(and of possibilities coupled with possible worlds)
Possibilities versus possible worlds page 1
Truth and falsity of sentences page 2
Urquhart’s initial results: relevance logic versus intuitionistic logic page 2
The discreet charm of the relevantist page 3
How can we think about Urquhart’s possibilities? page 3
Possibilities with worlds on the side page 4
How can we think of Urquhart’s world-coupled possibilities? page 4
What we may regret page 5
Possible worlds, as they appear in the semantics of modal logic, trade on our imagination schooled by Leibniz. They are complete and definite: the set of sentences that is true in a given world — that world’s description — leaves no question unanswered.
Possibilities versus possible worlds
This notion of a world soon had its rivals in various approaches to non-classical logic.
One of the first appeared in Urquhart’s 1972 semantics for relevant logic. In his informal commentary, the elements (not yet named “possibilities”) are something to be thought of as possible pieces of information. Urquhart emphasizes that his concept “contrasts strongly with that of a possible world [since] a set of sentences exhaustively describing a possible world must satisfy the requirements of consistency and completeness.” (p. 164). At the same time he sees his enterprise as generalizing the form familiar from possible world semantics:
“The leading idea of the semantics is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in relevant logics validity may be defined in terms of certain valuations on a semilattice-interpreted informally as the semilattice of possible pieces of information.” (p. 159)
Yet in most of the paper there is no, or little, role for the distinction between the description and the described. Indeed, Urquhart goes on quickly to specify that a piece of information, as he conceives of it, is a set of sentences. In later work in this area, his approach tends to be presented more abstractly, with the nature of the elements of the semilattice left as characterless as is the nature of possible worlds in the standard analysis of modal logic.
Lloyd Humberstone (1981) introduced “possibilities” as the general term for what may be “less determinate entities than possible worlds … [which] are what sentences … are true or false with respect to.” This term is now standard (witness e.g. the paper by Holliday and Mandelkern referenced in my recent posts), and I will use it.
When we think about possibilities, with the intuitive guide that they must correspond to partial descriptions of worlds, it is natural to see the possibilities as forming a partially ordered set (poset): x may have as much as or more information in it than y. That partial ordering relation is reflexive and transitive. Urquhart introduces in addition a join operation: if x, y are possibilities then so is (x ∪ y). The more sentences the more information, so (x ∪ y) has at least as much as, or more than, x or y. Urquhart adds the empty set of information, 0, which has the least information. A poset with this sort of operation defined on it is a semilattice (specifically, a join-semilattice).
Truth and falsity of sentences
Sentences may be evaluated as true or false ‘on the basis of’ given information. That is not as straightforward as it may look at first. Urquhart encourages us to think of it as the relation of premises to conclusion in an argument which commits no fallacies of relevance. His target, after all, is relevance logic, so that must be his main guide.
Given that, we cannot assume that if A is in x then A is true at x. Not so. There would have to be an argument from x to A, with all the sentences in x being relevant, playing an indispensable role, in that argument. While this notion of relevance, in the role it plays in the informal commentary, cannot be made more precise, what can be done is to show the form that any evaluation must take.
An evaluation is a function v that assigns truth-values to sentences relative to elements of the semilattice. Given such an assignment to atomic sentences, it is completed for conditionals by the clause
v(A → B, x) = T iff for all y, either v(A,y) = F or v(B, x ∪ y) = T
with the more friendly formulation being
iff for all y, if V(A, y) = T then v(B, x ∪ y) = T
with “if … then” understood in our metalanguage as the material conditional. Since (x ∪ x) = x, it follows that Modus Ponens is valid for the arrow.
Urquhart’s initial results: relevance logic versus intuitionistic logic
Urquhart proves quickly that the set of formulas involving just → which are valid in the sense of always receiving T relative to all elements of such a semilattice, are the theorems of Church’s weak theory of implication. That is also the implicational fragment RI of the relevance logic R.
Urquhart notes secondly that if we as an additional principle that
(*) if V(A, x) = T then V(A, x ∪ y) = T
then we leave relevance logic and arrive at the implicational fragment of intuitionistic logic.
That we get a characteristic irrelevancy is clear: if V(B, x) = T, then (given *), it will be the case for all y, that whether or not V(A, y) = T, V(B, x ∪ y) = T. In that case the irrelevancy [B →( A → B)] is valid.
The underlying difference between the two sorts of logic is that, if relevance is taken into account, then the structural rule
Weakening: if X entails B then X ∪ {A} entails B
is invalid. A relevance logic is a sub-structural logic.
The discreet charm of the relevantist
Relevance logics, which are Urquhart’s main target in this study, are, in many people’s eyes, weird. Something surprising surely springs to the eye when we note the omission of condition (*) in the semantics for RI. Although in the visualized picture (x ∪ y) contains all the information included in x, we are not to assume that if A is true at x then it is true at (x ∪ y). For the truth evaluation clause for A → B to make sense, we must think of (x ∪ y) as produced by adding the information in x to whatever makes A true at y. Nevertheless, the result (x ∪ y) may apparently lack some information, which was present in x, or have in it some information that renders some of the content of x impotent.
It is instructive to look at the model Urquhart constructs to show the completeness of RI. The elements of the semilattice are the finite sets of formulas of the language of RI, ∪ is set union, and 0 is the empty set. Then the evaluation defined is
V(B, {A(1), …, A(n)} = T if and only if A(1)→ … →. A(n) → B is a theorem of RI
In this notation (due to Church) a dot stands for a left hand parenthesis, and you have to imagine the right hand parenthesis. So A→. B → A is the same as A → (B → A). Since that is not a theorem, it is clear that in general V(A, {A, B}) may be F.
How can we think about Urquhart’s possibilities?
How can we think of this? I see two ways. The first is the one always evident in discussions of relevance logic: an irrelevant premise is a blemish, a blot on the escutcheon, anathema to natural logical thinking. So if some premises provide a good argument for global warming, say, the addition of a premise about the beauty of the Mona Lisa spoils the argument, removes its validity.
There is a second way, it seems to me. As a monologue or dialogue continues there are accepted devices for rendering something impact-less, though it was previously or elsewhere entered into the context. You may take back what you said. So if each element of the semilattice is a record, with times of entry noted, of things said in a conversation, then some of the content of x might be impact-less in the combination (x ∪ y).
It may also be interesting to think of what could happen here to the notion of updating, or conditionalizing. Suppose x is the information a person has, who then learns that A. So then his new information is x ∪ {A}. This will presumably mean that he now has some beliefs (counts as true) some statements he did not have as beliefs before. But we can also see that, due to the strictures on relevance, he will in general lose beliefs that he had.
We can imagine examples: someone believes that Verdi is Italian, and Bizet is French, and now is told (and accepts) that they are compatriots. He will clearly have to lose at least one of his beliefs about them, but which one? Or should he lose both the original beliefs; then should he at least retain that they are both from countries with Romance languages? Accommodating loss of beliefs has been not easy to handle in logical treatments of updating. Perhaps the advice to consider is that he should believe all and only what is relevantly implied by his information. That works even if his total information is made inconsistent by the addition.
Possibilities with worlds on the side
Lewis and Langford’s classic text distinguished the strict conditional “Necessarily, if A then B” from the ordinary conditional “If A then B”, in their creation of modern modal logic. That relation, between strict and ordinary, is intuitively also the relation between the conditional in relevance logic E and relevance logic R. (“E” for “entailment”, “R” for “relevant”.)
To elaborate the semantics for RI into one for EI Urquhart accompanies each possibility with a possible world. To determine the truth value of a sentence, he submits, “we may have to consider not only what information may be available, but also what the facts are”. So the new sort of model has as its elements pairs x, w, with x an element of the semilattice and w a member of a set of worlds. That set of worlds is equipped, in the familiar way, with a ‘relative possibility’ relation, which Urquhart stipulates to be reflexive and transitive.
Now the correct evaluation clause for the conditional is this:
v(A → B, x, w) = T iff for all y, and all worlds u which are possible relative to w,
if v(A,y, u) = T then v(B, x ∪ y, u) = T
No special symbol is introduced for necessity: “Necessarily A” is symbolized as “(A→ A)→ A”. The implicational fragment EI of E is sound and complete on this semantics.
How can we think of Urquhart’s world-coupled possibilities?
An obvious way is this: in the couple x,w the world w represents what is actually the case and the element x represents the information a certain privileged inhabitant has. Here “information” is to be read very neutrally: it may be true or false information, even inconsistent information, Now there may be a distinction in the language between purely factual statements, whose truth value is entirely determined by the world w and information-dependent statements whose truth value is at least in part determined by the possibility x. The conditionals are of the latter sort.
On this reading, a person who says, in sequence, “It rains” and “If it rains then it pours” is, by intention, if s/he understands her own language, first asserting that something is the case and then, after that, expressing his belief as to what things are really like in this vale of tears.
What we may regret
I have been hinting along the way that there may be interesting connections between Urquhart’s semantics for relevant logic and current discussions about conditionals – even if not immediately obvious. There are two disparities with current discussions, such as about epistemic modals, probabilities of conditionals, and the like. The first is that in the latter reference to roles for inconsistent information, let alone to relevance logics, is hard to find if not absent altogether. The second is that in the extensive literature concerning relevant implication that developed since the 1970s, reference to any natural language examples, let alone to work in philosophy of language, is scarce to the point of being negligible.
But the salience of similarities, between the exploration of liberal conceptions of possibilities and worlds suggest to me that they should perhaps be kept in mind.
SOURCES
Anderson, Alan R. and Nuel Belnap (1975) Entailment: The Logic of Relevance and Necessity. Princeton: Princeton University Press.
Humberstone, I. L. (1981) “From Worlds to Possibilities”. Journal of Philosophical Logic 10: 313-339.
Standefer, S. (2022). “Revisiting Semilattice Semantics”. In: Düntsch, I., Mares, E. (eds) Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs. Outstanding Contributions to Logic, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-71430-7_7
Urquhart, Alasdair (1972) “Semantics for Relevant Logics”. Journal of Symbolic Logic 37(1); 159-169.
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