jones must be either in Paris or in London then (Jones is not in London → Jones is in Paris).

We need not give the “cannot be conceived” as much weight as Thomason does here. On the contrary, it affords a way to signal, conversationally and contextually, some things that are to be kept constant, not bracketed.  The use of □ gives us a means of identifying statements made earlier that are meant to remain under supposition.  The necessity may just be contextual, signaling that something is to be a presupposition rather than a mere assertion in this argumentative monologue. 

Fifth suggestion.  This one I will not endorse.  Thomason’s rule “reit 4” engenders the principle is that if (A → B) and (B →A) are given, then (B → C) implies (A →C).  We may think of it this way:  the given establishes an equivalence relationship between A and B, and this equivalence is so strong that A and B are mutually replaceable as antecedents of conditionals.

Thomason’s “reit 4” fits with both Stalnaker’s and Lewis’ logics of conditionals, whose guiding idea is that conditional assertions are based on a ‘nearness’ relation among possible worlds.  It may be instructive to note how Thomason and others have tried to give it plausibility. After discussion of an example, Thomason writes:

Thus, reit 4 will be valid if we assume that there is economy in the choosing of situations, i.e. if we assume that in positing a condition we imagine a situation ß differing from a situation already imagined only if we are forced to do so in virtue of the fact that the condition is false in α. This rule therefore reflects a “law of least effort” in envisaging situations. Equally well, we can regard reit 4 as reflecting an orderliness in the choosing of situations; we can suppose that the choice of situations is dependent on a preferential arrangement of situations which can be described independently of the process of imagining.  (Thomason 1970: 410)

That is correct as an explanation of the rule. But it does not give a reason to think that the way we reason with conditionals is based on the relevant type of preferential arrangements.

Eva, Shear, and Fitelson are considerably less circumspect in their endorsement of the suggested principle, which they call Uniformity.  If we imagine the principle not to hold, they write that asentence of form (A → B) & (B →A) & (B → C) & ~(A →C) would be sometimes true:

Clearly, this would be a deeply strange and counterintuitive result. (Eva, Shear, and Fitelson 2020: 7)

As for me, I have always had my doubts about it. But that will be a discussion for another occasion.

SUMMARY.  It is clear that in actual reasoning under supposition we will rely on information not implied by the supposition, and much of that information will not be explicitly furnished.  The rules in the first four suggestions above, as well as Modus Ponens, may be taken as correct for this sort of reasoning, under all circumstances.  That is to say, challenges to someone’s actual reasoning under a supposition will be appropriate only for moves made not in accordance with those rules. The justification for such a challengeable move would presumably be for the agent to reveal information tacitly ‘held constant’.  In an ideal case, there would be no need for it, for if A is to be held constant in this way, that could be conveyed by entering (~A →A) as a premise at the beginning. 

3.   APPENDIX I.  The logical principles engendered by these rules

Here I will lay out more formally how these rules for reasoning under supposition engender certain logical implication relations among statements with conditionals.  In this form it will be easier to connect this topic with logics of conditionals.

From the outset classical logic is assumed for for &, ~,  and ⊃, as well as both Modus Ponens and Conditional Introduction for →.  But the latter is allowed only following the rules of permission introduced in the suggestions above.  

[I]  (A → B) logically implies (A ⊃ B)

The triple A, ~B, A → B is inconsistent, by Modus Ponens

[II]  A →A

Suppose A.  Repetition is allowed as a deduction, so infer A.  Discharge to form conclusion A → A.

[III]  If X ├  Y then (A → X) ├ (A →Y)

(Using  ├  for logical implication).   If (A → X) is given enter the supposition A.  By reit 1, insert X.  Since X ├ Y deduce Y.  Discharge to conclusion (A → Y).

[IV] A → B and A → C together imply A → (B & C)

Suppose that A → B and A → C are given; introduce supposition that A. By rule reit 1 we can write each of B and C below the supposition.   By logical reasoning we derive (B & C) and then discharge the supposition to conclude A → (B & C).

 [V]   A → (B & C) implies A → B and also A → C.

This follows at once frome [III].  

Given [IV] and [V] we can assert [III] in the more general form, with index set J finite:

[III*] If {X_i: i in J} together imply Y then {(A → X_i): i in J}  together imply (A → Y)

[VI]  □(A ⊃ B) logically implies (A → B)

This follows from the fourth of the suggested rules of permission. if □(A ⊃ B) is given, enter supposition A and insert  (A ⊃ B).  By classical reasoning deduce B; discharge to the conclusion (A → B).

APPENDIX II. The literature about supposition

The early paper by Rescher (1961) exemplifies what I called the first approach to supposition, namely to focus on its role in conversation and argumentation.  As the title indicates, Rescher’s concern is specifically with suppositions believed to be false, and he argues that there can be no rule for what previously given information can serve when reasoning from a supposition.  From this he concluded that the subject of counterfactual conditionals is not one for logic.  In his later (1964) Rescher extends his discussion to hypotheses, which he says are suppositions whose truth-value may be doubtful or undetermined.  In this way practical reasoning that is not belief-contravening is included in the scope of the book.   

Fisher (1989) is also within the first approach, though motivated by the question of how to classify suppositions among speech acts.  His initial example belongs to practical deliberation, and so could fit the notion of ‘provisional updating’.  That is not explored, and he quickly introduces examples of the other sorts. Fisher introduces doubts about any straightforward connection between suppositions and conditionals, though his reasons do not go beyond exhibiting awkwardness in examples of reading “Suppose A.  In that case B” as “If A then B”.  

Green (2000) points out that “supposition” may refer either to a statement offered for supposing, or the speech act of doing so, or the agent’s intentional or conative state or act in doing so, and argues that there are norms governing these. He relates these norms directly to the natural deduction Conditional Introduction rule, but does not go into how that needs to have restrictions on its application.  

Kearns (2006) presents a system of illocutionary logic, with the syntax of classical sentential logic augmented with illocutionary operators signifying assertion, denial, supposing true, supposing false.  The semantics has two tiers: in the first truth-values are assigned, and the second provides a commitment evaluation (partial; positive, negative).  When a supposition is introduced the agent/speaker remains committed to it until it is discharged (or otherwise removed). On this characterization of the agent’s activity it does seem apt to read a supposition as provisional updating.  When Kearns turns to conditionals, his theory is that there are no conditional statements, a conditional assertion is neither true nor false.  Rather, the act in question is the assertion of the consequent on the condition of the antecedent. A new illocutionary operator, signifying conditional assertion, is introduced.  Kearns does not take up embedded conditionals in such examples as “If this vase would break if dropped on the floor then it would break if thrown against the wall”, and there does not seem to be a place for it in the syntax he describes.  Even the simpler question of under what assumptions the antecedent “the vase would break if dropped on the floor” would entail the consequent “the vase would break if thrown against the wall” does not seem to be formulable here.  

The second approach, relating supposition to updating, guided Isaac Levi’s work on supposition (Levi 1996; Magnani 1998-2000).  The examples of practical deliberation lend themselves very well to the idea of suppositional reasoning as provisional belief updating. In my initial example the speaker introduces what would be, or could be, the occasion for a relevant updating of his beliefs:  finding out that it is going to rain during his vacation.  Spinning out the implications about his situation in that case and his preferences, he concludes that in that case  he will wish to spend his time reading.  These reflections are evoked precisely by his ignorance or uncertainty about what will happen, and the supposition does not contravene any of his beliefs.  So this is a special case.  It may be suggested that the difference is that in this case the conditional will be in the indicative rather than the subjunctive.  That may be so, though it won’t be very clear in actual usage, since the subjunctive is not much used in contemporary English. But if so, it would seem that supposition is, in most of the sorts of examples found, connected with the subjunctive, and hence less easy to see as provisional updating.

It is in Eva, Shear, and Fitelson (2020) that the second approach is most fully, and technically, explored. “Suppositions – or propositions provisionally accepted for the sake of argument – afford … tools for deliberation.  We use these tools to guide activities that are essential to intelligent behavior …”.  When a supposition introduced in the indicative, we are concerned with what we would expect if we learned that it was true.  When in the subjunctive, we align our evaluations with our judgments about how things would be if it were true.” They then explore four different theories that connect supposition to updating.  

While this work is too large for an exposition here, we may just note how it brings to light an interesting difference between qualitative accounts of straight updating and of provisional updating.  The former has the principle of Preservation: if your beliefs so far are consistent with new input S then add S to your beliefs.  The latter has the principle Compositionality:  if the set of worlds compatible with beliefs B is the union of the sets of worlds respectively compatible with B’ and B”,  then that relation (of union) also holds after updating on supposition S.  An example would be: I believe that Peter is either in France or in Italy, and I am asked “What if Peter were in a capital city?”  It seems I would naturally answer “Then he would be either in Paris or in Rome”.  Eva et al. construct a counterexample to this, for when the question is placed in indicative mood, with respect to a a notion of belief that is not full belief, i.e. not equated with (or implying) subjective probability 1.

REFERENCES

Eva, Benjamin; Ted Shear; and Branden Fitelson (2020) “Four Approaches to Supposition”. phisci-archive.pitt.edu/18412/7/fats.pdf

Fisher, Alec (1989) “Suppositions in Argumentation”.  Argumentation 3: 401-413.

Kearns, John T. (2006) “Conditional Assertion, denial, and supposition as illocutionary acts”. Linguistics and Philosophy 29: 455-485.

Levi, Isaac (1996) For the Sake of Argument: Ramsey Test Conditionals, Inductive Inference, and Nonmonotonic Reasoning.  Cambridge: Cambridge University Press.

Magnani, Lorenzo (1998-2000) “Suppositional reasoning, induction, and non-monotonic inferences” (Review of Isaac Levi).  Modern Logic 8: 159-165.

Rescher, Nicholas (1961) “Belief-Contravening Suppositions”.  The Philosophical Review 70: 176-196.

Thomason, Richmond H. (1970)  “A Fitch-style formulation of conditional logic”.  Logique et Analyse N. S. 13: 397-412.