Urquhart: semilattices of possibilities (2)

What exactly is so different and difficult about relevance logic? 

I will illustrate with Urquart’s 1972 paper that I discussed in the previous post. I’ll assume my post was read, but won’t rely on it too much. 

1.   A parting of the ways: two concepts of validity

At first, we feel we are on familiar ground. The truth condition for conditionals Urquhart offers is this:

v(A → B, x) = T iff for all y, if v(A, y) = T then v(B, x ∪ y) = T

We see at once that Modus Ponens is valid.  For if A → B is true at x, and A is true at x , then B is true at  x ∪ x, for that is just x itself.

But used to the usual and familiar, we’ll have one puzzle immediately

This semilattice semantics yields as valid sentences precisely the theorems of the implicational fragment of R.  

The first axiom of that logic is A → A.  So, what about v(A → A, x)?   

We might think that it must be T because if v(A, y) = T then v(A, x ∪ y) = T, because x ∪ y has in it all the information that x had and more.  But that is not so!  Urquhart points out emphatically that 

the information in x ∪ y may not bestow T on A, because dragging in y may have dragged in irrelevant information.

So A → A is supposed to be a valid sentence although it does not receive T from all valuations?  

Right!  The criterion of validity is a different one:  

a sentence is valid if there is an argument to it from no premises at all, from the empty set of premises. 

 (Being used to the usual and familiar, we would have thought that the two criteria would coincide …)

So Urquhart’ semilattice has an zero, 0, which is the empty piece of information.  And A is valid if and only if v(A, 0) = T for all evaluations v.

And the join operation obeys the semi-lattice laws, so for example (x ∪ 0) = (x ∪ x) = x.  

Now we can see that the first axiom is indeed valid.  The condition for A → A to be valid is the tautology: if v(A, 0) = T then for all x, if v(A, x) = T then v(A, x ∪ 0) = T.

So within this approach:

That a sentence is valid does not imply that it is True in every possibility.  Valid conditionals, specifically, are False in many possibilities.

And that is a significant departure from how possibilities are generally understood, however different they may be from possible worlds.

But it is right and required for relevance logic, where valid sentences are not logically equivalent.  In general A → A does not imply, and is not implied by B → B, since there may be nothing relevant to B in what A is about.

2.   Validity versus truth-preservation

What happens to validity of arguments?  The first, and good, news is that Modus Ponens for → is not just validity-preservation but truth-preservation, in the good old way, as I mentioned above.

Btu after this, in relevance logic we will depart from the usual notion of valid argument.  We can have instead:

The argument from A₁, …, A_n to B is valid (symbolically, X =>> A) if and only if A₁ →. →. A_n → B is a valid sentence.

That is different from our familiar valid-argument relation.  Some characteristics are the same.

By Urquhart’s completeness theorem, the valid sentences are the theorems of the implicational fragment of R.  This logic has, apart from the rule of Modus Ponens, the axioms:

1. A → A
2. (A →. A → B) → (A → B)
3. (A → B) → .(C → A) → (C → B)
4. [A → (B → C)] → [B → (A → C)] 

By axiom 4), the order of the premises does not matter.  Therefore =>> is still a relation from sets of sentences to sentences.    So, for example, the argument from A and B to C is valid exactly if A → (B → C) is a valid sentence, which is equally the case if B → (A → C) is a valid sentence.

But there are crucial differences.

3.   What it is to be a sub-structural logic

This consequence relation =>> does not obey the Structural Rules, and the consequence operator is not a closure operator.

Let  X╞A mean that the argument from sentences X to sentence A is valid: the semantic entailment relation.  The Structural Rules (which are the rules that can be stated without reference to specific features of the syntax) are these:

Identity            if A is in X then X╞A

Weakening      If X ⊆ Y and X╞A then Y╞A

Transitivity      If X╞A for each member A of Y and Y╞ B then X╞ B

 The corresponding semantic consequence operator is defined:  Cn(X) = {A: X╞A}.  If the Structural Rules hold then this is a closure operator. 

In relevance logic, Weakening is seen as a culprit and interloper:  extra premises may bring irrelevancies, and so destroy validity.  

And the new argument-validity criterion above does not include Weakening.  If A, …, N =>> B it does not follow that C, A, …, N =>> B.  

Here is an extreme example, that actually throws some doubt on the motivating intuitions about the role of irrelevancy.  Even A does not in general entail A → A in this sense. For:

v(A → (A → A), 0) = T only if:

for all y, if v(A, y) = T then v(A →A, y ∪ 0) = T, 

  …… then v(A → A, y) = T,

    …….. then for all z, if v(A, z) = T then v(A, y ∪ z) = T

and that does not follow at all.  For A’s being true at z and at y is no guarantee that A will be true at  y ∪ z.

So even A → (A → A) is not a valid sentence form.  

This can’t very well be because A has too much information in it, irrelevant to A → A. 

Rather, the opposite: A → A has too much information in it to be concluded on the basis of A.  We have to think of valid conditionals as not being ‘empty tautologies’ at all, but as carrying much information of their own.

4.   Attempting to look at this algebraically

In subsequent work referring back to Urquhart’s paper the focus is on the ‘join’ operation, and the approach is called operational semantics.  But the structures on which the models are based are still, unavoidably, semilattices. 

The properties of the join operation are these: it is associative and commutative, but also idempotent (x ∪ x = x), and 0 is the identity (x ∪ 0 = x).  So far this amounts to a semigroup.  But there is a definable partial order: relation x ≤ y iff x ∪ y = y is a partial ordering:

x ∪ x = x, so x ≤ x

        if x ∪ y = y and y ∪ z = z then x ∪ z = (x  ∪ (y ∪ z) = (x ∪ y) ∪ z = y ∪ z = z ; so ≤ is transitive.

This partial order comes free, so to speak, and that makes it a semilattice.

Can we find some ordering directly related to truth or validity?

Relative to any specific evaluation v we can see a partial order R in the sentences defined by:

ARB iff v(A → B,0) = T

Then we see that:

R is idempotent:  ARA

R is transitive: if v(A → B, 0) = T and v(B → C, 0) = T then v(A → C, 0) =T.

For suppose for all y, if v(A, y) = T then v(B, y ∪ 0) = T. Suppose also that all y, if v(B, y) = T then v(C, y ∪ 0) = T. Since y ∪ 0 = y, we see that for all y, if v(A, y) = T then v(C, y ∪ 0) = T.

So R s a partial order, defined in terms of truth-conditions, to be discerned on the sentences, relative to a valuation.  

But trying to find a connection between this ordering of sentences relative to v, and the order in the semilattice, we are blocked.  For example, define

If A is a sentence and v an evaluation then [[A]](v) = {x: v(A,x) = T} is the proposition that v assigns to A. 

The proposition that v assigns to A will most often not have 0 in it, so it is not closed downward.  Nor is it closed upward, for if x is in [[A]](v) it does not follow that x ∪ y is in [[A]](v). So the propositions, so defined, are neither the ideals nor the filters in the semilattice.

I have a feeling, Toto, that we are not in Kansas anymore …….

REFERENCES

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.