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.
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