Thompson and Mann: "Perceived Necessity Explains the Dissociation Between Logic and Meaning" (1995)

Under which conditions do people think that If A, then B can be paraphrased as A only if B? This paper by Valerie Thompson and Jacqueline Mann is an empirical investigation of the question, checking a couple of relevant parameters.

As it turns out, two factors play a major role: The temporal order between A and B, and whether we perceive A and B to be equivalent in the concrete case at hand.

By contrast, the type of discourse relationship between A and B plays no role. It thus doesn't matter whether the relationship between them is causation, permission, co-occurrence, definition, etc.

Independent Variables

Let me just fix some terminology. What I here call the discourse relationship is what Thompson and Mann call "pragmatic relations." I just dislike this term because it's not quite consistent with the jorgon of linguistics.

The two most important discourse relations that they are dealing with are causation and permission:

• Butter melts if it's heated. (causation)
• You may enter if you're over 18. (permission)

They introduce a couple more (p. 1557), but since disourse relationship turns out to have no effect, this is of a minor importance.

Second, when Thompson and Mann talk about "necessity" relationships, they are really talking about condtional perfection. This is the backwards conditional If B, then A that we sometimes infer when we hear the forward one:

• If water is heated to 100°C, it boils (… and vice versa).
• If it rains, the pavement will be wet (… but not necessarily vice versa).

The effect is a conflation of implication and bi-implication. This "logical" difference does, unsurprisingly, turn out to have an effect on the acceptability of paraphrases.

Lastly, the notion of temporal succession is the most interesting one, and it interacts in some non-trivial ways with the psychology undergraduates' intuitions about synonymy:

• If a plant has received enough care, it grows. (A before B)
• If a plant grows, it has received enough care. (A after B)

In terms of the relationships visible to classical logic, these sentence mean very different things: The first one rules out rules out externalities that could hinder growth even in the event of care; the second one rules out other sufficient causes of growth. However, from an intuitive perspective, the sentences seem to point towards the same underlying causal relationship.

Results

Thompson and Mann's main concern is whether their subjects think that a sentence of the form If A, then B is synonymous with A only if B, and whether it is synonymous with B only if A. As I mentioned above, it turns out that this depends strongly on whether the (inferred, perceived) temporal order of A and B, and the (inferred, perceived) equivalence of A and B.

Thompson and Mann used a super-weird scoring scheme in which their subjects had to assign a 1 to a perfect match and a 7 to a complete mismatch. "For ease of comprehension," they report the transformed score 8 – x instead of x (p. 1557; why didn't they just use the easy one in the first place?).

This gives means between 1 and 7. I've transformed these means into percentages to make it easier to see how far the various means are from the maximal and minimal scores. I did this by computing 100/7 * (y – 1) from the reported y = (8 – x). So let's look at a couple of snapshots from the results of Thompson and Mann's experiment 2b.

First, causal relationships with forward-moving time and no conditional perfection. An example of this is the following:

• If the car runs out of gas, then it will stall.
1. The car only runs out of gas if it stalls (13% — equivalent)
2. The car only stalls if it runs out of gas (50% — not equivalent)

In this case, subjects do not like the actually equivalent form (which suggests a modus tollens inference schema). Note that the percentages are the average scores for this class of sentences, not the specific example.

Now a causal relationship with backward-moving time, but still no conditional perfection:

• If the car drives, then there is gas in the tank.
1. The car only drives if there is gas in the tank (79% — equivalent)
2. There is only gas in the tank if the car drives (21% — not equivalent)

So this reversal of time completely turns the intuitions upside-down: Now, the equivalent paraphrase seems more consistent with the order of terms (STATE only if PRECONDITION), and the non-equivalent seems less natural.

If we put these two sets of statistics together, we get the following chart of acceptabilities:

Lastly, a forward-moving example with conditional perfection:

• If water is heated to 100°C, it boils.
1. Water is heated to 100°C only if it boils (26% — equivalent)
2. Water only boils if it is heated to 100°C (75% — not equivalent)

On a very coarse level, this is the pattern of forward-moving time without conditional perfection; there is an intensity effect, but no reversal of judgments.

The Role of Time

So it seems that the single most predictive factor about intuitions of synonymy and inference is the distinction between forward-moving and backward-moving time. Certain ways of construing a causal situation highlight the potential for following the actual causal direction in your thoughts, and other ways highlight the possibility of following the order of inference rather than the order of events.

If this is true, then it would have some consequences for how difficulty various inference types are, as well as how they errors will occur through "normalization." For instance, denial of the antecedent can be seen as a natural thought to have if we follow the order of events in a case where the literal meaning of the premises requires us to follow the order of inference.

Literature on the meaning of "only if"

I've been looking for some empirical studies of A only if B constructions. In the theoretical literature on natural language semantics, there is a number of models, but I want to know more about how they are actually understood. Fortunately, there seem to be some facts about that out there, too.

What Does It Mean, Allegedly?

The problematic issue with the only if construction is that it is supposed to be logically equivalent to a number of related constructions, even though non-logicians sometimes disagree with this. According to the classical convention, the following sentences thus all mean the same:

• It only thunders if it rains.
• If it thunders, it rains.
• If it doesn't rain, it doesn't thunder.

On the other hand, if we reverse the implication, we change the truth conditions:

• It only rains if it thunders.
• If it rains, it thunders.
• It it doesn't thunder, it doesn't rain.

If this was just a mere convention about logical language, all would be fine. The problem is, however, that these sentence forms are not used in the same situations, and they do not integrate equally well into all reasoning patterns in spite of their (alleged) equivalence.

The Performance Problem

One difference between the If A, then B and A only if B forms is that if form is generally more difficult to use in a modus tollens inference than the only if. At least, this is what Carlos Santamaría and Orlando Espino say (Santamaría and Espino 2002, p. 42). They're referring to three studies, including one by Jonathan Evans and M. A. Beck (Evans and Beck 1981).

The problematic case is thus the following inference:

If it thunders, it rains.
It doesn't rain.
–––––––––––––––––
It doesn't thunder.

This (clasically valid) inference should be performed more readily when served in this alternative, and supposedly equivalent formulation:

It only thunders if it rains.

I doesn't rain.

–––––––––––––––––––––

It doesn't thunder.

Cognitively, or perhaps in terms of actual natrual language semantics, this seems to indicate that A only if B works more like the contrapositive If not B, then not A than like its positive translation, If A, then B. Or at least, it seems to issue a conversational warrant closer to it.

It would be interesting to know if this alternative formulation comes with a corresponding decrease—are we trading of willingness to perform the straightforward modus ponens inference for higher rates of modus tollens? This would imply that the following inference generally is less accepted:

It only thunders if it rains.
It thunders.
–––––––––––––––––––––
It rains.

If the only if formulation really does works like a contrapositive, then this inference should appear to us like a modus tollens inference in terms of plausibility and difficulty. I do not know right now whether such an effect can actually be measured or not.

The Issue of Time

Another interesting proposal that Santamaría and Espino cite, also coming from Evans and Beck, is that there is a systematic interaction between our conception of temporal order and the choice of form.

Thus, even though If A, then B, and A only if B are supposedly logically equivalent, we get different patterns of acceptability or naturalness depending on whether A or B happened first. For A preceding B, we then (perhaps) have:

• If you bought on Tuesday, you're paying on Wednesday.
• (?) You bought on Tuesday only if you paying on Wednesday.

And for B preceding A:

• (?) If you're paying on Wednesday, you bought on Tuesday.
• You're only paying on Wednesday if you bought on Tuesday.

Of course, much clearer intuitions can be produced if we ruffle up the tenses a bit. But this, I think, relatively fair example to start the discussion from.

So, Causality?

Note that the issue of before/after interfaces with the concept of causality, which is notoriously bound up with implication, even if logicians and statisticians hate to admit this fact.

Possibly, the the only way we can really justify an inference from a later effect to a prior cause in the form of If EFFECT, then CAUSE is to objectify the cause and the effect by thinking about the observation of the effect and the deduction of the cause. In this way, we would straighten out the temporal sequence so that EFFECT could in fact precede CAUSE.

If this is true, it has a quite important consequence for the psychology of reasoning: We would then only be able to understand abductive inference by effectivly embedding a cause/effect relationship in a different and larger cause/effect relationship—namely the only in which a real or imagined person reasons from fire (the logical "cause") to smoke (the logical "effect").