Conundrum

May 27, 2010

Leonard & Goodman’s Axioms: Antisymmetry = Extensionality

In my PQ paper, I made the claim that the antisymmetry of parthood is intimately tied up with extensionality principles in mereology. I made some informal arguments (that were perhaps not as strong as I would have liked). I basically said that antisymmetry is like an extensionality principle for improper parthood. I also argued that rejecting antisymmetry causes failures of all extensionality principles, and that’s, in effect, all that one loses since we can recapture classical mereology given certain axiom sets.

Some axiomatizations of classical mereology are much more explicit about this than others. In fact, the standard axiomatization (Partial Order + Strong Supplementation + Unrestricted Fusions) actually hides this fact quite well, so much so that people have tended to blame Strong Supplementation for extensionality. But in fact, even Weak Supplementation generates extensionality principles in the presence of Tarski/Lewis style fusions and antisymmetry.

Leonard and Goodman’s axiomatization is actually quite elegant, rivaled perhaps only by Tarski’s axiomatization (Transitivity + Uniqueness of Fusions). And moreover, it is extraordinarily explicit about the role antisymmetry is playing in extensionality principles.

Leonard and Goodman take disjointness ( $\wr$ ) as their mereological primitive. Let’s start with some definitions:

Parthood: $x \leq y := \forall z ( z \wr y \rightarrow w \wr x)$
Proper Parthood: $x < y := x \leq y \wedge x \neq y$
Overlap: $x \circ y := \exists z (z \leq x \wedge z \leq y)$
Fusion: $\mathsf{Fu}(t,\varphi(x)) := \forall y (y \wr t \leftrightarrow \forall x (\varphi(x) \rightarrow y \wr x)))$

Here, x is a part of y whenever anything disjoint from y is also disjoint from x. One thing to not is that there is quantification built into the definition of parthood here. Proper parthood is normal (although all the same issues will come up when rejecting antisymmetry). Overlap is normal as well. L&G fusions are a lot like standard fusions, except it’s done in terms of disjointness instead of overlap (in fact, they’re equivalent given Axiom 2 below).

Now for the axioms:

Antisymmetry: $(x \leq y \wedge y \leq x) \rightarrow x = y$
Duality: $x \circ y \leftrightarrow \neg x \wr y$
Unrestricted Fusion: $\exists x \varphi(x) \rightarrow \exists z \mathsf{Fu}(z, \varphi(x))$

(1)-(3) guarantee classical mereology. But look carefully at the first axiom. A little unpacking shows that it is equivalent to:

$\forall z ( z \wr x \leftrightarrow z \wr y) \rightarrow x = y$

But that is just explicitly an extensionality principle for disjointness. Indeed, in the presence of the second axiom, we can contrapose the extensionality of disjointness to get the extensionality of overlap:

$\forall z (z \circ x \leftrightarrow z \circ y) \rightarrow x = y$

And again this is pretty uncontroversially problematic for the antiextensionalist. Just look at the LRD of the extensionality of overlap:

$\forall z (z \circ x \rightarrow z \circ y) \rightarrow x \leq y$

Weakening gives us:

$\forall z (z \leq x \rightarrow z \circ y) \rightarrow x \leq y$

Contraposing again:

$x \nleq y \rightarrow \exists z (z \leq x \wedge \neg z \circ y)$

But that’s just the Strong Supplementation axiom, the axiom that has gotten so much bad press for extensionality. On L&G’s definitions and axiom 2, antisymmetry implies strong supplementation. None of this is terribly deep. And if you work much in this area, you know that you require antisymmetry to prove extensionality principles from supplementation principles. But this fact can be hidden — and L&G’s axiomatization is a pretty clear way to draw this out.

May 3, 2010

Yep. Done.

As most of you already know, I defended my dissertation on Friday and passed. So, I’m officially Dr. Cotnoir. The defense went as well as I could have expected, including some killer questions.

A big thanks to my committee: Marcus Rossberg, Don Baxter, Achille Varzi, and Jc Beall (my advisor). In addition, I also want to thank the entire UConn philosophy faculty; but in particular, Tim Elder, Sam Wheeler, Lionel Shapiro, and Michael Lynch. Thanks also to Reed Solomon, a mathematician (but in my book, a philosopher nonetheless). Moreover, thanks to all my fellow grad students, past and present; but in particular, Colin Caret and Doug Owings.

As announced previously, my next step is to take up a postdoc at NIP. I’ll also be posting at their blog from time to time. Check it out!

April 19, 2010

Greenough’s Logic of Indeterminacy

Patrick Greenough has been visiting UConn for the last few weeks. He recently gave a talk to the Logic Group on deflationism and truth value gaps. He argued, roughly, that deflationists about truth cannot also be gap theorists about vagueness because gap theory is inconsistent with a certain sort of transparency platitude about truth. I’m not going to give that argument here, even though I think it’s a good argument. Along the way, however, Patrick put forward a new gappy logic that has some nice advantages over the typical gap theorist logics like Strong Kleene and Łukasiewicz’s logics.

The logic is really cool; and as far as I can see, it’s new. His presentation relied on proof theory mostly, although he did give some truth tables to get the feel of it. I wanted to write up a full semantics for it and then draw out some disadvantages of the logic. There is another logic that is extremely similar to Patrick’s that avoids these disadvantages, but unfortunately fails to yield the full T-scheme. As a result, I think this shows that the disadvantages of Patrick’s logic are actually necessary for achieving his desiderata.

The basic desiderata:

We want the T-biconditionals to be provable for all sentences; and the F-biconditionals (A is false iff not-A) to be provable too.
We want there to be truth value gaps.
We want to be able to (truly) express that there are truth value gaps using negation.

Let our set of values be $\{1, .5, 0\}$ . Let conjunction, disjunction, and negation be treated along Strong Kleene lines:

$\nu(\neg A) = 1 - \nu(A)$
$\nu(A \land B) = min\{\nu(A), \nu(B)\}$
$\nu(A \vee B) = max\{\nu(A), \nu(B)\}$

Of course, the material conditional $A \supset B := \neg A \vee B$ is terrible for truth theories wanting the Tarski biconditionals. It’s also not really a conditional since gappy sentences give failures of $\vdash A \supset A$ .

So, let’s add a conditional ( $\rightarrow$ ) that behaves better than that. Ordinarily, the main option is the Ł3 conditional, but Patrick’s not going there. Instead, he proposes the following (actually, I’m reading this clause off his truth table):

$\nu(A \rightarrow B) = \left\{ \begin{array}{l} 1 \mbox{ if } \nu(A) \neq 1\\ \nu(B) \mbox{ otherwise }\end{array} \right.$

This conditional doesn’t contrapose. But Patrick’s happy with that; he thinks Dummett’s arguments against truth value gaps essentially rely on contraposition.

Further, a gap theorist should use negation to characterize their theory. That is, one ought to be able to say of a gappy sentence that it is ‘not true and not false’ where ‘not’ here is just ordinary negation (in this case the De Morgan negation of Strong Kleene). In order to do that, one needs a strong truth operator:

$\nu(T(A)) = \left\{ \begin{array}{l} 1 \mbox{ if } \nu(A) = 1\\ 0 \mbox{ otherwise }\end{array} \right.$

Actually, using Patrick’s conditional, one can define a different (non-de Morgan) negation, and use it to define strong truth.

$T(A) := \neg( A \rightarrow \bot)$
$F(A) := T(\neg A)$
$N(A) := \neg T(A) \land \neg F(A)$ .

These definitions are equivalent to Patrick’s truth tables for strong truth, strong falsity, and gaps.

An argument from $X$ to $A$ is valid whenever there is no assignment $\nu$ s.t. $\nu(x) = 1 \ \forall x \in X$ and $\nu(B) \neq 1$ . Patrick claims the proof theory of the logic nice: you get a full deduction theorem plus his conditional satisfies conditional proof.

There are, however, some weird features of the logic. First, the only place Patrick’s arrow differs from the Ł3 one is the fact that $.5 \rightarrow 0 = 1$ . That’s kind of weird, but the motive is supposed to be that we’re concerned with truth-preservation, and not simply any drop in truth-value. Plus, given the above definition of validity, one gets a deduction theorem. I’m not convinced, however. It does seem to me that we care about drop in truth value precisely because we care about falsity preservation backwards. This is precisely what is at issue when we are reasoning via modus tollens, which fails for Patrick’s conditional.

Another oddity is the biconditional. The biconditional differs from the Ł3 in the following cases: $.5 \rightarrow 0 = 1$ and $0 \leftrightarrow .5 = 1$ . Patrick glosses that we are primarily concerned with parity of truth, and that we don’t really care about disparity of non-truth. I’m not so sure; one of the main issues is that substitutivity of equivalents will fail everywhere contraposition does. The Ł3 conditional does not have this problem.

There is a logic very near by that does not have these problems. Recall that the Łukasiewicz arrow is the residual of t-norm conjunction. Patrick’s arrow, on the other hand, is strikingly close to the residual of standard ( $min$ ) conjunction.

$\nu(A \Rightarrow B) = \left\{ \begin{array}{l} 1 \mbox{ if } \nu(A) \leq \nu(B) \\ \nu(B) \mbox{ otherwise }\end{array} \right.$

This arrow is actually the conditional from Godel-Dummett logic and has an important relationship to the intuitionistic conditional. The only difference between the Ł3 arrow, Patrick’s $\rightarrow$ , and $\Rightarrow$ is in the case mentioned above: $.5 \Rightarrow 0 = 0$ .

Indeed, the conditional satisfies modus tollens on Patrick’s definition of validity. Moreover, the biconditional is nicer, since the biconditional is never true when the truth values differ. Because the biconditional is evaluated as true only when the LHS and RHS have the same value, we get the substitutivity of equivalents.

We also get definitions of true, false, and neither for free.

$F(A) := A \Rightarrow \bot$
$T(A) := F(\neg A)$
$N(A) := \neg T(A) \land \neg F(A)$

Falsity here is essentially Godel-Dummett negation, which is a variant on intuitionistic negation. In fact, if we extended the set of truth values to the [0,1] interval in the real line with the above semantics, we can show that the logic is Godel-Dummett’s logic extended with De Morgan negation (for more on the logics in this area, see this cool paper by Hajek et. al.).

One issue is that we now have contraposition; which was desired to fail to get around Dummett’s argument against gaps. One could redefine validity accordingly: an argument is valid just when the minimum value of the premises is less than or equal to the value of the conclusion. This will cause contraposition to fail (just look at $1 \Rightarrow .5$ and $.5 \Rightarrow 0$ ). It will also (I think) give us a deduction theorem.

Unfortunately, however, using this definition of validity will again cause us to lose modus tollens (just look at the case where $\nu(A)= 1$ and $\nu(B) = .5$ ; the premises will both be .5 and the conclusion 0).

There is an even bigger problem lurking in the background, however. We don’t get the T-biconditionals using $\Rightarrow$ . The LRD is no problem. But when $\nu(A) = .5$ we know that $\nu(T(A)) = 0$ by the semantics for truth; hence, the conditional $A \Rightarrow T(A) = 0$ .

All this is to say that the weirdness surrounding the conditional (and biconditional) appears to be necessary for guaranteeing the desiderata in the current setting.

April 8, 2010

Quantification for Alethic Pluralists

I’ve talked a bit about alethic pluralism before, but in the earlier post I was just thinking about it propositionally. However, there are interesting issues that arise when you start thinking about quantification. If you’re an alethic pluralist, you might want to consider also being a satisfaction pluralist too. This has has been suggested independently by Gila Sher (here) and Stewart Shapiro (here), and I think it’s a natural thought. The basic idea is that we have different truth properties for each domain of inquiry because we have different satisfaction relations for each domain of inquiry. There are different ways an object can satisfy a predicate, depending on what you’re talking about. It means something very different to satisfy the predicate “is funny” than it does to satisfy the predicate “is prime”.

But what, exactly, would satisfaction pluralism look like? I think getting clear on this question leads to a couple of interesting new problems for the alethic pluralist to handle, and gets at the major questions at the heart of what alethic pluralism is.

Here’s a very simple way to think about modeling satisfaction pluralism. The initial idea is adapted from an old trick from Van Bentham. We approximate some of first-order quantification by reinterpreting propositional modal logic. Here’s the rough picture:

Take the semantics for classical normal propositional modal languages (ignore which access relation we have at the moment). Now, try to wrap your mind around thinking of propositions as one-place predicates expressing properties. We can think of worlds as objects in our domain. So, a proposition being true-at-a-world corresponds to an object satisfying a monadic predicate. We’re just exploiting the fact that the formal structure here is identical: propositions are sets of worlds, properties are sets of objects. Now, in the modal language, we have modal operators, which are really just quantifiers over (accessible) worlds. Since we’re thinking of worlds as objects, $\Box$ just amounts to $\forall$ and $\Diamond$ is really just doing the work of $\exists$ .

Got the picture? Now, how does this help with satisfaction pluralism? Well, that’s where access relations come in. Let’s think of domains as being individuated via collections of objects. In other words, each area of inquiry is primarily about specific kinds of things. So, we can think of the access relation in our models as expressing a “same kind of object” relation. Indeed, I think it makes sense to impose that access is an equivalence relation, effectively partitioning our set of objects into ‘domains’.

So our models will be Kripke frames: $\langle W, R, \nu \rangle$ where $W$ is a non-empty set of objects, $R \subseteq W \times W$ is our access-relation, $\nu : W \times \{p, q, r \ldots\} \rightarrow \{1, 0\}$ is a set of object-relative satisfaction relations. That is, $\nu$ takes us from an object-property pair, and tells us whether the property is ‘true-of’ that object.

Semantics for all the usual connectives will be ordinary. Stipulating that access is reflexive, symmetric, and transitive, we can exploit this to get domain-relative quantifiers $\boxminus$ and $\ominus$ .

$\nu(w, \boxminus A) = 1$ iff for all $w'$ such that $wRw', \ \nu(w', A) = 1$ .
$\nu(w, \ominus A) = 1$ iff for some $w'$ such that $wRw', \ \nu(w', A) = 1$ .

These guys will only range over the accessible objects. So when you’re quantifying over these objects, you’re always staying within the same equivalence class — staying within a domain. And we won’t vary the models in the way typical for modal logic — there is one “intended” access-relation that expresses truths about which kinds of objects are in the same domain.

We can also introduce domain independent quantifiers $\Box$ and $\Diamond$ which will get the typical all-worlds semantics.

$\nu(w, \Box A) = 1$ iff for all $w', \ \nu(w', A) = 1$ .
$\nu(w, \Diamond A) = 1$ iff for some $w', \ \nu(w', A) = 1$ .

The upshot of these clauses is that we can now quantify over all objects, regardless of domain. Since $\Box$ and $\Diamond$ are looking over ‘all-objects’, these are our global quantifiers. The logic of both types of quantification should turn out to be the same, but they have different scope.

Thinking about this got me thinking about some interesting issues: the account so far only works for unary properties. I’m not sure there’s a straightforward way to extend this to relations. And this raises a real issue. Call a relation is mixed if it is true of objects from different domains. It’s not clear what the satisfaction pluralist should say about mixed relations. (These problems are importantly related to the problem of mixed compounds faced by alethic pluralists — see here for more.)

One option is to claim that every mixed relation has only one salient object that the relation is about, and the satisfaction-relation in the domain of that salient object wins. But that’s philosophically kind of weakly motivated.

Another option would be to use both satisfaction relations of the objects involved to give the truth of mixed binary relations. We would then have to extended this up through n-ary relations. This is clearly the best option philosophically, but I have no idea how it should be modeled.

Finally, on the above account, it’s the objects that determine the domain; properties can be had in various domains. But some pluralists have suggested that domains are individuated by properties (or concepts), and that objects remain fixed throughout domains. Which account is right? Consider the following sentence: “There are infinitely many jokes, but Aaron knows only about 5 of them that are funny.” Clearly, mathematical concepts can be used in sentences about humor. This sort of issue gets right to the heart of a major issue: how do we individuate domains? And that is a core issue for the alethic pluralist.

March 26, 2010

Boundaries as gaps or gluts

There are a number of existing theories of boundaries of objects. But I think all of them fail to accommodate a basic intuition that I think is plausible. Consider Varzi’s example of the Mason-Dixon line:

Imagine ourselves traveling from Maryland to Pennsylvania. What happens as we cross the Mason-Dixon line? Do we pass through a last point p in Maryland and a first point q in Pennsylvania? Clearly not, given the density of the continuum; for then we should have to admit an infinite number of further points between p and q that would be in neither State. But, equally clearly, we can hardly acknowledge the existence of just one of p and q, as is dictated by the standard mathematical treatment of the continuum; to do so would be to assign the boundary between the two States to only one of the States, and either choice would amount to a peculiar privileging of one State over the other. And we cannot identify p with q, either, for we are speaking of two adjacent States, so their territories cannot have any parts in common. So, where is the Mason-Dixon line, and how does it relate to the two adjacent entities it separates?

There is a puzzle here. Granted that objects may be continuous (and that they can touch), here are some options.

Classical: A boundary is either part of an object or its complement, but not both.
Gappy: A boundary may not be part of an object nor part of its complement.
Glutty: A boundary may be part of both an object and its complement.
Coincidence: There are two coinciding boundaries: one that is part of an object, and one that is part of its complement.
Eliminativist: An object really has no boundary at all; boundaries are either abstractions (equivalence classes) of convergent series of nested bodies, or inherited by the boundary of some space-time receptacle of that object, etc.

With respect to the classical theory, I am over simplifying a bit. In the classical case where neither an object nor its complement is fully bounded, both objects are partly open and partly closed. As such, each has a proper
part that is closed, and therefore has a bordering part must therefore be open. The forced choice simply returns at a smaller scale. We will ignore this complication in the future. Also, I’m assuming that the empty object is a fiction. Since the empty object will be clopen, so will its complement (the universe), and so its boundary will be a glut.

I’m also lumping a whole host of heterogeneous theories under the label “Eliminativist”, many of which have rigorous formalizations. But for our purposes, let’s just assume that boundaries are real entities and ought to be included in our domain. Then we’re left with options (1)-(4).

To my knowledge, only options (1) and (4) have been formalized. (1) is the classical mereotopology called GEMTC by Casati and Varzi. Option (4) was explored in detail by Chisholm and formalized explicitly by Barry Smith (here). But I think both theories have drawbacks. In particular, one might be inclined toward the following symmetry intuition:

Symmetry: If there is no principled metaphysical difference between two connected objects, the boundary between them either belongs to both objects, or to neither.

It does appear to be somewhat arbitrary to assign boundaries to one over the other. But I don’t really want to spend a lot of time defending this intuition. I just want to see whether we can formalize a (non-eliminativist) theory that satisfies it.

I think we can. A topological space is a structure $\langle W, \mathcal{T} \rangle$ where:

$W \in \mathcal{T}, \ \emptyset \in \mathcal{T}$ ;
$\mathcal{T}$ is closed under finite intersections;
$\mathcal{T}$ is closed under arbitrary unions.

We say that any set $O \in \mathcal{T}$ is open. Let $X \subseteq W$ . Then the interior of a set (written $i(X)$ is the largest open subset of $X$ . Alternatively, a set is open whenever it is identical to its interior.

Elements $C$ s.t. $W - C \in \mathcal{T}$ are called closed; the set of all closed sets in $W$ we call $\mathcal{T}_C$ . $\mathcal{T}_C$ has the dual properties of $\mathcal{T}$ .

$W \in \mathcal{T}_C, \ \emptyset \in \mathcal{T}_C$ ;
$\mathcal{T}_C$ is closed under arbitrary intersections;
$\mathcal{T}_C$ is closed under finite unions.

The closure of a set (written $c(X)$ is the smallest closed set containing $X$ . Alternatively, a set is closed if it is identical to its closure.

A set $X$ is regular open if $X = i(c(X))$ and a set $Y$ is regular closed if $y=c(i(y))$ . Now, according to the classical theory, set-theoretic complement (-) is Boolean and boundaries are defined thus: $\partial(X) = c(x) \cap c(W-x)$ . And these facts gives rise to the widespread failures of the symmetry intuition.

The plan is to give models of a mereotopology with non-Boolean complementation so that boundaries behave as we’d like. So here’s what to do.

The powerset of any set with the usual set theoretic operations is a complete Boolean algebra. If we take $\wp(W)$ , the powerset of our set with topology $\mathcal{T}$ , we can use the standard set theoretic operations and some of the topological ones to yield interpretations of most of the needed mereological operators.

Domain: $\wp(W)$
Binary product: $x \cap y$
Binary sum: $x \cup y$
Fusions: $\bigcup X$ where $X$ is any subset of the domain
Universe: $W$
Parthood: $x \subseteq y$
Interior: $i(x)$
Closure: $c(x)$
Boundary: $c(x) \cap c(W - x)$

However, we do not treat mereological supplementation as Boolean. We define a new complementation operator $\sim$ thus.

$\sim(x) = \left\{ \begin{array}{l} W - c(x) \mbox{ if } x \mbox{ is regular open } \\ W - i(x) \mbox{ if } x \mbox{ is regular closed } \\ W - x \mbox{ otherwise} \end{array} \right.$

Given this definition of complementation, we have examples where the boundary between objects and their complements are part of neither. We also have examples where the boundary is part of both. We also get a form of double complementation: $\sim\sim x = x$ . (Indeed, this is why we only go non-classical with regular open and regular closed sets.)

One interesting thing is that objects and their complements aren’t always connected. In particular, regular open objects are never connected to their complements given the standard mereotopological definition of connection:

Connection 1: $x C y \ \mathrm{iff} \ x \circ^* y \vee x \circ^* c(y) \vee^* c(x) \circ^* y$

Here $\circ^*$ signifies non-trivial overlap, where two objects non-trivially overlap whenever their product is not the empty object (in the models: $x \cap y \neq \emptyset$ ).

To address this, we can borrow a trick from some of our Eliminitivist friends who have the same issue. They simply redefine connection as:

Connection 2: $x C y \ \mathrm{iff} \ c(x) \circ^* c(y)$

March 18, 2010

Validity for Strong Alethic Pluralists?

Alethic pluralism is the view that there is more than one kind of truth. Why would anyone want to be an alethic pluralist? Because, if we are, we can agree with Wright (1998):

Almost all the areas which have traditionally provoked the realist/antirealist debate – ethics, aesthetics, intentional psychology, mathematics, theoretical science, and so on – turn out to traffic in truth-apt contents, which moreover [. . . ] we are going to be entitled to claim to be true.

There is a standard distinction between two types of alethic pluralism: strong and weak.

Weak Pluralism: there is a universal truth property that all true propositions satisfy, in addition to the many other truth properties restricted to a domain of discourse.
Strong Pluralism: there is no universal truth property; there are only restricted truth properties.

People have raised issues for both types of pluralists. But the main problem facing strong pluralism is how to handle mixed inferences, like:

Torture is wrong.
The United States tortured prisoners at the Abu Ghraib prison.
So, the United States did something wrong.

Intuitively, (1) and (3) are supposed to be morally true, while (2) is supposed to be descriptively true. So there is no truth property that is preserved over the inference. Tappolet (1997) claims the strong pluralist faces a dilemma: either (i) reject that mixed inferences are valid, or (ii) reject that validity is necessary truth preservation.

Perhaps you think (ii) isn’t so bad — you may prefer a proof-theoretic characterization of validity or something. That’s fine, but you shouldn’t be forced to be a proof-theorist by virtue of being a pluralist. So, the challenge is to give a fairly standard — semantic — account of validity according to which no single property is ‘preserved’.

I think the challenge can be met. One idea is to appeal to an algebraic semantics, and the corresponding definition of validity.

Start by using n-tuples as truth values, where n is the number of domains of discourse. A 1 in i-th place just means: the proposition has the truth property for domain i. A 0 in i-th place just means: the proposition does not have the truth property for domain i.

We say a value is atomic iff only a single place gets a 1 (if any). So every true atomic proposition is true in just one domain. Atomic propositions get atomic values; compounds get values based on connectives. So how do we treat the connectives? Negation toggles 1 and 0 in each domain. Disjunction is the component-wise maximum. And conjunction is component-wise minimum. Call any valuation function satisfying these constraints $\nu$ .

Now, let’s abbreviate $\langle 0, 0, \ldots, 0\rangle = \bot$ and $\langle 1, 1, \ldots, 1\rangle = \top$ . It’s easy to check that $A \wedge \neg A = \bot$ and $A\vee \neg A = \top$ for any proposition at all.

In fact, we can just define the usual algebraic ordering on the set of values thus:

Order: $\nu(A) \leq \nu(B)$ iff $\nu(A \wedge B) = \nu(A)$

This just gives us a partial order on which conjunction is the greatest lower bound, and disjunction is the least upper bound. So, we have a complemented lattice. It’s also pretty easy to check that we have distribution, and so we’ve really got a Boolean algebra on our set of values.

Now, define consequence thus:

Validity: $A_1, \ldots, A_n \vDash B$ iff $\nu(A_1 \wedge \ldots \wedge A_n) \leq \nu(B)$

On this definition, we get the classical propositional consequence relation. The algebraic definition of validity is a standard – semantic – idea. But according to the above definition, no single truth property plays a privileged role. Every truth property is equally involved in the definition of validity in a strong pluralist friendly way.

Some pluralists have acknowledged that some domains should be governed by a non-classical logic (e.g. Lynch’s 2009 book). We can accommodate that suggestion by extending the idea: values are n-tuples of 1, 1/2, and 0. Negation toggles 1 and 0 but is fixed on 1/2. Conjunction, disjunction, and validity are the same. The result is a weak De Morgan logic called ‘S3′ which is both paracomplete and paraconsistent. (It’s the intersection of LP and K3, or alternatively, the logic you get by adding $A \wedge \neg A \vdash B \vee \neg B$ to FDE.)

That’s cool, but what I really want is a way to handle intuitionistic logic. Does anyone know of any easy way to extend Heyting algebraic semantics to n-tuples like this? There’s bound to be some standard construction taking two Heyting algebras into a new Heyting algebra, but I couldn’t find one and haven’t bothered to try to construct one myself.

Anyway, the approach above is kind of nice, I think. Any thoughts?

UPDATE: As mentioned in the comments, the simplest non-Boolean Heyting algebra won’t get the full intuitionistic logic: 1, 1/2, 0 where conjunction and disjunction are exactly the same as before. We define $\nu(A \rightarrow B) = \nu(B)$ if $\nu(A) > \nu(B)$ and 1 if $\nu(A) \leq \nu(B)$ . Here $\nu(A) \leq \nu(B) \ \mathrm{ iff } \ a_i \leq b_i$ for each $a_i \in \nu(A), b_i \in \nu(B)$ . So negation gets treated as $\nu(\neg A) = \nu(A \rightarrow \bot)$ per usual. Apparently, Dummett proved that the logic you get from defining semantics over linear Heyting algebras is intuitionism plus adding the axiom of conditional excluded middle. If we want the intuitionistic consequence to come out, we’ll need to use arbitrary Heyting algebras.

March 16, 2010

News

I’m back from the hiatus. Since my last post some things have happened.

I’ve just about put the finishing touches on my dissertation. I’m defending at the end of the semester, and will graduate May 8.
I’ve got a new paper forthcoming. My paper on alethic pluralism and the semantic paradoxes is set to appear in the OUP volume “Truth and Pluralism: Current Debates”, edited by Nikolaj Pedersen and Cory Wright.
I went on the job market. Not the best year to go on the market, but somehow I ended up with fantastic opportunity. I’m very pleased to say that I’ve accepted one of the postdoctoral research fellowships for the Northern Institute of Philosophy at the University of Aberdeen.

Now that all of this is somewhat settled, I’m ready to begin posting again regularly.