Wolfgang Schwarz

Pragmatic non-equivalence despite semantic equivalence

Bergen, Levy, and Goodman (2016) assert that "the rational speech acts model, and neo-Gricean models more generally, cannot derive distinct pragmatic interpretations for semantically equivalent expressions".

In the previous post, I gave a counterexample. I presented an RSA model that explains why 'pockets' is interpreted as plural and 'a pocket' as singular, even though the two expressions are semantically equivalent.

Now Bergen et al don't just claim that RSA models can't do this. They prove it, through "Lemma 1" and "Lemma 2" in section 4.1.1.

What's going on here?

Bergen et al rely on certain assumptions about what an RSA model (and a neo-Gricean model more generally) must look like. In particular, they assume that hearers treat every utterance as an alternative to every other utterance. My counterexample assumes that 'a pocket' and 'pockets' have different alternatives.

It is often thought that the space of alternatives must be restricted in order to avoid the "symmetry problem": if 'all' and 'some but not all' were both alternatives to 'some', the Gricean recipe for computing implicatures would predict that 'some' implicates 'not all' as well as 'all'. Since the latter implicature does not arise, 'some but not all' must be excluded from the alternatives of 'some'.

Bergen et all note (in section 3.2) that an RSA model can predict the attested implicature of 'some' without ruling out 'some but not all' as an alternative, by assuming that speakers prefer simpler utterances. From this example, they hastily generalize to the claim that the RSA approach provides a general solution to the symmetry problem. This, in turn, makes them assume that proper RSA models do not involve any constraints on the alternatives.

I agree that it would be nice if RSA models didn't need any such constraints. This would be an argument in favour of the RSA approach. But I'm not sure it can be done. Even if it can, I don't think this should be a defining feature of RSA models. As far as I know, there are independent reasons to believe that language interpretation involves a restricted set of alternatives. (See e.g. Fox and Katzir (2011).) If our language module makes certain alternatives salient, why couldn't these play a role in pragmatic reasoning?

In my model of the plurality implicature, I assumed that a level-2 hearer who hears an utterance U only compares U with a restricted set of alternatives, inferring that the level-1 speaker chose U because it is among the best of these alternatives.

But this assumption isn't essential to the derivation. We can tweak the model so that it doesn't use any external constraints on the alternatives.

Recall that we want to explain why 'pockets' is interpreted as plural and 'a pocket' is not, even though both utterances are true iff there is at least one pocket. The key step in the derivation is that a level-1 hearer might reason as follows, upon hearing 'a pocket':

(i) The speaker chose 'a pocket'. They could have chosen 'several pockets'. This wouldn't have been a lot more complex. It's a plausible alternative. They would probably have chosen it if it had led to significantly greater hearer accuracy, as it would have done if there were several pockets. So there probably aren't several pockets.

By contrast, they might reason as follows when they hear 'pockets':

(ii) The speaker chose 'pockets'. They could have chosen 'several pockets'. But that's quite a bit more complex. It's not an obvious alternative. So it's not obvious they would have chosen it even if there were several pockets. So maybe there are several pockets.

In my model, I simply stipulated that 'several pockets' is an alternative to 'a pocket', but not to 'pockets'. But as the informal reasoning above shows, we could try to derive these facts from considerations of complexity.

Let's make the inference a little more concrete. Suppose the level-2 hearer is unsure about how much the level-1 speaker cares about complexity. Upon hearing 'a pocket', they might reason as follows.

(i) The speaker said 'a pocket'. So they don't have a strong preference for simplicity; else they would have said 'pockets'. This means that they would have said 'several pockets' if that had led to significantly greater hearer accuracy. It would have done so if there were several pockets. So there probably aren't several pockets.

Upon hearing 'pockets', they might reason differently:

(ii) The speaker said 'pockets'. They may well have a strong preference for simplicity. In that case it would be unlikely that they would have said 'several pockets' even if there were several pockets. So maybe there are several pockets.

What drives the implicature, on this account, is uncertainty about the speaker's utility function. The speaker's choice of an utterance tells us something about their utility function. That the speaker uttered U shows that they were willing to pay the cost of U. If a slightly more costly alternative would have been significantly better in other respects, they should have chosen that instead.

Here's a simple implementation of this idea.

var states = ['0 pockets', '1 pocket', '2+ pockets'];
var meanings = {
    'no pockets': function(state) { return state == '0 pockets' },
    'pockets': function(state) { return state != '0 pockets' },
    'a pocket': function(state) { return state != '0 pockets' },
    'several pockets': function(state) { return state == '2+ pockets' }
}
var complexity = {
    'pockets': 2,
    'a pocket': 3,
    'several pockets': 4,
    'no pockets': 3
}
var hearer0 = Agent({
    credence: Indifferent(states),
    kinematics: function(utterance) {
        return function(state) {
            return evaluate(meanings[utterance], state);
        }
    }
});
var speaker1 = function(observation, is_chatty) {
    return Agent({
        options: keys(meanings),
        credence: Indifferent([observation]),
        utility: function(u,s){
            var q = learn(hearer0, u).score(s);
            var c = is_chatty ? 0 : complexity[u]/2;
            return q - c;
        }
    });
};
var hearer2 = Agent({
    credence: Indifferent(Cross({'state': states, 'is_chatty': [true, false]})),
    kinematics: function(utterance) {
        return function(s) {
            var speaker = speaker1(s.state, s.is_chatty);
            return sample(choice(speaker)) == utterance;
        }
    }
});
//showKinematics(hearer2, keys(meanings))
var speaker3 = function(observation, is_chatty) {
    return Agent({
        options: keys(meanings),
        credence: Indifferent([observation]),
        utility: function(u,s){
            var q = marginalize(learn(hearer2, u), 'state').score(s);
            var c = is_chatty ? 0 : complexity[u]/2;
            return q - c;
        }
    });
};
//showChoices(speaker3, states, true)
var hearer4 = Agent({
    credence: Indifferent(Cross({'state': states, 'is_chatty': [true, false]})),
    kinematics: function(utterance) {
        return function(s) {
            var speaker = speaker3(s.state, s.is_chatty);
            return sample(choice(speaker)) == utterance;
        }
    }
});
showKinematics(hearer4, keys(meanings))

speaker1 has two parameters: observation represents the speaker's knowledge of the state; is_chatty represents their tolerance of complexity: if its value is true, the speaker's utility is just a function of the hearer's accuracy; if its value is false, the speaker also prefers simpler utterances. The two versions of speaker1 differ over what they do in state '1 pocket'. The chatty version is indifferent between 'pockets' and 'a pocket', the non-chatty version prefers 'pockets' . Both choose 'several pockets' whenever the state is '2+ pockets'.

hearer2 performs a joint inference over the state and the speaker's chattiness. If they hear 'a pocket', they can infer that the speaker is chatty and that the state is '1 pocket'. If they hear 'pockets', they can only infer that there's at least one pocket.

Since uttering 'a pocket' is a sure way of conveying '1 pocket', both the chatty and the non-chatty version of speaker3 use it if the state is '1 pocket'. The chatty version uses 'several pockets' if the state is '2+ pockets', the non-chatty version slightly prefers 'pockets'.

As a result, hearer4 infers that the state is '1 pocket' if they hear 'a pocket', without learning anything about the speaker's chattiness. If they hear 'pockets', they infer that the state is '2+ pockets' and that the speaker is non-chatty. At level 6 (not shown in the code), the inference about the speaker's chattiness would disappear and we only have the interpretation of 'a pocket' as '1 pocket' and of 'pockets' as '2+ pockets'.

I'm not sure whether this is a useful model. But it's at least a proof of concept. It shows that Bergen et al's assertion only holds under highly restrictive conditions. The RSA models that figure in Bergen et al's Lemma 1 and Lemma 2 don't just rule out external constraints on alternatives. They also assume that the exact shape of the speaker's preference for shorter utterances is common knowledge. This is surely not a realistic assumption! Relaxing it is enough to break the prediction that semantically equivalent expressions can't get distinct pragmatic interpretations.