negation

Modality and Negation before 1932

March 20, 2026March 20, 2026 Bas van FraassenLeave a comment

Lewis and Langford’s Symbolic Logic (1932) was the culmination of work over the previous half-century. This is a well-studied development, and I do not have anything new to say. But I would like to add some comments on the role of negation.

1. Exclusion, choice, and strong negation

When Gerrit Mannoury discussed Intuitionism he placed much weight on the distinction between exclusion negation and choice negation. The latter is a form of negating a proposition that presupposes a definite contrast class, with a meaning that is something like “not this, but one of the others”. For example, “the apple is not ripe” would typically be understood as asserting that the apple is a real apple in one of the stages before ripeness. The context would supply that definite contrast class. If we read that statement with “not” as exclusion negation, the contrast would be totally indefinite: the apple is raw, or it is rotten. or it is a fake porcelain apple, or it is a painted apple in a Rembrandt still life, or ….

That distinction shed some light on the Intuitionists’ approach to infinity. But Intuitionist negation, as it appears in Heyting’s logic, does not fit either. Within mathematics, as the Intuitionists present it, a statement can be negated only on the basis that it leads to a self-contradiction. In the logic, the negation of A is A → f, where f is the falsum, the absolute absurdity.

I will call this third form strong negation. Outside Intuitionism, or at least outside its original presentation in terms of proofs and refutations, we can explain it this way:

to assert ¬A, in the sense of a strong negation, is to assert that A is not possibly true.

Strong negation appears in modal logic generally, and specifically in Lewis’ early work.

However, the principles that govern strong negation are saliently different from the familiar ones characteristic of choice negation.

2. Choice negation is ‘classical’

In lattice theory the complement x’ of element x is the largest element y such that x ∧ y = 0 (‘x and y are disjoint’), if it exists.^[1] That it is the largest means that if z is any element disjoint from x then z ≤ x’. Clearly, if we have several negations in the language they cannot all be like that.

In view of the many options that have been broached on this subject, it may be best to mention some suggested characteristics in various treatments of negation, and to ask which are satisfied in specific cases.

I will call a negation classical if its corresponding operation ‘ on a lattice of propositions meets three conditions:

x ∧ x’ = 0 (disjointness)

(x’)’ = x (involution)

x ≤ y iff y’ ≤ x’ (antitone)

So when negation corresponds to an operation of that sort on a lattice of propositions, then negation obeys the principles

(A & ¬A) implies the falsum (Non-Contradiction)

¬¬A entails, and is entailed by A (Double Negation)

A entails B if and only if ¬B entails ¬A. (Contraposition)

Choice negation is classical. For example, if the contrast class is a range of colors then “the apple is not-not red” is true iff the apple has a color in this range that is not one of the colors other than red, hence is red. And, taking e.g scarlet to imply red, it is also the case that being other than red (in the intended range) implies being other than scarlet.

We may also note that if negation is defined by the two-valued truth-table, it is a choice negation, hence classical.

Is Exclusion negation classical? I’ll discuss that in the Appendix.

Strong negation is not classical. That the law of Double Negation does not hold in Intuitionistic logic is well known. Contraposition is closely related to Reductio ad Absurdum of which one form (negation elimination) fails in Intuitionistic logic. Below we’ll see how Contraposition fares in normal modal logic.

3. A quick note on Lewis’ relation to MacColl

Lewis’ theory of strong negation and the conditional is the same as MacColl’s, whose work he acknowledges. (Lewis 1918, p. 292: “The fundamental ideas of the system are similar to those of MacColl’s Symbolic Logic and its Applications”.)

I will go chronologically backward for a moment to discuss Lewis first and then the earlier work by MacColl.

(I will adapt their notation, or use my own, as convenient.)

4. C. I. Lewis chose strong negation as modal primitive

Lewis published his Survey of Symbolic Logic in 1918, and there presented his logic of strict implication with as modal primitive a strong negation connective ~.

Notation: I will now keep ¬ for just the truth-functional (material) negation.

The sentence ‘~p’ is to be read as ‘it is impossible that p’. Then the strict conditional → is defined by:

p → q =_def ~(p & ¬q)

Lewis’ strong negation is intuitively at least the same as the Intuitionist’s. For by the above definition,

p → (q & ~q) = ~(p & ¬ (q & ¬q)) = ~p

given that logically equivalent formulas are mutually substitutable everywhere (and in normal modal logic the falsum is equivalent to any self-contradiction).

But Lewis was not guided by any precise semantics for modality, and his vague intuitions led him astray. Specifically he assumed that strong negation obeys Contraposition. He had the, at first sight so innocent looking, principle

2.21 (~q → ~p) → (p → q)

Two years later Lewis published an emendation (Lewis 2020). Emil Post had pointed out to him that this principle led to the theorem

~p = ¬ p

thus collapsing strong negation into material negation. Lewis commented “Mr. Post’s example which demonstrates the falsity of 2.21 is not here reproduced, since it involves the use of a diagram and would require considerable explanation.”

With our current preference for □ as primitive, we understand his 2.21 as

2.21*. □(¬◊q ⊃ ¬◊p) → □(p ⊃ q)

In the Appendix I will sketch a possible world model that gives us a counterexample to 2.21* [Hint: focus on the case of ¬ ◊q being false.]

Dropping the ill-fated 2.21 did not do great damage to the logic of strict implication, which then eventually appeared in Lewis and Langford.

5. Oskar Becker to C. I. Lewis: nesting and iteration

But the theory of strict implication was not unchanged in other ways. In 1930 there had appeared Oskar Becker’s monograph (Becker 1930) which discussed iterations and nestings of modal operators, with questions about how they are to be understood. Lewis and Langford acknowledged this in their preface and wrote their famous Appendix II in response.

Becker had proposed the following principles as possible additions to Lewis’ logic:

C10. □□p = □p

C11. ◊p → □◊p

C12. p → □◊p (which Becker called the Brouwersche Axiom)

Lewis and Langford then defined five modal logics, S1-S5, in which (our familiar) S4 is S3 plus C10, and S5 is (our familiar) S4 plus C11 and C12.

But Becker’s questions about how C10.-C12. are to be understood were not very well answered. We’ll see below how MacColl made a determined effort to establish the truth conditions for one case of an iterated modality, one that involves negation.

6. Tracing it all back to Hugh MacColl

From a distance MacColl’s symbolism looks rather like arithmetic. He introduces five modalities: true, false, certain, impossible, and variable. Each has a Greek letter as its symbol, with ε (epsilon) for certainty and η(eta) for impossibility, for example.

But there seems at first to be a curious ambiguity. The proposition that A is certain or necessary is symbolized A^ε. But then we also see ε entering as a propositional constant, for the absolutely certain, the ‘top’ or ‘unit’ of the algebra.

I think we can understand that this is not just a notational quirk. In arithmetic, consider the sentences “3 + 2 = 5” and “3² = 9”. The numeral “2” appears first as the name of a number, and then, in superscript, as the name of the squaring function. We can give a good account of this: the symbol “3²” the denotes the value of a certain function (exponentiation) applied to the ordered couple <3, 2>. Superscripting is a convention to symbolize exponentiation.

modal statements, strong negation

In the same way we should understand MacColl’s formulation of modal statements. In his commentary and examples they are clearly taken as de dicto: the assertion that it is certain that A is the assertion that A has a certain modal property, namely certainty. The symbol “A^ε” denotes the proposition which is the value of a certain function (we may call it propositional exponentiation) applied to the ordered couple <A, ε>. The function is symbolized by the superscripting convention.

In the same way the assertion that A is impossible is symbolized A^η. This is strong negation, it is what Lewis later symbolized as ~A.

propositional constants

But impossibility too is itself a proposition, the ‘bottom’ of the algebra, and we have the laws (MacColl uses ‘ for ordinary negation and . for conjunction):

A . ε = A (ε is the unit, the top, the tautology)

A . η = η (η is the bottom, the absurdity)

(A . A’)^η (Law of Non-Contradiction)

A^ε . (A’)^ε = η

6.3 the strict conditional

MacColl defines a conditional:

A : B =_def (A . B’)^η (It is impossible that A and not B)

That is exactly the definition that Lewis then chose for his strict conditional. MacColl, who has the connective + for disjunction, points out that, equally, A : B = (A’ + B)^ε, it is certain that either not-A or B.

6.4 iterated and nested modal operators

Especially interesting is iterated propositional exponentiation, to symbolize nested modalities, a subject which, as we saw above, did not return in the literature till 1930. MacColl writes (section 9, page 7):

The symbol A^BC means (A^B)^C; it asserts that the statement A^B belongs to the class C, in which C may denote true, or false, or possible, &c. Similarly A^BCD means (A^BC)^D, and so on. (MacColl 1906: 7)

So for example, (A^ε)^η is the statement that it is impossible that A is certain: what we would write as ~◊□A.

MacColl returns to this in section 22, where he writes “But, it may be asked, what is meant by statements of the second, third, &c., degrees, when the primary subject is itself a statement?” What follows in that section, and in many other pasages, points to an understanding of logic as pertaining to information processing. There is the suggestion that, for example, A^η may be a revision if A, made when new data arrive that are incompatible with A. But I will leave that aside for now.

There is an interesting discussion of a nested modality in a later paper (MacColl 1910). Here he finds what he takes at first blush to be an antinomy. In his system the symbol θ stands for the modality variable, it is his term for what is possible but not certain, or equivalently, what is neither impossible nor certain. So it could be defined:

A^θ = (A^ε)’.(A^η)’

MacColl then writes:

The symbol A^θ^θ, in my system, is short for (A^θ)^θandasserts that the statement A^θ is a variable. The antinomy consists in the conflict of two arguments, of which the one professes to prove that the second-degree proposition A^θθ is an impossibility or self-contradiction; while the other professes to prove that it is not. (MacColl 1910: 196, with the initial “A^θ” corrected to “A^θθ”)

A first blush its natural reading looks entirely intelligible, or at least perfectly grammatical:

“It is neither certain nor impossible that it is neither certain nor impossible that A”.

But it becomes puzzling when we try to see under what conditions, or for what sort of statement A, this would be true or definitely false.

To prove the first option in the antinomy MacColm argues, first of all:

A will itself be either certain, or impossible, or neither certain nor impossible. If A is either certain or impossible then A^θ is clearly false.

That seems correct, but then MacColl argues:

if A is neither certain nor impossible then it is certain (and hence not neither certain nor impossible) that A^θ. And so again, A^θθ is false.

Therefore A^θθ is false under all conditions, hence impossible.

The reasoning about the case in which A is itself neither certain nor impossible, is not obviously correct. The inference appears to rely on an unstated modal principle, perhaps one as strong as the S5 principle, that statements asserting or denying certainty or possibility are certain if true. With such an assumption added, the first horn of the dilemma is established.

Accordingly, MacColl’s contrary argument, that A^θθ is possible, is not needed if we can push back to a prior question:

whether, or under what conditions, A^θ is certain if true.

The most reasonable attitude to take would seem to be that there are many modes of modality, pertaining to different subject matters – in some cases S5 is the correct logic and in other cases not.

7. APPENDIX.

[1] Is Exclusion negation classical?

Suppose we assert that the apple is not red, while intending no definite contrast whatsoever. One option may be to assert that nevertheless, all the ways the apple might or could be, ways that do not involve its being red, form a class. Then, just as for choice negation it will follow that not being in any of those ways implies being red.

This ostensibly simple view may have difficulties, with vagueness and the vagueness of vagueness, or with Russel-type paradoxes, or with more general questions about whether the class could or could not be a set.

If we are uneasy with the idea of reifying ways things might or could be as a definite class, then we will certainly begin to doubt inference by Reductio ad Absurdum (as Intuitionists do), and almost certainly Contraposition and Double Negation as well, whenever the negation in play is exclusion negation.

In a many-valued logic that defines the connectives in terms of a ‘many-truthvalue-table’, it is possible to have a negation that is not classical. For example, if ¬1 = 0, ¬2 = 1, ¬0 = 1 then ¬¬2 ≠ 2. We could reckon this as an exclusion negation (‘value other than the designated value’), though a simple case. In a language with truth-value gaps, where conjunction and disjunction are typically not functional (not compositional), it may also be possible to have a non-functional negation.

It is perhaps a failing in formal semantics to take negation (in our language in use) for granted as understood and unequivocal.

[2] Counterexample to Lewis 2.21: (~q → ~p) → (p → q) in normal modal logic

In our preferred notation, and assuming the Duality □¬ = ¬◊, we can write 2.21 as

*. □{□(¬ ◊q ⊃ ¬ ◊p) ⊃ □(p ⊃ q) }

Recall that in a normal modal logic possible world model M = <W, R>, the sentence □A is true at world w in W iff A is true in all worlds in R(w).

I will sketch a model, describing only just enough to show that it provides a counterexample to *.

The worlds w₁, w₂, w₃, w₄ in W are related as follows:

w₂ is in R(w₁)
w₃ and w₄ are in R(w₂)
For any world w in W, if w is in R(w₂) then R(w) = R(w₂)
p and ¬q are true in w₃
p and q are true in w₄

Argument

q is true in w₄
◊q is true in all members of R(w₂)
(¬ ◊q ⊃ ¬ ◊p) is true in all members of R(w₂)
□(¬ ◊q ⊃ ¬ ◊p) is true in w₂
(p & ¬q) is true in w₃
□(p ⊃ q) is false in w₂ since w₃ is in R(w₂)
□(¬ ◊q ⊃ ¬ ◊p) ⊃ □(p ⊃ q) is false in w₂, by 4. and 6.
□{□(¬ ◊q ⊃ ¬ ◊p) ⊃ □(p ⊃ q) } is false in w₁ since w₂ is in R(w₁)

Note that, for simplicity, I made clause c. stronger than it needs to be to justify line 2. Apart from that I have left <W, R> with as little constraint as possible.

8. REFERENCES

Becker, Oskar (1930) “Zur Logik der Modalitäten”. Jahrbuch für Philosophie und Phänomenologische ForschungXI: 497-548.

Gabbay, Dov M. and John Woods (2006) Handbook of the History of Logic. Vol. 7: Logic and the Modalities in the Twentieth Century. Amsterdam: Elsevier.

Lewis, Clarence I. and C. H. Langford (1932) Symbolic Logic. New York: The Century Company.

Lewis, Clarence Irving (1920) “Strict implication – an emendation”. The Journal of Philosophy, Psychology, and Scientific Methods 17: 300-302.

MacColl, Hugh (1906) Symbolic Logic and ‘its Applications. London: Longmans, Green and Co.

MacColl, Hugh (1910) “Linguistic misunderstandings (I)”. Mind, n.s., 19: 186–199.

Read, Stephen (1998) “Hugh MacColl and the algebra of strict implication”. Nordic Journal of Philosophical Logic 3: 59-84.

Wolenski, Jan (1998) “MacColl on Modalities”. Nordic Journal of Philosophical Logic 3:133-140.

^[1] In a Boolean algebra it exists, is unique, and x ∧ x’ = 0, x v x’ = 1.

On Tarski’s Calculus of Systems (2)

January 21, 2026 Bas van FraassenLeave a comment

The relation between negation and infinity loomed large in the Intuitionist critique of mathematics. But when we come to Intuitionistic logic, it turns out to be all about the conditional. Negation comes in just by means of a definition: ~A =_def A → f.

It is in the logic’s realizations that the intimate relation between negation and infinity comes to the fore. Recall from the previous post that the lattice of theories of any distributive logic is a complete Heyting lattice with zero element Cn(𝜙), and

T → T’ is the weakest (largest) theory X such that T ∩ X ⊆ T’;

T → T’ = ⊕{T’’: T ∩ T’’ ⊆ T’)

The pseudo-complement ¬T of T, if it exist, must be the weakest element X which is contrary to T, that is, it is such that T ∩ X ⊆ Cn(𝜙). So we instantiate the above: ¬ T is the weakest element X such that T ∩ X ⊆ Cn(𝜙)), hence:

¬ T = T → Cn(𝜙). Equivalently, ¬ T = ⊕{T’: T ∩ T’ ⊆ Cn(𝜙)}.

So much, so general. How does it happen that the Laws of Excluded Middle and of Double Negation fail in Intuitionistic logic, and their corresponding principles fail in the calculus of systems?

To continue, let L be classical propositional logic, with & and ~ as primitive connectives and let S be the set of all sentences. Thus S = Cn(S) is the unit (top) of the lattice. For brevity, if A is a sentence I will write “Cn(A)” for “Cn({A})”.

Note: Contrariety, the condition T ∩ T’ ⊆ Cn(𝜙), is different from mutually inconsistency. With p, q atomic sentences, Cn(𝜙) is contrary to but consistent with Cn(p), Cn(p) and Cn(~p) are both mutually inconsistent and contrary. But Cn(p & q) and Cn(~p & q) are mutually inconsistent, their join is S, but not contrary for their intersection contains non-theorem q. Contrariety is this:

Theorem 1. T ∩ T’ ⊆ Cn(𝜙) iff for every non-theorem A in T there is a consistent extension of T’ that includes ~A.

To prove this we can appeal to the general features of theories in classical logic:

any consistent theory is part of a maximal consistent theory
every theory is the intersection of all its maximal consistent extensions
for all sentences A, if T is a maximal consistent theory then T contains either A or ~A
if T does not imply A then T has a maximal consistent extension that includes ~A

Proof.

Let A be any L-non-theorem in T. Suppose that T’ has no consistent extension that includes ~A. Then A is in all the maximal consistent extensions of T’, hence in T’, hence in T ∩ T’, which is thererfore not included in Cn(𝜙).
Suppose that for every L-non-theorem A in T, T’ has a consistent extension which includes ~A. Then if A is an L-non-theorem in T, A is not in T’, and hence not in T ∩ T’. If A is an L-non-theorem that is not in T then it is also not in T ∩ T’. Therefore T ∩ T’ contains only L-theorems, and is Cn(𝜙).

Lemma 1. Cn(~A) ⊆ ¬Cn(A)

If E is in Cn(A) there is a sentence B such that E is equivalent to A v B. For each such sentence Cn(~A) has a consistent extension that includes ~(A v B).

If ~A entails B then it too entails A v B, so then A v ~A entails A v B, hence A v B is an L-theorem.
If ~A does not entail B then Cn(~A) has a maximal consistent extension that includes ~B, which includes ~A & ~B.

So by Theorem 1, Cn(A) ∩ Cn(~A) ⊆ Cn(𝜙). Hence Cn(~A) ⊆ CnA) → Cn(𝜙) = ¬Cn(A).

Theorem 2. If T is finitely axiomatizable then T Θ ¬T = S

Because of the classical conjunction rules, if T is finitely axiomatizable then there is a sentence A such that T = Cn(A). By the lemma, Cn(A) Θ Cn(~A) ⊆ Cn(A) Θ ¬Cn(A). But Cn(A) Θ Cn(~A) = Cn(Cn(A) ∪ Cn(~A)) = S.

It follows then that violations of Excluded Middle can only be by theories that are not finitely axiomatizable.

To pursue this we need to improve on the above lemma.

Lemma 2. T ∩ T’ ⊆ Cn(𝜙) iff for every non-theorem A in T there is a maximal consistent extension of T’ that includes ~A.

This follows at once from Theorem 1.

Lemma 3. If T ∩ T’ ⊆ Cn(𝜙) then T’ ⊆ ∩ Cn({~A: A is in T})

Suppose T ∩ T’ ⊆ Cn(𝜙) and that A is an L-non-theorem of T. Let KT’(A) be the set of all consistent extensions of T’ that include ~A. So ∩KT’(A) ⊆ Cn(~A). Let MT’ be the set of all maximal consistent extensions of T. Then for each L-non-theorem A of T:

T’ = ∩MT’ ⊆ ∩KT’(A) ⊆ Cn(~A).

Since T’ has a consistent extension that includes ~A for each L-non-theorem in T it follows that T’ ⊆ ∩{Cn(~A): A is an L-non-theorem in T}. From this the lemma follows because if A is an L-theorem then Cn({~A}) = S, which has no effect on the intersection.

Theorem 3. ¬T = ∩({Cn{~A}: A in T}).

First, it is clear that for every L-non-theorem A in T, ∩({Cn{~A}: A in T}) has a consistent extension that includes ~A. Therefore ∩({Cn{~A}: A in T}) ⊆ ¬T by theorem 1 and definition.

Secondly, T ∩ ¬T ⊆ Cn(𝜙), so by Lemma 3, ¬T ⊆ ∩ Cn({~A: A is in T})

Corollaries: Cn({~A}) = ¬Cn(A), ¬Cn(𝜙) = S, ¬S = Cn(𝜙)

This theorem gives us a way to identify the pseudo-complement of theories that are not finitely axiomatizable.

Example. Let AT be the infinite list of atomic sentences p₁, …, p_m , … and T = Cn(AT). Let AT_fin be the set of finite conjunctions of atomic sentences. Every member of T is equivalent to a finite conjunction C of atomic sentences disjoined with some other sentence A (e.g. p v q).

So ¬T = ∩ Cn({~(C v A): C in AT_fin})

As an example, consider the L-non-theorem (p₁ & ~p₂). Could it belong to ¬T? It does not belong either to Cn(~p₁) or to Cn(~p₂). Nor does it belong to Cn(~q) for any q in AT_fin other than p₁ or p₂.

We can generalize this reasoning to cover any L-non-theorem. Since L is classical propositional logic, we can think in terms of truth-table rows or their generalization:

a finite state-description is a consistent conjunction p^*₁ & …& p^*_m of atomic sentences each of which either has or does not have an appended negation sign.

In classical propositional logic, A is an L-non-theorem if and only if there is a finite state-description B such that B├_L~A. For any such B, since B is finite and AT is infinite, there is an atomic sentence q which does not appear in B. But then B is not in Cn(~q), hence not in ¬T. Therefore there is no derivation of ~A in ¬T. Hence ¬T = Cn(𝜙).

This gives us a counterexample to both Excluded Middle and Double Negation, with T = Cn(AT):

T ⊕ ¬T = T and T ≠ S

¬ ¬T = ¬ Cn(𝜙) = S and S ≠ T

The following analogy strikes me as apt. Remember that ¬T is also the join of its contraries. There are uncountably many infinite state descriptions, all inconsistent with each other. So we can look to continua for analogies. In a geometric space, the join of a family of subspaces is the least subspace that contains all. Suppose P is a plane, its subspaces are the straight lines through the origin. Take away one such straight line: the join of the remaining ones is still the entire plane.

Probability statements (2) Compounds

May 10, 2020 Bas van FraassenLeave a comment

There is never any difficulty in adding truth-functional connectives to a set of statements, but doing so doesn’t give us any new insight into their character or structure. The better approach is to ask: are there, in this very class of statements itself, already ones that count as conjunctions, disjunctions, and the like? (What is the ‘internal’ logic of this sort of discourse?)

Conjunction

A simple example is (P(A) > x & P(A) < y), which can also be written as: P(A) ε (x, y). But that is the special case, of two statements about the probability of a single proposition. What about such combinations when different propositions are involved?

Theorem: If C is a family of convex sets (finite, countable or uncountable), then the intersection of the members of C is a convex set.

Proof: That the intersection ∩C is convex is trivially so if the intersection is empty, or has just one member. If ∩C has more than one member, consider any two p, p’ of its members: these belong to each member of C, and hence any of their convex combinations are also members of each member of C. Hence all the convex combinations belonging to all members of C, and hence of ∩C, are in ∩C.

This Theorem shows that it is fine to introduce the usual sort of conjunction in to the language, for then the set of measures that satisfy both of two elementary statements will also be convex. So if Q and R are elementary statements then (Q & R) is the statement such that |Q&R| = |Q| ∩ |R|, and this is again an elementary statement.

Disjunction

The same ease is not to be found for disjunction. The union of two convex sets is not in general convex. Anyway, we already know that disjunction does not behave like a truth-function when it comes to probability. In fact, it does not make sense to ask whether p satisfies (P(A) = r or P(A) = s) , as opposed to asking whether it satisfies either (P(A) = r) or satisfies (P(A) = s). At most we can ask whether p satisfies what the two ‘have in common’.

Can we find an operation on elementary statements that has the main characteristics we require of disjunction, in general?

Requirement. The general concept of disjunction of two statements, Q, R, in any kind of language, requires that it must be the logically strongest statement that is implied by both Q and R, and thus itself implies all that is implied by both Q and R.

In the first post I defined entailment for elementary statements. What we should look at therefore is this situation. Suppose that S is a statement S

Q entails S and R entails S

What is the relation that |S|bears to |Q| and |R|?

Theorem. If Q, R, S are elementary statements, Q entails S, and R entails S, then all convex combinations of members p of |Q| and p’ of |R| belong to |S|.

For note that if Q and R entail S then both |Q| and |R| are part of |S|. Therefore, iff is in |Q| and p’ is in |R| then both belong to |S|. Since |S| is convex it will also contain all the convex combinations of p and p’.

The smallest set that fits the Requirement above is therefore the convex hull of the union |Q| ⋃ |R|, that is the set of all convex combinations of members of those two sets. That is the smallest convex set which contains both. Since that is more than just the union, it does not correspond to a truth-functional disjunction. So let’s introduce a special symbol:

Definition. The join of convex sets X and Y is (X ⊕ Y) = {ap +(1-a)p’: a ε [0,1], p in X, p’ in Y}.

That is precisely the convex hull of X ⋃Y. Following upon this we can introduce a statement connective of ‘disjunction’ to the language, which will combine elementary statements into other elementary statements. Without expecting any confusion from this, I will use ⊕ equally for the statement connective and for the operation on convex sets:

| Q ⊕ R| = |Q| ⊕ |R|.

Really, there is no negation. In a specific case we can make up the negation, but it will typically not be an elementary statement. For example what would be the negation of P(A) = 0.5?

Its contraries are P(A) < 0.5 and P(A) > 0.5. Each of these is an elementary statement. But there is no truth-functional ‘or’ that would combine them into the contradictory of P(A) = 0.5, at least not one that would produce an elementary statement.

The sort of disjunction we do have produces something, but not the contradictory of P(A) = 0.5. In fact,

(P(A) < 0.5) ⊕ (P(A) > 0.5)

is satisfied by the 50/50 convex combination p” of p and p’ which assign 0.25 and 0.75 to A respectively, and p”(A) is 0.5. So we have arrived at a tautology, this disjunction has the same semantic value as |P(A) ε [0,1]|.

The underlying reason is of course that there is no largest convex subset of [0,1] disjoint from [0.5]. The two convex sets disjoint from [0.5] are [0.0.5) and (0.5,1] which are as large as can be, so there is no largest.

New Question: are there other forms of that elementary statements can have? What about other sorts of probability talk, such as about odds or conditional probabilities?

That will be the topic of the next post.