Matter Thinks

Material Logic October 9, 2009

Posted by Michael in Logic, Ontology.
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Here’s an interesting exercise: what happens when you disquote one step too far?

So if instead of stopping at

'P' → P

you go on to

P → ?

what do you end up with?

My intuition about disquotation is that disquoting is a step towards reality. Maybe disquoting again would bring you the rest of the way. So if you disqoute a reference to a thing you get that thing.

Unless of course that thing doesn’t exist. Disquoting is not magic. Disquoting a reference to a thing that doesn’t exist gets you nothing.

So let’s try it with a proposition: “There exists an elephant”

'There exists an elephant' → There exists an elephant
There exists an elephant → [an actual elephant]

One thing that jumps out here is that you can’t play this game without an elephant. But, there’s no rule that says the elephant must actually be in your possession. The elephant must exist somewhere. If we both believe that to be a fact, that’s good enough for us to take it as a valid proposition.

So what happens when something doesn’t exist?

'There exists a unicorn' → There exists a unicorn
There exists a unicorn →

What happens is that you can’t make a statement, because you can’t find a unicorn.

Unless you’re Marco Polo, in which case:

'There exists a unicorn' → There exists a unicorn
There exists a unicorn → [a rhinoceros]

Note that even if you can make a statement, there’s no guarantee that what exists is what you say exists — only that it exists.

What I find highly intriguing about this is that in the final disquoting step you move from a bivalent logic to a monovalent logic — instead of being true or false, a statement merely, but literally, exists. There is no opposite, because in this unusual logic if something doesn’t exist it can’t be a statement.

Yes, logic. I think of this final step as a kind of logic, a funny kind of logic but still a formalizable, verifiable, arguable logic. I call it material logic.

Material logic makes claims of existence. That’s all it does. There are no explicit properties in material logic, other than the property (if you consider it that) of existence. But this is precisely where propositional logic leaves off (I have classical logic in mind, but I think it applies to intuitionistic and other logics as well).

In classical logic you can propose that an elephant exists, and you might even be able to prove the proposition, given the existence of certain evidence, and perhaps even prove things about the evidence, based on other evidence — a causal chain, you might say — but at some point you have the original physical evidence. What makes it “physical” is that it physically exists. But this claim of existence must simply be accepted and can’t be proven. This is an unaviodable consequence of the representation problem: ultimately, we can only logically analyze our mental representations of things, but the connection between the mental representation and the physical thing lies outside of logic. Naming a thing doesn’t prove the thing exists.

The value of material logic is that it deals exclusively with the last step, the one which classical logic takes as a given — claims of physical existence. And, as the exercise above shows, it’s possible to trace a path from propositional logic to material logic, thereby extending the reach of logical analysis into the existential realm. This is not by itself a solution to the representation problem, or a way to prove physical existence, but it is, I believe, a tool for achieving progress towards these goals, by providing a better way to talk about physical existence and its relationship to mental states. Better in two ways actually: closer to what’s actually going on, and easier for us to reason about.

Material logic is, essentially a formalization of object mapping — the mechanism by which mental representations are established, according to the object-oriented ontology I have proposed. To make it useful, we need to take it a little farther than what I’ve described up to this point (but not much). First, however, let’s review what we have so far.

• A statement in material logic is a statement that something exists. There are no names, categories or properties in material logic, so the statement doesn’t say what exists, just that something exists.
• A statement in material logic consists of something that exists.
• There is no way in material logic to state that something doesn’t exist. So material logic is monovalent (there is no negation).
• Since every statement says that something exists, and every statement consists of something that exists, every statement in material logic is correct — you simply can’t utter an incorrect statement.

Material logic’s infallibility and monovalence mean that you can’t use material logic to claim truth or falsity. But certain truth claims are also existence claims, and they can provide a bridge between propositional logic and material logic.

However, if we want to think about material logic we first face a bit of a problem: material logic is unparseable. Take the case of operators. Operators in material logic must necessarily look different than those in propositional logic — in fact they won’t look like anything at all, because an operator is an abstraction and material logic cannot express abstractions. All we can do materially is express the result of applying an operator, if there is one. And since there may be many ways a particular result came to be, you can’t see what the statement was just by looking at the result.

Luckily, we aren’t required to talk about material logic in material logic. Clever creatures that we are, we can come up with readable, analyzable notation that represents material logic without being itself material.

Continuing with the convention I used above, let’s represent the mapping of object x onto the real world like this:

[x]

We can say that [x] is the materialization of x, or, conversely, that x is the representation of [x].

We can describe the basic relationship between propositional logic and material logic as follows:

∃x → [x]
[x] → ∃x

or simply

∃x ↔ [x]

Now we’re in a position to define operations. I’ll define three: intersection, composition and subtraction.

The intersection of two materializations consists of the substance they share. So, for example, the intersection of my head and my skeleton would be my skull. We can express intersection with familiar notation:

[x] ∩ [y]

Intersection implies some degree of identity: if two objects intersect, then they are in part or in whole the same in substance.

The composition of two materializations consists of the substance belonging to either or both. The union operator seems appropriate:

[x] ∪ [y]

The third operation, subtraction, consists of the substance that belongs to one materialization but not another, and is represented like this:

[x] - [y]

One shortcoming with the above notation is that not every instance of it corresponds to a statement in material logic. In particular, because two of the operations, intersection and subtraction, are not closed, it is possible to describe an operation that does not yield a materialization and is therefore not a materiological statement.

In this regard it’s useful to remember that an expression like [x] – [y] is merely a stand-in for the material result of the operation, providing the result exists; otherwise the expression is nonsense. Some expressions, such as [x] – [x], are inherently nonsense. Here classical logic can come to the rescue. Precisely because [x] – [x] is not a statement in material logic, we can state categorically, in classical logic, that ∃([x] – [x]) is false.

There is one more very important step we need to take with material logic for it to be of most use: we must incorporate motion and change. This will also bring us face-to-face with limits to our knowledge, gaps that cannot be eradicated. It’s a big question, and I’ve only begun to put together an answer, but here are some of the main points:

• Motion and change correspond to transformations in material logic. We can catalog and study such transformations.
• We experience persistence by detecting continuity in sense data, and we experience motion and change by detecting changes in sense data. The computation behind these mental processes corresponds to finding the transformation that best fits a series of materializations.
• However, what we experience in this way is neither persistence of substance nor persistence of objects, it’s the persistence of the mapping between them.

Substance may change in form but lasts forever, as far as we know. Abstract objects are themselves timeless and changeless. An object exists physically if and only if, and only as long as, it’s mapped to substance. My hope is that material logic will enable new insights into this mapping, and thereby into what it really means to say that a thing exists.