Mutually Exclusive, Collectively Exhaustive

A natural number is either even or odd.

It cannot be both.

And with respect to parity, there is no third possibility.

Structures of this kind play a major role in reasoning.

Some possibilities exclude one another.

Together they are taken to exhaust a topic completely.

Reasoning by cases depends on such structures.

So do sample spaces in probability and statistics.

Sherlock Holmes famously described a similar pattern: "When the impossible has been eliminated, whatever remains, however improbable, must be the truth."

Differential diagnosis, decision trees, proof by contradiction, and many other forms of reasoning rely on possibilities that are taken to be mutually exclusive and collectively exhaustive.

From the perspective of conviction formation, this is curious.

How do such possibilities become established?

How does the conviction arise that two possibilities exclude one another?

How does the conviction arise that there are no further possibilities?

And what sustains such convictions?

Even And Odd

The natural numbers can be imagined as pebbles placed into an empty field.

Zero corresponds to the empty field.

Each additional pebble represents one further natural number.

Now pair the pebbles. It is fine if no pairs can be formed.

If you have pairs, remove them all.

If no pebble remains, then the number is called even.

If exactly one pebble remains, the number is called odd.

There is no third outcome.

If two pebbles remain, they themselves form a pair.

If three remain, two form a pair and one remains.

No matter how many pebbles are present, repeated pairing eventually leaves either zero or one pebble.

The conviction that every natural number is either even or odd, and there are no other possibilities, rests on a simple procedure that can be inspected directly.

The structure of the natural numbers together with the pairing procedure establishes both exclusivity and exhaustiveness.

Colors

Consider the visible color spectrum.

It runs roughly from violet through blue, green, yellow, and orange to red.

Suppose a witness reports that the suspect wore a plain-colored shirt.

The shirt cannot simultaneously be blue and red.

In that sense the possibilities exclude one another.

Yet the situation is already different from parity.

Where exactly does orange end and red begin?

There may be cases where observers disagree.

One might describe the shirt as orange-red.

Another as reddish orange.

The boundary is less sharp than the boundary between even and odd.

Still, the overall structure remains usable.

The shirt may have been difficult to classify precisely, but it was clearly not bright green, deep blue, or violet.

The uncertainty concerns the boundary, not the entire structure.

A second question appears.

Are these all the colors there are?

Ordinary experience strongly suggests that they are.

Rainbows, light prisms, pigments, displays, clothing, paintings, and daily perception continually reinforce the familiar spectrum.

Yet the conviction is not as straightforward as before.

Other animals possess different visual systems.

Future modifications of human perception might reveal distinctions we do not currently experience.

Novel visual experiences can already be produced under unusual physiological conditions.

The possibility cannot simply be ruled out that future observers might have a different, evolved color space.

Nevertheless, the structure remains stable in practice.

It has accumulated a long history of successful use.

Experience repeatedly supports it.

Nothing in ordinary life exerts significant pressure against it.

The conviction remains strong, but it is sustained differently than in the case of parity.

Pluto

For much of the twentieth century, Pluto was counted among the planets.

Schoolchildren learned a list of nine planets.

My very educated mother just served us nine pizzas.

Then astronomers began discovering additional objects in the outer solar system.

Some resembled Pluto closely.

A choice emerged.

Either the category "planet" would have to expand to include many of these newly discovered objects, or the criteria for being a planet would have to change.

Pluto was eventually reclassified.

The sky did not change.

The object did not disappear.

The category changed.

The very educated mother now serves nachos.

The conviction that "planet" and "non-planet" formed an exhaustive partition of certain celestial objects remained.

But the support structure behind the classification shifted.

Definitions were revised.

New discoveries created pressure on existing categories.

The partition remained.

Its boundaries moved.

The example illustrates that classifications can remain useful and stable while their criteria change.

Physicalism And Non-Physicalism In Regards To Consciousness

Consider a different opposition.

The philosophical explanations of consciousness are often presented as either physical or non-physical.

There are philosophical schools in each camp.

At first glance, the structure resembles the previous examples.

Yet physics itself has undergone repeated conceptual revolutions.

The physical world of Aristotle differs from that of Newton.

Newton's differs from Einstein's.

Einstein's differs from the picture emerging from contemporary quantum field theory.

What counts as physical has repeatedly changed. The meaning of physics and physical was completely redefined.

This raises a question.

Is physicalism a stable position?

That depends in part on what is meant by "physical".

If "physical" means whatever is described by current physics, the position has a comparatively clear content. But every major revision of physics places pressure on that content.

A different possibility is to define the physical through our best future physics.

Whatever ultimately exists, if it becomes part of a successful future physical theory, counts as physical.

Under this interpretation the category becomes remarkably flexible.

Suppose consciousness requires entities unknown to current physics.

Physics expands.

Suppose information turns out to be fundamental.

Physics expands.

Suppose something entirely unexpected is discovered.

Physics expands again.

The position remains intact.

But the support for the partition has changed.

In the first case, it rests on confidence in current physics.

In the second, it rests on confidence in a future theory whose contents are largely unknown.

Whether that confidence is justified is not the concern here.

What matters is that the conviction supporting the partition differs substantially from the support found in the previous examples.