conditionals

Logicians have always known that there were “funny ifs” out there, but it is only in the last couple of decades that any serious progress has been made in figuring out the logic (mainly the semantics) of these connectives (if that is what they are). While they have not reached a consensus (or even officially decided that there are several related notions here which each needs an elaboration – the usual situation with modal notions), the list of types they have isolated may be useful to have before us. To be sure, most of this does not affect anything in the language per se and very little of it affects our use of it (some points about negations might be significant).

Logicians have – with a few notable exception – preferred the material conditional for their everyday “if… then - - -.” This is equivalent to the inclusive disjunction of the consequent (“then” part) and the denial of the antecedent (“if” part). It is fully truth functional, and of the four situations this allows, is false only when both the antecedent is true and the consequent is false. Otherwise it is true. In addition to this specific false case, the material condition supports the inferences modus ponendo ponens (from “if p then q” and “p” to infer “q”) and modus tollendo tollens (from “if p then q” and “not q” to infer “not p”) and contraposition (from “if p then q” to “if not q then not p”). But it is true whenever its antecedent is false and also whenever its consequent if true. Obviously this will not do for the usual subjunctive conditionals, where the antecedent is assumed false to begin with but the conditional may be either true or false (independently even of the truth or falsity of the consequent).

So the subjunctive conditional is not truth-functional in the usual way (nor is the indicative conditional: “if it is raining I will take an umbrella” which at least does not assume the antecedent is false). Nor do free logics and supervaluations (truth functional but allowing sentences without values in a given world but perhaps getting values after all if they have the same value in every world in which they have a value) help much. Typically, a subjunctive conditional would be valueless in a case where the antecedent is false, but that is not very useful, since we are interested in true ones and false ones. And, at least if we are just running through logical possibilities, as we typically are in working with truth values directly, clearly the conditional will not get the same value in every case where it has a value (where the antecedent is true). So, the connective involved is not truth-functional at all and we have to consider something more that merely logical possibilities to find out its nature.

Well, that last bit, while hinted at in the free logic case, is really established by the suggestion that what is involved is strict implication: not just that if p then it just happens that q, but that “if p then q” is necessary. The trouble with this – in even the weakest systems that have a chance of working for what we want – is that a strict implication is true just if the antecedent is impossible (necessarily false) or the consequent is necessarily true, modal version of the problem with material implication, of which strict implication is just the modal version.

The moral is that subjunctive conditionals must involve some facts, not just logic. Given the role that such conditionals purportedly play in science, the obvious suggestion is that such a conditional holds just in case it is we can infer the consequent from the antecedent together with all the laws of nature. But this is immediately inadequate, since many of the most interesting cases begin “if the laws of nature were different is this way, then…” Adding this to the laws of nature would immediately give a contradiction, from which anything at all can be inferred (or nothing can be, if we keep some free logic running here). But even beyond this formal problem there is the fact that this approach will not work for fairly simple cases: “If my daughter ate pecans, her throat would swell shut.” The consequent here does not follow from the antecedent plus the laws of nature. To get to the consequent we need also some facts about my daughter’s body: that it has a histaminic response to juglanin (or all of the details that account for that).

So, we must include some facts about the world in addition to the (probably false) p to establish “if p were then q would be.” But what facts? As few as possible, definitely excluding ~q, is the usual answer. In possible worlds terms (as the discussion now is almost completely) this means an accessibility relation (what tells what worlds are possible relative to some starting world) that is ordered by similarity to the real world. So the conditional is true if q is true in the closest world in which p is true. Ah, but suppose there is no the closest world; what if there are a number of worlds that are equally close (ordering does not have to be linear, after all). One world changes one little thing beside p, another changes another, but (on some componential analysis of similarity) they are equally similar (each just one thing off).

Well, then, we take the class of all the equally similar p-worlds that are closest to the real world: the conditional is true if q is true in all of these. But it may be that the ordering is not only not linear but is also not discrete. So between any world and the real world there is always another world and it might be that even the p-worlds are dense. Then there is not such thing as a set of closest p-worlds even. OK; so now we take all the p-worlds more similar than some bound range. This can be done in a number of ways, setting different conditions on the bound. Indeed, the conditions – or at least the bound – may vary with the circumstances under which the conditional is asserted. All of these come down to, one way or another, that the conditional depends upon the truth of q in selected p-worlds, with the selection intended to give some sense to the notion of a sufficiently similar world to this one.

But this notion of sufficiently similar is seen by some as too coarse; a finer one looks not at similarity to the whole world but rather similarity to what is relevant in the immediate situation of the utterance (with two wings, depending upon whether “the utterance” is taken to mean the conditional sentence itself or the event of its being uttered). Thus, the worlds may be wildly different from the real world so long as they are the same in all the relevant aspects. This is again a kind of selection process for worlds, but is meant to model different conditions. In particular, factors relevant to q as well as to p may be involved. What these conditions are – in any such theory – will depend on the actual conditional, of course, but – in the utterance dependent case – perhaps also on the conversation in which the conditional is asserted. In most cases, what the relevant conditions are will appear in the (real or imagined) challenges to the claim: if one response is “Oh, but what if …” the … has to have been accommodated in the conditions.

A particular way of establishing similarity for a large number of cases rests on the fact (in I-E languages) that subjunctive mode draws its markers at least from the tense system. If we imagine that time (and its history) are ever branching into the future, that there are as many immediate successors to a given instant as you might want (or the similar notion for dense and continuous orderings), then to worlds (instants) are somewhat similar if they are on branches coming from the same instant and are more similar the more recent the separation was. The conditional, “if p were then q would be” when p is true on a branch from the last time p could have become true which is as far from that juncture as the present (presumably non-p) is and q is true at that same time on that same branch. This corresponds to many common subjunctive conditionals: “If I had known then what I know now, I would not have opened the door” “If Hinckley had been a better shot, Reagan would have died in 1982.” And so on. As the tenses-like forms suggest, the pattern is usually to pick a past event (or rather missed event) as p and an event future to that, though still in a parallel past (no so far along the in the future in its path as now is on this one), both also (one past and one future) to some critical past point (my coming to the door, Hinckley shooting Reagan). Perhaps most conditionals cannot be forced into that mold. Even some that are tensed do not, going instead to a past event and an even further past one: “I Reagan had died in 1982, Hinckley would have been a better shot.” The consensus is that most such conditionals are, like this one, clearly false (though attempts to explain this have often led to validating really awful conditionals of this sort -- the simple answer seems usually to be that generally, a given event might come about from a variety of causes while the effects of an event are narrower: Hinckley’s shot basically either kills Reagan or does not, but Reagan might have died – even at about the same time – from any number of causes, not just the shot).

But even aside from these, many interesting conditionals don’t fit this pattern or do so only in rather unhelpful ways. “If the gravitational constant were 2% greater, then 8 million more people would have broken bones this year” requires going back at least to the formation of the planet, if not the Big Bang and so involves a whole different evolutionary history which we can only guess at (although it does seem that if people evolved, they would have evolved with correspondingly stronger bones). Similarly, conditionals that start off like “If Socrates were a 17th century Irish washerwoman” are not helped by considering alterations in either the 17th century or the -5th as a basis for an alternate history up to “now.” (I’ve never been able to stick with these long enough to discover whether they are about reincarnation or whether the involve a haeceity for Socrates at least that is completely independent of his actual biography yet matches enough of it – but which parts? – to make the consequent interesting: “then there would be at least one bald snub-nosed washerwoman” “then s/he would have been killed early on for asking damned-fool questions” “then s/he would have accepted flight rather than remaining to be executed” “then s/he would have had none of the properties usually associated with Socrates” Go figure!)

All these varied background stories lead to a surprisingly small number of possible logics for subjunctive conditionals. All of them have modus ponens, all agree that p & ~q anywhere in the area of interest spikes the whole conditional. All use some selection of the following rules for (subjunctive) conditionals (RC): (> is the subjunctive conditional, -> the material one, => the strict one)

RCEC: From p ó q to infer (r>p ó r>q) (Equivalent Consequents)

RCK: From (p1 & …& pn) > q to infer [[r >p1) & … & (r>pn]] => (r > q)

(Konjunction)

RCEA: From p ó q to infer (p>r) ó (q>r) (Equivalent Antecedents)

RCE: From p => q to infer p>q

They then aim at validating all tautologies and some selection of the following theses:

Transitivity: [[p>q) & (q>r]] => (p>r)

Contraposition: (p>~q) => (q>~p)

Strengthening Antecedent: (p>q) => [p & r)>q)

ID: p>p

MOD (p>q) => (p => q)

CSO [[p>q) & (q>p]] => [[p>r) ó (q>r]]

CV [[p>q) & ~(p>~r]] => [p & r)>q)

CEM (p>q) v (p>~q)

CS (p & q) => (p>q)

CC [[p>q) & (p>r]] => (p>(q & r]

CM (p>(q & r] => [[p>q) & (p>r]]

CA [[p>q) & (r>q]] => [p v r) => q)

SDA [[p v q)>r) => [p>r) & (q>r]]

The well-developed systems take the following sets:

VW: RCEC, RCK, ID, MOD, CSO, MP, CV

SS: RCEC, RCK, ID, MOD, CSO, MP, CA, CS

VC: RCED, RCK, ID, MOD, CSO, MP, CV, CS

C2: RCEC, RCK, ID, MOD, CSO, MP, CV, CEM

In addition to subjunctive conditionals there are several other conditionals that require some attention. The “might if” conditional, “if p were, then q might be” is generally conceded to be just `~(p>~q)’ (however `>’ is defined). On the “even if” conditional, there are three possibilities discussed. Either “p would be even if q were” is `p & (q>p)’ (which would combine with CS to mean that any time ‘p & q’ is true, so is “p even if q” would be) or it is just `q>p’ uttered when p is true or it is `(q>p) & (~q>p).’

For indicative conditionals like “if it rains I will take my umbrella” the easiest solution is that this is what the material conditional is for. Of course, taking that line – which works out right in the easy cases – makes the first three conditionals on the list of theses true and we may not want some of them. Contraposition in particular is a problem, since, “if it is after 3 a.m., it is not much after 3 a.m.” is sometimes true, whereas “if it is much after 3 a.m., then it is not after 3 a.m.” is pretty clearly false. Further, while I might deny, “If the butler didn’t do it than I did it” since I are sure that I didn’t do it, we might accept the equivalent “Either the butler did it or I did it” if we were sure the butler did it. If we want to save this handy theory, we have to distinguish between the truth conditions of the sentence and the conditions for its assertion. Thus, although the troublesome conditionals here may be strictly true, they may also be inappropriate to utter in the indicted situation: in the butler case that it is inappropriate to assert a weaker sentence

(~p v q, p -> q) when we can assert a stronger one that implies it (~p, q). That is the examples seem wrong only because they are inappropriate, not because they are false. If we allow truth-value gaps, the situation becomes somewhat tidier still, since a material conditional in that case is simply valueless when its antecedent is false. Or we can say that we just don’t care what the value is then and ignore embarrassing cases where the antecedent is false.

The second possibility is to take indicative conditionals as what some theory of subjunctive conditionals that you think is wrong thinks are subjunctive conditionals. The favorite for this is C2 with the near world selection involving the shared presuppositions that have been built up in the conversation up to the point where the conditional is uttered, which are not involved in establishing the n3ear worlds for the subjunctive case. The other possibility is CV, which is arrived at first by dealing only with probabilities: pr(p>q) = pr(p & q)/pr(p), the conditional probability of q on p. `p>q’ is true just in case pr(q/p) than some set value (the acceptability level again – something like this can be worked out for fuzzy logics as well, of course – which value shifts with the situation). When this is transferred to validity (true in all models/worlds), the system turns out to be CV, which has nothing overtly to say about probabilities.

None of these systems works for all cases and none of them really says much about deciding in a given case whether the conditional is true. Happily, we are doing language building not logic here and so all we need come up with are ways of expressing the various conditionals in Lojban. I would like to suggest the following.

For indicative conditionals, we should go with the otherwise fairly useless material conditional so built into the language. We obviously take it in the “don’t really give a damn when the antecedent is false” sense and also accept the pragmatic restrictions on what can be uttered (appropriately).

For the “might if” conditional we should accept the identification that is general in logic.

For the “even if” conditional we should take the basic “p even if q” to be `(q>p)’ when p is thought to be true and (assuming ~q is thought more likely that q) `(~q>p) & (q>p)’ otherwise.

Which all assumes we something for subjunctive conditionals. For that I suggest that usage has already suggested {da’i} for the antecedent. As a UI, {da’i} can go anywhere, but usage suggests either sentence initial (or just before a guhek for compound sentences) or in the tense location, after the first argument and before the selbri. This is actually a stronger move than merely subjunctive conditional, for {da’i} can be taken as setting up a hypothetical world which them persists until explicitly dropped with {da’inai}. The conditional is a special case with only one sentence in the scope of the {da’i} (or longer discussions are elaborated conditionals with long conjunctions of claims as consequents). This may make it difficult to express some of the theses of conditional logic, but these are mainly to be used, not stated. We will find ways to say them when we need them. We do need to work out how to interpret new {da’i}s within the scope of existing ones, although this seems pretty obvious – they become in effect either a conjunction under a single {da’i} or more literally go off to a world alternate to the alternate world already reached. What the consequences of either way of doing this are is not immediately clear (and probably depends upon which conditional logic we use – they were bound to come in somewhere).

Advantages: have a uniform way of saying things we have been having trouble saying for 50 years. Disadvantage: we will have to clean up all those earlier attempts (but people did more cursing than creating in this area, so there may not be much).

(All of this material is shamelessly cribbed from Donald Nute, “Conditional Logic” pp 387 - 439 of Volume II of The Handbook of Philosophical Logic (first edition, as it now is), Kluwer, Dordrecht, 1984. This paper has a good beginning bibliography if you want to chase this topic down.)