A motivation for this, which will show up in Application 2, is to show that it is tenable to hold that in general, typically, conditionals are true only if they are certain. I do not propose this for conditionals in natural language. But I think it has merits in certain contexts in philosophy of physics, notably interpretation of the conditionals that appear in Einstein-Podolsky-Rosen and Bell Inequality arguments.
[1] The algebra page 1
[2] The language: first step in its interpretation page 1
[3] The algebra: filters and ideals page 2
[4] The language and algebra together: specifying a truth-filter page 2
[5] The language, admissible valuations, validity and the consequence relation page 3
APPLICATION 1: probability space models page 4
APPLICATION 2: algebraic logic of conditionals with probability page 5
NOTE: I will explain this approach informally, and just for the simple case in which we begin with a Boolean algebra.
The languages constructed will in general not be classical, but in this case validity of the classical sentential logic theorems will be preserved, even if other classical features are absent.
But this approach can be applied starting with some other sort of algebra.
[1] The algebra
Let us begin with a Boolean algebra A, with the operations ∩, ∪, -, relation ⊆, top K, and bottom Λ. From my choice of symbols you can see that I find it useful to think of it as an algebra of sets. That will be characteristic of some applications. But this play no role for now, it just helps the imagination.
I will use little letters p, q, r, … to stand for elements of A.
I have left open here whether there are other operations on this algebra, such as modal operators. Application 2 will be to a Boolean algebra with modal operator ==>.
[2] The language: first step in its interpretation
As far as the algebra is concerned, all elements have the same status. But we can introduce distinctions from outside, by choosing a language that can be interpreted in that algebra. When we do that each sentence E has a semantic value [[E]], which is an element of A, and we call it the proposition expressed by that sentence.
So let us introduce a language L. It has atomic sentences, the classical (‘Boolean’) connectives &, v, ~. It may have a lot more. The interpretation is such that
[[~E]] = -[[E]] (the Boolean complement)
[[E & D]] = [[E]] ∩ [[D]]
[[E v D]] = [[E]] ∪ [[D]]
and there will of course be more clauses if the language has more resources for generating complex sentences.
The atomic sentences, together with those three classical connectives, form a sub-language, which I will call Lat. This this is a quantifier-free, modal operator free, fragment of L. I tend to think of the members of Lat as the empirical sentences, the language of the data, but again, that is at this point only a mnemonic.
The set of propositions expressed by sentences in Lat I will call A0, that is {[[E]]: E is in Lat}, and it is clearly a Boolean algebra too, a sub-algebra of A. In general A will be much larger than A0.
[3] The algebra: filters and ideals
What about truth and falsity? I will take it that the true sentences in the language together form a theory, that is, a set closed under the language’s consequence relation — which clearly includes the consequence relation of classical sentential logic. I take it also that this theory is consistent, but do not assume that must be complete.
The algebraic counterpart of a theory is a filter: a set F of elements of A such that, if p ⊆ q and p is in F then so is q, and if r, q are both in F then so is (r ∩ q). A filter is proper exactly if it does not have Λ as a member. That corresponds to consistency.
The filter that consists of the propositions expressed by the members of a consistent theory is a proper filter. Obviously all filters contains K.
A set of elements of A is an ideal exactly if: if p ⊆ q and q is in G then so is p, and if r, q are both in G then so is (r ∪q). The ideal is proper if K is not in it. Obviously any ideal contains Λ.
Filter F has as counterpart an ideal G = {-p: p is in F}, where -p is the complement of p in A. This corresponds to what the theory rules out as false.
[4] The language and algebra together: specifying a truth-filter
Now we are ready to talk about assigning truth-values. Remember that the language L already has an interpretation [[.]] into the algebra of propositions A. What we need to do next then is to select the propositions that are true, and then assign value T to the sentences that express those propositions.
Well, I will show a way how we can do that; but there are many ways. I would like the ‘empirical sentences’ all to get a truth-value. In addition there may be a class of sentences that also should get truth-values, for some reason. They could be selected syntactically (in the way Lat is), or they could be selected as the ones that express a certain sort of proposition. The latter would be a new way of doing the job, so that is what I will outline.
Step 1 is to specify a proper filter T on A, which will be the set of propositions that we will specially specify as true, regardless of whether they belong to A0. Its corresponding ideal U is then the set of propositions that we will specially specify as false.
Step 2 is to specify a filter T0 on A0, as the set of true propositions which are values of ‘empirical sentences’, and indeed we want T0 to be a maximal proper filter on A0. Then its corresponding ideal U0 on A0 is a maximal proper ideal, and A0 is the union of T0 and U0. So every proposition in A0 is classified as true or false.
There is one important constraint on this step. Clearly we do not want any proposition to be selected as true in one step and false in the other step. So the constraint is:
Constraint on Truth Filtering. T0 does not overlap U.
It follows then also that U0 does not overlap T.
The final step is this: T* is the smallest filter that contains both T and T0. We designate T* as the set of true propositions in A. This is the truth-filter. Its corresponding ideal U* is the set of false propositions in A.
This is an unusual way of specifying truth conditions, not least because there will in general be propositions that belong neither to T* nor to U*: in general, bivalence fails.
We need to show that T* is a proper filter.
Lemma. For every proposition p in T* there is a proposition q in T and a proposition r in T0 such that q ∩ r ⊆ p.
It is easiest to prove this via the relation between filters and theories. Let Z be the least theory that contains theories X and Y: thus Z is the set of sentences implied by X ∪ Y. Implication, in our context, is finitary, so if A is in Z then there is a finite set of sentences belonging to X ∪ Y whose conjunction implies A.
Suppose now that T* is not proper. Then there is a proposition p such that both p and -p are in T*. They cannot both be in T nor both in T0. The Constraint on Truth Filtering implies that if p is in T0 then -p is not in T, so -p must a proposition that is not in either T or T0. Similarly if p is in T then -p cannot be in T0 so it must be in neither T nor T0. So we see that either p or -p belongs to neither T nor T0, but must be in the part of T* that is ‘implied’ by meets of elements taken from T and from T0.
By the Lemma there must be propositions q and r in T and T0 respectively such that (q ∩ r) ⊆ p, and also q’ and p’ in T and T0 respectively such that (q’ ∩ r’) ⊆ -p . But then there is a proposition s = (q ∩q’) in T and a proposition t = (r ∩ r’) in T0 such that (s ∩ t) ⊆ (p ∩ – p) = Λ.
In that case t ⊆ -s, while t is in T0 and -s belongs to U. And that is not possible, given the Constraint on Truth Filtering.
Therefore T* is a proper filter.
[5] The language, admissible valuations, validity and the consequence relation
Time to look into the logic in language L when the admissible assignments of truth-values are all of this sort!
What we have described informally now is the class of algebraic models of language L. The sentences E in L have as semantic values propositions [[E]] in A. A is a Boolean algebra with a designated filter T* and designated ideal U* = {-p: p is in T*}. An admissible valuation of L is a function v such that for all sentences E of L:
- v(E) = T if and only if [[E]] is in T*
- v(E) = F if and only if [[E]] is in U*.
This function is not defined on other sentences: those other sentences, if any, do not have a truth-value.
So an admissible valuation is in general a partial function on the set of sentences of L.
Validity
Boolean identities correspond to the theorems of classical sentential logic. If E is such a theorem then [[E]] = K, which belongs to every filter, and hence E is true.
This holds for any model of the sort we have described, so all theorems of classical sentential logic are valid.
Deductive consequence
E1, …, En imply F exactly if, in each such model, if [[E1]], …, [[En]] are all in T* then [[F]] is in T*.
In classical sentential logic E1, …, En imply F exactly if (E1 & … & En) ≡ (E1 & … & En & F) is a theorem. So then ([[E1]] ∩ …∩ [[En]]) = ([[E1]] ∩ …∩ [[En]] ∩ [[F]]).
It follows that if [[E1]], …, [[En]] are all in a given filter then so is [[F]].
Therefore all such classically valid direct inferences (such as Modus Ponens) are valid in L.
Natural deduction rules
Those which involve sub-arguments can be expected to fail. For example, (E v ~E) is valid, but it is possible that E lacks a truth-value, and so we would expect the Disjunctive Syllogism to fail.
We’ll see examples below.
APPLICATION 1: probability space models
The structure S = <K, F, P> is a probability space exactly if K is a non-empty set, F is a field of subsets of K (including K), and P is a probability function with domain F.
A field of sets is a Boolean algebra of sets. So we can proceed as above.
First there is a language LS, and if E is a sentence of LS then [[E]] is a measurable subset of K, that is to say, a set in F, a member of the domain of P. And as before we have a fragment LSat which is the closure of the set of atomic sentences under the Boolean connectives. The range of [[.]] restricted to LSat is a subfield — a Boolean subalgebra — F0 of F.
The set TS = {p is in F : P(p) = 1} is a proper filter. That is so because P( Λ) = 0, P(p) is less than or equal to P(q) if p ⊆ q, and P(p ∩ q) = 1 if and only if P(p) = P(q) = 1.
Similarly, there is a corresponding proper ideal US = {p is in F : P(p) = 0}.
Just as above, TS is the beginning, so to speak, of the set of true propositions. To determine an appropriate set of true propositions in F0 we begin with X = US ∩ F0 That is a proper ideal as well, within that subalgebra. Every such proper ideal can be extended (not uniquely) to a proper maximal ideal US0 on F0. This we choose as the set of false propositions in that subalgebra, and the corresponding maximal filter TS0 on F0 is the set of true propositions there.
And now, to complete the series of steps we are following, we define TS* to be the least filter on F which contains both TS and TS0. The general argument above applies mutatis mutandis to show that TS* is a proper filter — our truth filter in this setting.
Unless LSat is the whole of LS we will now have truth-value gaps: there will be non-empirical sentences that receive some probability intermediate between 0 and 1, and these are neither true nor false.
As before, there is no doubt that the axiomatic classical sentential logic is sound here. However there are natural deduction rules which are not admissible. For example, if something follows from each of P(p) = 1 and P(q) = 1 it may still not follow from P(p v q) = 1. For example, if we are going to toss a coin then Probability(Heads) = 1 entails that the coin is biased, and Probability(Tails)= 1 also entails that the coin is biased. But Probability( Heads or Tails) =1 is true also if the coin is fair.
APPLICATION 2: algebraic logic of conditionals with probability
This is an example of a probability space model, in which the algebra is a Boolean algebra with a binary modal operator ==>. It begins with a ‘ready to wear’, off the shelf, construction, which I’ll describe. And then I will apply the recipe developed above to give a picture of a language in which conditionals, typically, are true only if they have probability 1, and and false only if they have probability 0.
I am referring to the logic CE, which is like Stalnaker’s logic of conditionals, but weaker (van Fraassen 1975; see also my preceding blogs on probabilities of conditionals).
The language has the Boolean connectives plus binary connective –>. A structure M = <K,F, s> is a model of CE exactly if K is a non-empty set (the worlds), F is a field of subsets of K (the propositions), and s, the selection function, is a function which maps K x F into the subsets of K, with these properties:
- s(x,A) ⊆ A
- if x is in A then s(x,A) = {x}
- s(x, A) has at most one member
- s(x, A) = Λ only if A = Λ
The truth conditions for &, v, ~ are as usual, and for –> it is:
A –> B is true in world x if and only if s(x,A) ⊆ B
equally: [[A –> B]] = {x is in K: s(x, [[A]]) ⊆ [[B]]}
and we can see that there is therefore an operator on F , for which I’ll use the symbol ==>:
[[A –>B]] = [[A]] ==> [[B]].
This differs from Stalnaker’s semantics only in not imposing the further restriction on the selection function that it must derive from an ordering. We may intuitively refer to s(x, A) as the world nearest to x that is in [[A]], but this “nearest” metaphor has no content here.
When this language is thus interpreted in model M, the propositions form a Boolean algebra with operator ==>, which has the properties:
(I) [p ==> (q ∪ c)] = [(p ==> q) ∪ (p ==> c)]
(ii) [p==> (q ∩ c)] = [(p ==> q) ∩ (p ==> c)]
(iii) [p ∩ (p ==> q)] = (p ∩ q)
(iv) (p ==> p) = K ( “necessity” )
(v) (p ==> -p) = Λ unless p = Λ (“impossibility”)
Let us call this a CE algebra.
A probability model for CE is a structure <K, F, s, P> such that <K, F, s > is a model for CE and P is probability function with domain F such that for all p, q in F
P(p ==> q) = P(q | p) when defined
This condition is generally called Stalnaker’s Thesis (or more recently, just “the Thesis”). Stalnaker’s logic of conditionals could not be nontrivially combined with this thesis but CE could. As it happens, CE has a rich family of probability models.
Thus, if <K, F, s, P> is a probability model for CE then S = < K, F, ==>, P> is a probability space model in the sense of the previous section, with some extra structure.
Now we can proceed precisely as in the preceding section to define a truth filter T* on the algebra of propositions. As empirical statements we take the closure of the set of atomic sentences under just the Boolean connectives, that is the sentences in which there are no occurrences of –>. The image of this language fragment by the map [[.]] is the relevant, privileged Boolean subalgebra F0 of F in which every proposition is classified as true or false, as a first step.
In addition the propositions which have probability 1 are true. And finally, anything implied by true propositions is true — all this understood as coming about as shown in the preceding section. Thus all theorems of CE are valid, and inference by modus ponens is valid.
As to sentences of form (A –> B), they are typically true only if P(A | B) = 1. I say “typically” because we cannot rule out that the proposition [[A]] ==> [[B]] is a member of F0. For the model of CE could be a model of a stronger theory, perhaps one that entails (implausibly!) that “if it is lit then it burns” is the meaning of “it is flammable”. But typically that will not be the case, so typically (A –>B) will be classified as true only if P([[B]] | [[A]]) = 1.
REFERENCES
van Fraassen, B. C. (1976) “Probabilities of Conditionals”, in W. Harper and C.A. Hooker (eds.) Foundations of Probability and Statistics, Volume l. Reidel: 261-308.
Bas van Fraassen's philosophy blog 