Tag: imperatives

Truthmaker semantics for the logic of imperatives

Bas van FraassenLeave a comment

Seminal text:  Nicholas Rescher, The Logic of Commands.  London: 1966

  1. Imperatives: the three-fold pattern 1
  2. Denoting imperatives 2
  3. Identifying imperatives through their truthmakers 2
  4. Entailment and logical combinations of imperatives 3
  5. Starting truthmaker semantics: the events. 4
  6. Event structures 4
  7. The language, imperatives, and truthmakers 5
  8. Logic of imperatives 6
    APPENDIX. Definitions and proofs 7

In deontic logic there was a sea change when imperatives were construed as default rules (Horty, Reasons as Defaults: 2012).  The agent is conceived as situated in a factual situation but subject to a number of ‘imperatives’ or ‘commands’.  

Imperatives can be expressed in many ways.  Exclamation marks, as in “Don’t eat with your fingers!”, may do, but are not required.  Adapting one of Horty’s examples, we find in a book of etiquette:

  • One does not eat food with one’s fingers
  • Asparagus is eaten with one’s fingers 

These are declarative sentences.  But in this context they encode defeasible commands, default rules.  Reading the book of etiquette, the context in question, we understand the conditions in which the indicated actions are mandated, and the relevant alternatives that would constitute non-compliance. 

In this form of deontic logic, what ought to be the case in a situation is then based on the facts there plus the satisfiable combinations of commands in force.[1]  

1.   Imperatives: the three-fold pattern

Imperatives have a three-fold pattern for achievement or lack thereof:

  • Success: required action carried out properly
  • Failure:  required action not carried out properly or not at all
  • Moot:    condition for required action is absent 

In the first example above, the case will be ‘moot’ if there is no food, or if you are not eating.  Success occurs if there is food and it is not eaten with the fingers, Failure if there is food and it is eaten with the fingers.

Whenever this pattern applies, we can think of that task as having to be carried out in response to the corresponding imperative.  There are many examples that can be placed in this form.  For example, suppose you buy a conditional bet on Spectacular Bid to win in the Kentucky Derby. Doing so imposes an imperative on the bookie.  He is obligated to pay off if Spectacular Bid wins, allowed to keep the money if she loses, and must give the money back if she does not run.

2.  Denoting imperatives

An imperative may be identified in the form ‘When A is the case, see to it properly that B’.  This way of identifying the imperative specifies just two elements of the three-fold pattern, Success and (the opposite of) Moot.  

But the opposite of Moot is just the disjunction of the two contraries in which the condition is present.  Therefore it is equally apt to represent the imperative by a couple of two contraries, marking Success and Failure.  Doing so gives us a better perspective on the structure of imperatives and their relation to ‘ought’ statements.  

So I propose to identify an imperative with an ordered pair of propositions <X, Y>, in which X and Y are contraries.  Intuitively they correspond respectively to Success (and not Moot), and Failure (and not Moot).  

3.  Identifying imperatives through their truthmakers

Our examples point quite clearly to a view of imperatives that goes beyond truth conditions of the identifying propositions.  What makes for success or failure, what makes for the truth of the statement that the imperative has been met or not met, are specific events.

That Spectacular Bid wins, or that you close the door when I asked you to, are specific facts or events which spell success.  That I eat the asparagus with a fork is a distinct event which spells a failure of table etiquette.

Consider the command 

(*)   If A see to it that B!

as identified by its two contraries, Success and Failure.  For each there is a class of (possible) events which ‘terminate’ the command, one way or the other.  

The statement “Spectacular Bid wins” states that a certain event occurs, and encodes a success for the bookie’s client.  The statement that encodes Failure is not “Spectacular Bid does not win”. Rather it is “Spectacular Bid runs and does not win”, which is, for this particular imperative the relevant contrary.  

To symbolize this identification of imperatives let us denote as <X| the sets of events that make X true, and as |X> the set of events that make the relevant contrary (Failure) true.[2]  The imperative in question is then identified by an ordered couple of two sets of events, namely  (<X|,  |X>).  I will abbreviate that to <X>.  

In (*), <X> is the imperative to do B if A is the case, so X = the statement that (A and it is seen to that B), which is made true by all and only the events in set <X|.  Its relevant contrary in this particular imperative is the statement that (A but it is not seen to that B), and that relevant contrary is whatever it is that is made true by all and only the events in set |X>.

4. Entailment and logical combinations of imperatives

There is an obvious sense in which E, “Close the door and open the window!” entails F, “Close the door!”  Success for E entails success for F.  But that is not all.  Failure for F entails failure for E.   The latter does not follow automatically from the former, if there is a substantial Moot condition: not winning the Derby does not, as such, imply losing.

So entailment between imperatives involves two ‘logical’ implications, going in opposite directions, and we can define:

Definition.  Imperative A entails imperative B exactly if <A| ⊆ <B| and |B> ⊆ |A>.

“Open the door!” is a ‘strong’ contrary to “Close the door!”.  There is a weaker contrary imperative:  if someone looks like he is about to close the door, you may command “Do not close the door!”.  

Negation.  In the logic of statements, the contradictory is precisely the logically weakest contrary.  For example, yellow is contrary to red and so is blue, but to be simply not red is to be either yellow or blue or … and so forth.

So I propose as the analogue to negation that we introduce

<┐A>:             <┐A| = |A>  and |┐A> = <A|

Whatever makes ┐A true is what makes A false, and vice versa. Here the symbol “┐” does not stand for the usual negation of a statements, because  imperatives generally have significant, substantial conditions.  The relevant contrary to Success is not its logical contradictory (that would be: either Failure or Moot) but Failure (which implies not-Moot), and that is whatever counts as Failure for the particular imperative in question. 

Conjunction.  “Close the door and open the window” we can surely symbolize as <A & B>.  Success means success for both.  In addition, failure means failure for one or the other or both.  So there is no great distance between conjunction of Success statements and the ‘meet’ operation on imperatives:

<A & B>:             <A & B| = <A| ∩ <B|,  |A & B> = |A> ∪ |B>.

Disjunction.  Similarly, dually, for disjunction and the ‘join’ operation:

<A v B>:             <A v B| = <A| ∪ <B|,  |A v B> = |A> ∩ |B>

We can already see that some familiar logical relations are making an appearance.  

[1]  <A & B> entails <A>, while <A> entails <A v B>.

For example, <A & B| ⊆ <A| and |A> ⊆ |A & B>.

(All proofs will be provided in the Appendix.)

We could go a bit further with this.  But answers to the really interesting questions will depend on the underlying structure of events or facts, that is, of the truthmakers.

5. Starting truthmaker semantics:  the events.

Events combine into larger events, with an analogy to conjunction of statements.  So the events form a ‘meet’ semilattice.  Important are the simple events: 

Postulate:  Each event is a unique finite combination of simple events.  

Is it reasonable to postulate this unique decomposability into simple events?  

At least, it is not egregious.  Think of how we specify a sample space for probability functions:  each measurable event is a subset of the space.  The points of the space may have weights that sum up to the measure of the event of which they are the members.  Two events are identical exactly if they have the same members.  

In any case, the idea of truthmakers is precisely to have extra structure not available in possible worlds semantics.

Combination we can conceive of as a ‘meet’ operation.  Besides combining, we need an operation to identify contraries among events, in order to specify Success and Failure of imperatives.

Definition.  A event structure is a quadruple E = <E, E0, ., ° >, where E is a non-empty set, . is a binary operation on E, E0 is a non-empty subset of E, and °  is a unary operation on E0, such that:

  • ° is an involution: if a is in E0 then a° ≠ a and a°°  = a
  • . is associative, commutative, idempotent (a ‘meet’ operator)
  • If e and e’ are elements of E then there are elements a1, …, ak,  b1, …, bm of E0 such that e = a1… ak,  and e’=b1…bm  and e = e’ if and only if { a1, …, ak }= { b1, …, bm }

This last clause implies along the way that if e is an element of E then there is a set a1, …, an of elements of E0 such that e = a1 … an. That is part, but only part, of what the Postulate demands, and would not by itself imply unique decomposability. 

The involution operates solely on simple events.  A particular imperative could have a simple event b to identify Success; in that case simple event b°  will be identify its Failure.  

6.  Event structures

The following definitions and remarks refer to such an event structure E.

Definition.  e ≤ e’ if and only if there is an event f such that e’.f = e. 

Analogy: a conjunction implies its conjuncts, and if A implies B then A is logically equivalent to (A & C) for some sentence C.  

The definition is not the standard one, so we need to verify that it does give us a partial order, fitting with the meet operator.

[2]  The relation ≤ is a partial ordering, and e.f is the glb of e and f.

That is, we have the familiar semilattice laws:  e.g. if  e ≤ e’ and f is any other event then f.e ≤ e’.

So <E, ., ≤ > is a meet semilattice.  Note also that if a and b are simple events then a ≤ b only if a = b.  For if b.f = a, the Postulate implies that b = f = a.

So far we have a relation of contrariety for simple events only.  For events in general we need to define a general contrariness relationship.

Definition. Event e is contrary to event e’ if and only if there is an event a in E0 such that e ≤ a and e’ ≤ a° .

Contrariety is symmetric because a°°  = a.  

At this point we can see that the logic we are after will not be classical.  For contrariety is not irreflexive.  

That is because (a.a°) ≤ a and (a.a°) ≤ a°, so (a.a°) is contrary to itself.  But (a.a°) is not the bottom of the semilattice.  If a, a°, and b are distinct simple events then it is not the case that (a.a°) ≤ b.  For if b.f = a.a°  and f = a1 … an then the Postulate requires {b, a1, …, an} = {a, a°} so either b = a or b = a° .

It is tempting to get rid of this non-classical feature.  Just reducing modulo some equivalence may erase the distinction between those impossible events, a.a°  and b.b° .  Such events can never occur anyway.  

But there are two reasons not to do so.  The first is that the history of deontic logic has run on puzzles and paradoxes that involve apparent self-contradictions.  The second is more general.  We don’t know what new puzzles may appear, whether about imperatives or related topics, but we hope to have resources to represent whatever puzzling situation we encounter. Erasing distinctions reduces our resources, and why should we do that?

7. The language, imperatives, and truthmakers

More formally now, let us introduce a language, and call it LIMP.  Its syntax is just the usual sentential logic syntax (atomic sentences, &, v, ┐).  The atomic sentences will in a specific application include sentences in natural language, such as ‘”One does not eat with one’s fingers”.  The interpretations will treat those sentences not as statements of fact but as encoding imperatives.  In each case, the interpretation will supply what a context (such as a book of etiquette) supplies to set up the coding.

An interpretation of language LIMP in event structure E = <E, E0, ., ° > begins with a function f that assigns a specific event to each atomic sentence in each situation.  Then there are two functions, < | and | >, which assign sets of truth-makers to each sentence:  

  • If A is atomic and a = f(A) then <A| = {e in E:  e ≤ a} and |A> = {e in E:  e ≤ a°}.
  • <┐A| = |A> and |┐A> = <A|
  • <A & B| = <A| ∩ <B|,  |A & B> = |A> ∪ |B>
  • <A v B| = <A| ∪ <B|,  |A v B> = |A> ∩ |B>

Definition.  A set X of events is downward closed iff  for all e, e’ in E, if e ≤ e’ and e’ is in X then e is in X.

[3]  For all sentences A, <A| and |A> are downward closed sets.

Now we can also show that our connector ┐, introduced to identify the weakest contrary to a given imperative, corresponds (as it should) to a definable operation on sets of events.

Definition.  If X ⊆ E then X⊥ = {e: e is contrary to all elements of X}.

I will call X⊥ the contrast (or contrast class) of X.

Lemma.  X⊥ is downward closed.

That is so even if X itself is not downward closed.  For suppose that f is in X⊥.   Then for all members e of X there is a simple event a such that f ≤ a and e ≤ a°.  Thus for any event g, also g.f.e ≤ a while e ≤ a°.  Therefore g.f is also in X⊥.

[4]  For all sentences A, <┐A| = |A> = <A|⊥  and |┐A> = <A| = |A> ⊥ .

The proof depends De Morgan’s laws for downward closed sets of events:

Lemma.  If X and Y are downward closed sets of events then 

(X ∩ Y) ⊥  = X ⊥ ∪ Y ⊥   and (X ∪ Y) ⊥ = X ⊥ ∩  Y ⊥.

In view of [4], there is therefore an operator on closed sets of events that corresponds to negation of imperatives:

Definition.  If A is any sentence then  <A> ⊥  = (<A| ⊥ , |A> ⊥ ).

[5]   <A> ⊥ =  <┐A>

This follows at once from [4] by this definition of the ⊥ operator on imperatives.

8. Logic of imperatives

We will concentrate here, not on the connections between sentences A, but on connections between their semantic values <A>.  These are the imperatives, imperative propositions if you like, and they form an algebra.  

Recall the definition of entailment for imperatives.  It will be convenient to have a symbol for this relationship:

Definition.   <A> ⇒ <B> exactly if <A| ⊆ <B| and |B> ⊆ |A>. 

 The following theorems introduce the logical principles that govern reasoning with imperatives.

[6]  Entailment is transitive.

To have the remaining results in reader-friendly fashion, let’s just summarize them.

[7] – [11] 

  • Meet.
    • <A & B> ⇒ <A>, 
    • <A & B> ⇒ <B>
    • If <X> ⇒  <A> and <X> ⇒  <B> then <X> ⇒ <A & B> 
  • Join.
    • <A> ⇒ <A v B>
    • <B> ⇒ <A v B>
    • If <A> ⇒ <X> and <B> ⇒ <X> then <A v B> ⇒ <X>
  • Distribution:  <A &(B v C)> ⇒ <(A & B) v (A & C)>.
  • Double Negation. <A> ⇒ < ┐ ┐ A>  and < ┐ ┐ A>  ⇒ <A>.
  • Involution.  If <A> ⇒ <B> then <┐B> ⇒ <┐A>.
  • De Morgan.
    • < ┐ (A & B)> ⇒ < ┐A v ┐B> and vice versa
    • < ┐ (A v B)> ⇒ < ┐A & ┐B> and vice versa.

COMMENTS.   In order for these results to make proper sense, each of the connectors ┐, &, v needs to correspond to an operator on imperatives, modeled as couples of downward closed sets of events. This was shown in the previous section.

The logic of imperatives is not quite classical.  We can sum up the above as follows: 

The logic of imperatives mirrors FDE (logic of first degree entailment); the imperatives form a De Morgan algebra, that is, a distributive lattice with De Morgan negation. 

APPENDIX.  Definitions and proofs

Definition.  Imperative A entails imperative B exactly if <A| ⊆ <B| and |B> ⊆ |A>.

[1]  <A & B> entails <A>, and <A> entails <A v B>.

For <A & B| = <A| ∩ <B| ⊆ <A| while |A > ⊆  |A|> ∪ |B> = |A & B>.  Similarly for the dual.

Postulate:  each event is a unique finite combination of simple events.  

Definition.  A event structure is a quadruple E = <E, E0, ., ° >, where E is a non-empty set, . is a binary operation on E, E0 is a non-empty subset of E, and ° is a unary operation on E0, such that:

  • ° is an involution: a° ≠   a and a°°  = a,  if a is in E0
  • . is associative, commutative, idempotent (a ‘meet’ operation)
  • If e and e’ are elements of E then there are elements a1, …, ak,  b1, …, bm of E0 such that e = a1… ak,  and e’=b1…bm  and e = e’ if and only if { a1, …, ak }= { b1, …, bm }

[2]  The relation ≤ is a partial ordering, and the meet e.f of e and f is the glb of e and f.

For  e ≤ e because e.e = e (reflexive), and if e = e’.f and e’ = e”.g then e = e”.f.g (transitive).  

(Perhaps clearer:  For if e = e’.f  then e.g = e’.f.g, so if e ≤ e’ then e.g ≤ e’, for all events g.)

            Concerning the glb: 

First, e.f ≤ e  because there is an element g such that e.f .g = e. g, namely g = f.  

Secondly suppose e’ ≤ e, and e’ ≤ f.  Then there are g and h such that e.g = e’ and f.h = e’.  In that case e’ = g.h.f.e, and therefore  e’ ≤ e.f. 

Definition. Event e is contrary to event e’ if and only if there is an event a in E0 such that e ≤ a and e’ ≤ a° .

Contrariness is symmetric because a°°  = a.  But it is not irreflexive for (a.a°) ≤ a and (a.a°) ≤ a°.   

Lemma 1. If a and b are simple events then a ≤ b only if a = b.  

That is because decomposition into simple events is unique.  For suppose that a.f = b. Then there are simple events c1, …, ck such that  f = c1….ck and a.f = a. c1, …, ck = b, which implies that a = c1 = … = ck = b.

Interpretation of the imperatives expressed in language LIMP, in event structure E = = <E, E0, ., ° >, relative to function f from atomic sentences to simple events. Then there are two functions, < | and | >, which assign sets of truth-makers to each sentence:  

  • If A is atomic and a = f(A) then <A| = {e in E:  e ≤ a} and |A> = {e in E:  e ≤ a° }.
  • <┐A| = |A> and |┐A> = <A|
  • <A & B| = <A| ∩ <B|,  |A & B> = |A> ∪ |B>
  • <A v B| = <A| ∪ <B|,  |A v B> = |A> ∩ |B>

Definition.  A set X of events is downward closed iff  for all e, e’ in E, if e ≤ e’ and e’ is in X then e is in X.

[3]  For all sentences A, <A| and |A> are downward closed sets.

Hypothesis of induction: this is so for all sentences of length less than A.

Cases.

  1. A is atomic.  This follows from the first of the truth-maker clauses
  2. A has form ┐B.  Then <B| and |B> are downward closed, and these are respectively |┐A> and <┐A|.

A has the form (B & C) or (B & C).  Here it follows from the fact that intersections and unions of downward closed sets are downward closed.

Definition.  If X ⊆ E then X⊥ = {e: e is contrary to all elements of X}

Lemma 2.  X⊥ is downward closed.

Suppose that e is in X⊥.  Then for all e’ in X, there is a simple event a such that e ≤ a and e’ ≤ a .  This implies for any event f, that f.e ≤ a and e’ ≤ a .  Hence f.e is also in X⊥.

[4]  For all sentences A, <┐A| = |A> = <A|⊥  and |┐A> = <A| = |A> ⊥ .

Hypothesis of induction: If B is a sentence of length less than A then <┐B| = |B> = <B|⊥  and |┐B> = <B| = |B> ⊥ .

Cases.

  1. A is atomic, and f(A) = a.  Then by the first truth-maker clause, all elements of |A> are contrary to all of <A|.  Suppose next that e is contrary to all of <A|, so e is contrary to a, hence there is a simple event b such that a ≤ b and e ≤ b° .  But then a = b, so e ≤ a° , hence e is in |A>. Similarly all elements of <A| are contrary to all elements of |A>, and the remaining argument is similar.
  2. A has form ┐B.  Then by hypothesis <┐B| = |B> = <B|⊥ .  And <┐┐B| = |┐B> by the truthmaker conditions, and |┐B> = <B|, and the hypothesis applies similarly to this.   
  3. A has form (B & C)

We prove first that <┐A| = |A> = <A| ⊥

<A| = <B| ∩ <C|,  while <┐A| = |B & C> = |B>  ∪ |C>.  If e is in <┐A| then it is in  |B>  ∪ |C> so by hypothesis e is contrary either to all of <B| or to all of <C|, and hence to their intersection. 

Suppose next that e is in <A|⊥ = (<B| ∩ <C|)⊥ .  To prove that this is <┐A| = <┐(B & C)| = |B & C> = |B> ∪ |C> = <B|⊥ ∪ <C|⊥  it is required, and suffices,  to prove the analogues to De Morgan’s Laws for downward closed sets.  See Lemma below.

We prove secondly that  |┐A> = <A| = |A> ⊥ .  The argument is similar, with appeal to the same Lemma below.

(4) A has form (B v C).  The argument is similar to case (3), with appeal to the same Lemma below.

Lemma 3.  De Morgan’s Laws for event structures:   If X and Y are downward closed sets of events then  (X ∩ Y) ⊥  = X ⊥ ∪ Y ⊥   and (X ∪ Y) ⊥ = X ⊥ ∩  Y ⊥.

Suppose e is in X ⊥.  Then e is contrary to all of  X, hence to all of X ∩ Y, hence is in (X ∩ Y) ⊥. Similarly for e in Y ⊥.  Therefore (X ⊥ ∪ Y ⊥ ) ⊆ (X ∩ Y) ⊥.

Suppose on the other hand that e is in (X ∩ Y) ⊥.  Suppose additionally that e is not in X⊥.  We need to prove that e is in Y ⊥.  

Let e’ be in X and not contrary to e.  Then if e’’ is any member of Y, it follows that e’.e’’ is in X ∩ Y, since X and Y are both downward closed.  Therefore e is contrary to e’.e’’.  We need to prove that e is contrary to e”.

Let b be a simple event such that e ≤ b and e’.e” ≤ b°.   By our postulate, e’ and e’’ have a unique decomposition into finite meets of simple events.  So let e’ = a1…ak  and e’’= c1…cm, so that e’.e” = a1…ak.c1…cm.  Since e’.e” ≤ b°, there is an event g such that a1…ak.c1…cm = e’.e’’= g.b°.   The decomposition is unique, so b° is one of the simple events a1, …, ak, c1, …, cm.  Since e is not contrary to e’, it follows that none of a1, …, ak is b°.  Therefore, for some j in {1, ..,m}, cj = b°, and therefore there is an event h such that e” = h. b°, in other words, e” ≤ b°.  Therefore e is contrary to e”.

So if e is not in X ⊥ then it is in Y ⊥, and hence in X ⊥ ∪ Y ⊥.

The argument for the dual equation is similar.

In view of the above, there is an operator on closed sets of events that corresponds to negation of imperatives:

Definition.  If A is any sentence then  <A> ⊥  = (<A| ⊥ , |A> ⊥ ).

[5]   <A> ⊥ =  <┐A>

(<A| ⊥ , |A> ⊥ ) =  (<┐A|, |┐A>), in view of [4].

Definition.   <A> ⇒ <B> exactly if <A| ⊆ <B| and |B> ⊆ |A>. 

 The following theorems introduce the logical principles that govern reasoning with imperatives.

[6]  Entailment of imperatives is transitive.

Suppose <A> ⇒ <B> and <B> ⇒ <C>.  Then <A| ⊆ <B| and <B| ⊆ <C|,  hence <A| ⊆ <C|.  Similarly, |C> ⊆|A>.

[7]  <A & B> ⇒ <A>, and if <X> ⇒  <A> and <X> ⇒  <B> then <X> ⇒ <A & B>, Also  <A> ⇒ <A v B>, and if <A> ⇒ <X> and <B> ⇒ <X> then <A v B> ⇒ <X>

First, <A| ∩ <B| ⊆ <A| and |A> ⊆ |A> ∪ |B>, hence  <A & B> ⇒ <A>.  

Secondly, suppose that X is such that <X| ⊆ <A| and <X| ⊆ <B| while |A> ⊆ |X> and |B> ⊆ |X>.  Then <X| ⊆<A| ∩ <B| = <A& B| while |A & B> = |A> ∪ |B> ⊆ |X>.  Hence <X> ⇒ <A & B>.

The dual result for disjunction by similar argument.

[8]  Distribution:  <A &(B v C)> ⇒ <(A & B) v (A & C)>.

<A &(B v C)| = <A| ∩ <B v C| = <A| ∩ (<B| ∪ <C|) = [<A| ∩ <B| ] ∪ [<A| ∩ <C|)] = <(A & B) v (A & C|. Similarly for the other part.

[9] Double Negation:  <A> ⇒ < ┐ ┐ A>  and < ┐ ┐ A>  ⇒ <A>.

< ┐ ┐ A| = |┐ A> = <A|  and |┐ ┐ A> = <┐ A| = |A>

[10]  Involution.  If <A> ⇒ <B> then <┐B> ⇒ <┐A>.

<┐B> ⇒ <┐A> exactly if <┐B| ⊆  <┐A|, i.e.  |B> ⊆  |A>,   and  |┐A> ⊆  |┐B>, i.e. <A| ⊆ <B|.  But that is exactly the case iff <A> ⇒ <B>    

[11]  De Morgan.  < ┐ (A & B)> ⇒ < ┐A v ┐B> and vice versa, while < ┐ (A v B)> ⇒ < ┐A & ┐B> and vice versa.

< ┐ (A & B)| = |A & B> = |A> ∪ |B> = < ┐A| ∪ < ┐B| = < ┐A v ┐B|.  Similarly for  |┐(A & B> .  Therefore < ┐ (A & B)> = < ┐A v ┐B>.

Similarly for the dual.

7.                          REFERENCES

Curry, Haskell B.  (1963) Foundations of Mathematical Logic. New York: McGraw-Hill.

Lokhorst, Gert-Jan C. (1999) “Ernst Mally’s Deontik”. Notre Dame Journal of Formal Logic 40 : 273-282.

Mally, Ernst (1926)  Grundgesetze des Sollens: Elemente der Logik des Willens. Graz: Leuschner und Lubensky

Rescher, Nicholas (1966)  The Logic of Commands.  London: Routledge and Kegan Paul

NOTES

[1] Rescher traces this analysis of ‘ought’ statements to Ernst Malley (1926) who coined the name Deontik  for his ‘logic of willing’. Since the logic of imperatives we arrive at here is non-classical, note that Lokhorst (1999) argues that Mally’s system is best formalized in relevant logic.

[2] We can use Dirac’s names for them, “bra” and “ket”, with no reference to their original use.

Deontic logic: value-rankings

Bas van FraassenLeave a comment

The historical opening chapter of the Handbook of Deontic Logic and Normative Systems shows that, in various forms, this has been a typical way to connect ‘ought’ statements with values:

[O] It ought to be that A if and only if it is better that A than that ~A

as well as

[Cond O] It ought to be that A, on supposition that B if and only if it is better that (B & A) than that (B & ~A)

But in addition, deontic logics typically include the law that carries logical implication into the derivation of ‘ought’ statements:

[IMP] If A implies B then (It ought to be that A) implies (It ought to be that B)

(and the similar law for conditional ‘ought’ statements), important to keep the logic within the range of normal modal logics.

But do [O] and [IMP] go well together? That depends on the character of the value ranking which defines the ‘better than’ relation among propositions. Specifically, it requires that

[MON] If A implies B, and A is better than ~A, then B is better than ~B.

Problem: I will give examples below of ‘better than’ relations which do have property MON but which are intuitively unsatisfactory. But a ranking by expectation value does not have property MON.

Ranking by expectation value does not have MON: For example, a bank robber is confronted by the police. His best option (by expectation value) is to surrender. What about the option to (surrender or resist arrest)? This is America! If he resists arrest he will likely be shot. That lowers the expectation value considerably. (Reminiscent of Ross’ paradox, also about IMP.)

Solution: There is something right about [O] and [CondO], namely that value rankings have an important role to play in the understanding of ‘ought’ statements. But there is also something wrong about [O] and [CondO], namely that they presuppose that it is just, and only, value rankings that must determine the status of ‘ought’ statements.

But let me first give examples of rankings that do have MON and say why I find them unsatisfactory. In my own essay on deontic logic as a normal modal logic (1972) I gave this definition:

Ought(A) is true exactly if opting for ~A precludes the attainment of some value which it is possible to attain if one opts for A

or less informally,

Ought(A) is true in possible world h exactly if, there are worlds satisfying A which have a higher value than any worlds that satisfy ~A.

Very unsatisfactory! Today I opted not to buy a lottery ticket, thereby precluding a million dollar windfall (larger than anything I could get otherwise) and so I was wrong. I ought to have bought that ticket! As gamblers say, when you talk about prudence, “Yes, but what if you win!” Sorry, gamblers — this is not a good guide to life …

Jeff Horty offered a more sophisticated formulation in his 2019 paper (p. 78, the Evaluation Rule) as his explication of the Meinong/Chisholm analysis, which would in a normal modal logic context amount to:

Ought(A) is true in world h exactly if A is true in all worlds h’ possible relative to h, such that there is no world h” which satisfies ~A and has a value higher than h’.

Except for the relativization of possibility, this is like the preceding. Horty rightly rejects this as unsatisfactory, using the example of a forced choice between two gambles which has the same expectation value, but of which one carries no risk of loss. (One has outcomes 10 and 0, the other has only outcome 5 with certainty.) It is certainly not warranted to say that we ought always to make the gamble with the higher prize but higher risk.

There are surely other value rankings to to try, and I thought of this one:

Ought(A) is true in world h exactly if there is a one-to-one function f mapping the worlds that satisfy ~A into the worlds that satisfy A, such that for all worlds h in the domain of f the value h is less than the value of f(h).

This one too has property MON. Informally put, it means that whatever outcome you get if you opt for ~A, you realize you might have done better by choosing for A.

But imagine: gamble ~ A has with certainty one of the outcomes 5, 7, or 9 dollars, while gamble A has with certainty one of the outcomes 1, 10, 12, 14 dollars. However, to make gamble A you have to buy a ticket for $4. So your net outcomes for A are loss of $3, or win of 6, 8, 10. Clearly by the above principle you should take gamble A, for 5 < 6 < 7 <8 < 9 < 10. But is that really the right thing to do? If all the outcomes are equally likely then ~A has expectation value (21/3) and A has expectation value (21/4), which is less.

In previous posts I have discussed how Horty goes beyond this.

Now I just want to explain how the use of expectation value ranking works well with what I proposed some weeks ago in the post Deontic logic: two paradoxes” (which I gave a more precise formulation in the next post, “Deontic logic: Horty’s new examples”.) By “works well” I mean that the principle [IMP] is valid.

My proposal was that in setting up the framework for deontic logic, , we need to include both imperatives and values. So I envisage the agent as first of all recognizing the imperatives in force in his situation (‘if you have sinned, repent!’). The agent’s next step is to take account of the satisfaction regions for those imperatives (or better, maximally consistent sets of them). Then the value-ranking is applied to those satisfaction regions, and the ones that count are the ones that get highest value (the optimal regions). Next:

It ought to be that A if and only if there is an optimal region that implies A.

(This can be extended to conditional oughts in the way Horty does: go look at the alternative situation in which the agent has the condition added to his knowledge.)

When entered at this point, it does not matter whether the ranking has property MON. For what ever the ranking is, if an optimal region is part of A, and A is part of B, then an optimal region is part of B.

The story for choices, decisions and action planning is similar. It is not that the agent ought to do what is best, but rather that he has to make a best choice (the moral of Horty 2019). Suppose it is already settled that I will gamble, and I have a choice between several gambles. Now what ought to be the case (about what I do, about my future) is whatever is implied by my making a best choice. And I propose that the best choices are those which are represented by propositions (the choices themselves, not the possible outcomes of those choices) which have highest expectation value.