Tag: epistemic logic

Deontic logic: Horty’s gambles (2)

Bas van FraassenLeave a comment

In the second part of his 2019 paper Horty argues that there is a need to integrate epistemic logic with deontic logic, for “ought” statements often have a sense in which their truth-value depends in part on the agent’s state of knowledge.

I agree entirely with his conclusion. But is the focus on knowledge not too strict? Subjectively it is hard to distinguish knowledge from certainty — and apart from that, when we don’t have certainty, we are still subject to the same norms. So I would like to suggest that rational opinion, in the form of the agent’s actual subjective probability, is what matters.

Here I will examine Horty’s additional examples of gambling situations with that in mind. I realize that this is not sufficient to demonstrate my contention, but it will show clearly how the intuitive examples look different through the eyes of this less traditional epistemology.

Horty’s figure 4 depicts the following situation: I pay 5 units to be offered one of two gambles X1, X2 on a coin toss. My options will be to bett Heads, to bet Tails, or Not To Gamble. But I will not know which gamble it is! You, the bookmaker will independently flip a coin to determine that, and not tell me the outcome. In the diagram shown here, X1 is the gamble on the left and X2 the gamble on the right.

On Horty’s initial analysis, if in actual fact I am offered X1 then I should bet Heads, since that has the best outcome. But as he says, rightly, I could not be faulted for not doing that, since I did not know whether I was being offered X1 or X2.

Even if the conclusion is the same, the situation looks different if the agent acts on the basis of the expectation values of the options available to him. The alternatives depicted in the diagram are equi-probable (we assume the coins are fair). So for the agent, who has paid 5 units, his net expectation value for betting Heads (in this situation where it is equally probable that he is betting in X1 or in X2) is the average of gaining 5 and losing 5. The expectation value is 0. Similarly for the option of betting Tails, and similarly for the option of Not Gambling: each has net expectation value 0. So in this situation it just is not true that the agent ought to take up any of these options — it is indifferent what he does.

Horty offers a second example, where the correct judgment is that I ought not to gamble, to show that his initial analysis failed to entail that. Here is the diagram, to be interpreted in the same way as above — the difference is in the value of the separate possible outcomes.

Reasoning by expectation value, the agent concludes that indeed she ought not to gamble. For by not gambling the payoff is 5 with certainty, while the expectation value of Betting Heads, or of Betting Tails, is 2.5.

So on this analysis as well we reach the right conclusion: the agent ought not to gamble.

Entirely in agreement with Horty is the conclusion that these situations are adequately represented only if we bring epistemology into play. What the agent ought to do is not to be equated with what it would objectively, in a God’s eye, be best for her to do. It is rather what she ought to do, given her cognitive/epistemic/doxastic situation in the world. But she cannot make rational gambling decisions in general if her knowledge (or certainty) is all she is allowed to take into account.

It would be instructive to think also about the case in which it is known that the coin has a bias, say that on each toss (inlcuding the hidden first toss) it will be three times as likely as not to land heads up. Knowledge will not be different, but betting behavior should.

Deontic logic: Horty’s gambles (1)

Bas van FraassenLeave a comment

In “Epistemic oughts in Stit Semantics” Horty’s main argument is that an epistemic logic must be integrated in a satisfactory deontic logic. This is needed in order to account for a sense of what an agent ought to do hinges on a role for knowledge (“epistemic oughts”).

That argument occupies the second part of his paper, and I hope to explore it in a later post. But the first part of the paper, which focuses on a general concept of what an agent ought to do (ought to see to) is interesting in itself, and crucial for what follows. I will limit myself here to that part.

I agree with a main conclusion reached there, which is that the required value ordering is not of the possible outcomes of action but of the choices open to the agent.

However, I have a problem with the specific ordering of choices that Horty defines, which it seems to me faces intuitive counterexamples. I will propose an alternative ordering principle.

At a given instant t an agent has a variety V(h, t) of possible futures in history h. I call V(h, t) the future cone of h at t. But certain choices {K, K’, …} are open to the agent there, and by means of a given choice K the agent may see to it that the possible futures will be constrained to be in a certain subset V(K, h, t) of V(h, t).

The different choices are represented by these subsets of V(h, t), which form a partition. Hence the following is well defined for histories in V(m): the choice made in history h at t is the set V(K, h, t) to which h belongs; call it CA(h, t), thinking of “CA” as standing for “actual choice”.

In the diagram K1 is the set of possible histories h1 and h2, and so CA(h1,t) = K1 = CA(h2, t). (Note well: I speak in terms of instants t of time, rather than Horty’s moments.

And the statement that the agent sees to it that A is true in in h at t exactly if A is true in all the possible futures of h at t that belong to the choice made in history h at t. Briefly put: CA(h, t) ⊆ A.

The Chisholm/Meinong analysis of what ought to be is precisely what it is maximally good to be the case. Thus, at a given time, it ought to be that A if A is the case in all the possible future whose value is maximal among them. So applied to a statement about action, that means: It ought to be that the agent sees to it that A is true in h at t exactly if all the histories in the choice made in history h at t are of maximal value. That is, if h is in CA(h, t) and h’ is in V(h, t) but outside CA(h, t) then h’ is no more valuable than h.

But this analysis is not correct, as Horty shows with two examples of gambles. In each case the target proposition is G: the agent gambles, identified with the set of possible histories in which the agent takes the offered gamble. This is identified with: the agent sees to it that G. Hence the two choices, K1 and K2, open to the agent in h at t are represented by the intersection of V(h, t) with G and with ~G respectively.

In the first example the point made is that according to the above analysis, it is generally the case that the agent ought to gamble, since the best possible outcome is to win the gamble, and that is possible only if you gamble. That is implausible on the face of it — and in that first example, we see that the gambler could make sure that gets 5 units by not gambling, which looks like a better option than the gamble, which may end with a gain of 10 or nothing at all. While someone who values gambling for its own risk might agree, we can’t think that this is what he ought to do. The second example is the same except that winning the gamble would only bring 5 units, with a risk of getting 0, while not gambling brings 5 for sure. In this case we think that he definitely ought not to gamble, but on the above analysis it is not true either that he ought to gamble or ought not to gamble.

Horty’s conclusion, surely correct, is that what is needed is a value ordering of the choices rather than of the possible outcomes (though there may, perhaps should, be) a connection between the two.

Fine, but Horty defines that ordering as follows: choice K’ (weakly) dominates choice K if none of the possible histories in K are better than any of those in K’. (See NOTES below, about this.) The analysis of ‘ought’ is then that the agent ought to see to it that A exactly if all his optimal choices make A definitely true.

Suppose the choice is between two lotteries, each of which sells a million tickets, and has a first prize of a million dollars, and a second prize of a thousand dollars. But only the second lottery has many consolation prizes worth a hundred dollars each. Of course there are also many outcomes of getting no prize at all. There is no domination to tell us which gamble to choose, but in fact, it seems clear that the choice should be the second gamble. That is because the expectation value of the second gamble is the greater.

This brings in the agent’s opinion, his subjective probability, to calculate the expectation value. It leads in this case to the right solution. And it does so too in the two examples above that Horty gave, if we think that the individual outcomes were in each case equally likely. For then in the first example the expectation value is 5 in either case, so there is no forthcoming ought. In the second example, the expectation value of gambling is 2.5, smaller than that of not gambling which is 5, so the agent ought not to gamble.

So, tentatively, here is my conclusion. Horty is right on three counts. The first is that the Chisholm/Meinong analysis, with its role for the value ordering of the possible outcomes, is faulty. The second is that the improvement needed is that we rely, in the analysis of ought statements, on a value ordering of the agent’s choices. And the third is that an integration with epistemic logic is needed, ….

…. but — I submit — with a logic of opinion rather than of knowledge.

NOTES

John Horty “Epistemic Oughts in Stit Semantics”. Ergo 6 (2019): 71-120

Horty’s definition of dominance is this:

K ≤ K’ (K’ weakly dominates K) if and only if Value(h) ≤ Value(h’) for each h in K and h’ in K’; and K < K’ (K’ strongly dominates K) if and only if K ≤ K’ and it is not the case that K’ ≤ K.

This ordering gives the right result for Horty’s second example (Ought not to gamble), while in the first example neither choice dominates the other. But the demand that all possible outcomes of choice K’ should be better than any in K seems to me too strong for a feasible notion of dominance. For example if the values of outcomes in one choice are 100 and 4, while in the other they are 5 and 4, this definition does not imply that the first choice weakly dominates the other, since 5 (in the second) is larger than 4 (in the first) — while intuitively, surely, the first choice should be advocated.

An oblique look at propositions (1) a scheme

Bas van FraassenLeave a comment

I want to explore a simple scheme, one that has many instances, and seems to fit much about what one might naturally call propositions, given all the literature in the history of logic, and philosophical logic, where they have appeared.

At the end I will give examples of different languages/logics which can be viewed as based on instances of this scheme. In the next post I will display a minimal logic (‘structural’, in that the familiar Structural Rules for valid arguments hold, and later I plan to post some more about different logics in the same vein.

The scheme

This scheme has four ingredients; a set of states, a set of propositions, and two relations, that I will denote with the symbols » and ≤.

The first relation is one between states and propositions. To keep it simple I will read “x » p” as “in state x, p is true” or just “p is true in x”. That will not always be the most natural reading, other glosses such as “x makes p true” may be more apt, so let’s keep a mental reservation about just what it means.

The second relation is subordination, or being subordinate to. It has both a simple form and a general form. The simple form is easiest to think about, and can be defined explicitly from its more general form, but in some cases it is precisely the general form that matters.

In the simple form it is a relation between states:

x is subordinate to y exactly if all propositions that are true in y are also true in x.

It is easy to deduce from this statement that being subordinate to is a partial ordering of states, which is why I settled on the usual symbol ≤ . That it is reflexive and transitive follows not from any characteristics that states or propositions could have, but just from the logic of “all” and “if … then” in its definition.

Just as a teaser, we can immediately imagine a state that is subordinate to all states, and thus makes all propositions true — a sign of inconsistency — so let us call it (if it exists!) an absurdity. If it is unique I will denote it by the capital letter Φ.

The more general form of subordination relates states to sets of states:

x is subordinate to W exactly if all the propositions that are true in all of W’s members are also true in x.

The simple form of the relation is definable: x ≤ y exactly if x ≤ {y}. In its general form we can deduce that subordination has the following characteristics:

  • if Φ exists then Φ ≤ W, because Φ is subordinate to all states
  • If x is in W then x ≤ W, for if a proposition is true in all members of W then it is true in any given member of W
  • If U is a subset of W and x ≤ W then x  ≤ U, because if p is true in all of W and U is part of W then p is true in all of U
  • If all members of U are subordinate to W, and x ≤ U, then x ≤ W (a little more complicated, see Appendix)

This is a bit redundant, but all the better to remind us of analogies to, for example, the relation of logical consequence, or semantic entailment. But let’s not be too quick to push familiar notions into this scheme, it may have very different sorts of instances.

Five instances of the scheme

The suggested ways to gloss the terms can be followed up with initial suggestions about interpretation. I will list five, chosen to be importantly different from each other.

(A) Modal logic, possible world models. Here we can take states to be possible worlds and propositions to be sets of worlds. This is a trivial instance of the scheme: x » p iff x ≤ p iff x is a member of p, and x ≤ y iff x = y. As we know though, this instance becomes interesting if structures are built on it, say by adding binary or ternary relative possibility relations.

(B) Epistemic logic, beyond modal logic. Here we can take the states to be minds, characterized by associated bodies of information. That is, minds are related to propositions by holding them true, or taking them a settled, or believing. Then the simple ordering is not trivial because x ≤ y would be the case just if x believes all that y believes, but perhaps x believes a lot more. And the absurdity Φ would be the utterly credulous mind which believes everything. In that state, as Arthur Prior said about this sort of thing, all logische Spitzfindigkeit would have come to an end.

It could be more complicated, but it stays simple if we think of the propositions as just sets of states (i.e. of minds). Proposition p is identifiable as the set of minds who believe that p. Then something interesting could play opposite to the absurdity: those minds who are as agnostic (unbelieving) as is possible: call them Zen minds:

a state x is a Zen mind precisely if x believes all and only those propositions that all minds believe.

So all states are subordinate to a Zen mind; a Zen mind is at the pinnacle of meditative detachment.

Questions to tackle then: about how minds can grow to have more beliefs (updating), or even reason from suppositions to have conditional beliefs.

(C) Truth making. The phrase “makes true” suggests a still different way, appealing to a more or less traditional notion of fact, the sort of thing that is or is not the case:

Tractatus 1: “The world is everything that is the case. The world is the totality of facts, not of things.”

After Wittgenstein’s Tractatus, Bertrand Russell took this up in his fabulous little book The Philosophy of Logical Atomism. Facts can be small, like the fact that the cat is on the mat. But they can be big, consisting of bundles of small facts. Say, a, b, c are small facts, and a.b, a.c, b.c, a.b.c are bigger facts made up of them. Being the case is a characteristic of facts; the big fact a.b is the case iff both a and b are the case.

A proposition could then be a set of facts, and we can say that p is made true by x ( x » p) exactly if y is part of x for some y in p. So proposition U = {b, b.c} is made true by b, as well as by b.c, but also by a.b, and by all the other bigger facts that contain either of its members as parts, like a.b.c, a.b.c.d, etc.

This scheme becomes interesting when we face the challenge that Raphael Demos, one of the students, posed when Russell was lecturing at Harvard: what happens to negation, are there negative facts, could there be? And that in turn points to the treatment of negation in what Anderson and Belnap, studying relevant logic, called tautological entailment.

(D) Logic of opinion. We get to something quite different if we take the states to be probability measures on a space, that consists of K, a set of worlds, and F, a family of subsets of K on which these states are defined. Let’s say that if Q is in F then there is an associated proposition, namely the set of probability measures x such that x(Q) = 1. We can define x » A, for such a proposition A, to be true if and only if x is a member of A.

Now x ≤ y is not trivial. Suppose x(A ∩ B) = 1 and y(A) = 1 but y(C) <1 for any proper subset of A. Then all the propositions that are true in y will also be true in x, but not conversely.

And there is an intriguing angle to this. One measure x can be a mixture (linear combination) of several other measures, for example x = ay + (1 – a)z. In that case A will be true in x if and only if it is true in both y and z. So then we see a case of subordination of states to sets of states: x ≤ {y, z} if x is a mixture of y and z. And more generally, all the mixtures of states that make a proposition true are subordinate to that proposition. So the propositions are convex sets of probability measures.

(E) Quantum logic. Rather different from all of these, but related to the previous example, there is a geometric interpretation, introduced by von Neumann in his interpretation of quantum mechanics. The states can be vectors and the propositions subspaces or linear manifolds — so, lines and planes that contain the origin, three-spaces, four-spaces, and so on. One vector x can be a superposition (linear combination) of some others; for example, x = ay + bz + cw, and we can make a similar point, like the one about mixtures of probability functions. But superpositions are quite different from mixtures. And the linear manifolds form a lattice that is non-Boolean.

With these five examples we have suggestions that our simple scheme will relate in possibly interesting ways to alethic modal logic, epistemic logic, truth-maker semantics (which points to relevance logic), probabilistic semantics (which has been related to intuitionistic logic), and quantum logic. Seems worth exploring …

Note on the literature

There is lots of literature on all five of the examples, but I’ll just list some of my own (no need to read them, as far as these posts are concerned; but they are all on ResearchGate).

(B) “Identity in Intensional Logic: Subjective Semantics”  (1986) (C) “Facts and Tautological Entailments” (1969) (D) “Probabilistic Semantics Objectified, I” (1981) (E) “Semantic Analysis of Quantum Logic” (1973)

                     

APPENDIX

Characteristic 4. of subordination was this:

If all members of U are subordinate to W, and x ≤ U, then x ≤ W

Remember how states relate to propositions, reading “x » p” as “p is true in x”.

Suppose that b≤ U. That is:

1. For all q (If all z in U are such that z » q, then b » q)

Now suppose that all members of U are subordinate to W:

2. For all z in U { For all q (If all y in W are such that y » q, then z » q)}

The first two “all’s” can change position, and we can write this as

3. For all q: All z in U are such that [(If all y in W are such that y » q, then z » q)

And in the conditional in the middle part, we note that z does not appear in the antecedent, so that is the same as:

4. For all q: If all y in W are such that y » q, then all z in U are such that [( z » q)]

But that putting this together with 1. we arrive at

5. For all q (If all y in W are such that y » q, then b » q)

that is to say, b is subordinate to W.