From Propositions to Consistency

1. A Propositional World

Consider five concepts: a cat, a dolphin, a fish, a lizard, and a bird. We may ask simple yes-or-no questions about them: Does it feed its young with milk? Does it live underwater? Does it have four legs?

Concept	$p_1$: milk	$p_2$: underwater	$p_3$: four legs
cat	1	0	1
dolphin	1	1	0
fish	0	1	0
lizard	0	0	1
bird	0	0	0

Let $C$ denote the set of concepts and $P$ the set of propositions. Each proposition $p\in P$ is a Boolean-valued function on $C$:

$$p:C\to\{0,1\}.$$

We call

$$W=(C,P)$$

a propositional world.

A proposition tells us whether something is true or false for each concept in $C$.

2. What Does Consistent Mean?

Now choose a set of propositions

$$P'\subseteq P.$$

For example, take the three propositions “feeds its young with milk,” “lives underwater,” and “has four legs.” A basic question immediately appears:

$$\boxed{\text{Is }P'\text{ consistent?}}$$

Our everyday intuition is that a set of propositions is consistent when its propositions can all be true together.

The three propositions above seem perfectly consistent. There is no apparent contradiction in imagining an animal that satisfies all three.

Consistency means the possibility of joint truth.

3. A Natural Guess

How can we turn this intuition into mathematics?

Since every proposition is a function on $C$, the most immediate idea is to look for a concept $x\in C$ for which every proposition in $P'$ is true.

This suggests

$$P'\text{ is consistent}\quad\stackrel{?}{\Longleftrightarrow}\quad\exists x\in C:\ p(x)=1\text{ for all }p\in P'.$$

The reasoning is natural: if the propositions can all be true together, perhaps there should be something for which they are all true.

Joint possibility seems to suggest a common realization.

4. The Guess Fails

Return to our three propositions: milk, underwater, and four legs.

Looking at the table, no concept in $C$ satisfies all three:

$$\nexists x\in C:\ p(x)=1\text{ for all }p\in P'.$$

But this does not make the three propositions inconsistent. Nothing contradictory prevents us from imagining an animal that lives underwater, has four legs, and feeds its young with milk.

The problem is therefore the proposed equivalence. We confused what can be jointly true with what is already jointly realized in $C$.

One direction remains valid:

$$\exists x\in C:\ p(x)=1\text{ for all }p\in P'\quad\Longrightarrow\quad P'\text{ is consistent}.$$

But the converse need not hold.

What is realized in $C$ may be only part of what is consistent.

5. The Consistency Family

This failure suggests a change of viewpoint. Instead of trying to define consistency entirely through the concepts already contained in $C$, we record directly which sets of propositions are consistent.

Let

$$\mathrm{Cons}\subseteq\mathcal P(P),$$

where

$$P'\in\mathrm{Cons}$$

means that the propositions in $P'$ can all be true together.

There is one immediate structural requirement. If $P'$ is consistent, then removing some propositions cannot create a contradiction. Hence

$$P'\in\mathrm{Cons},\quad Q\subseteq P'\quad\Longrightarrow\quad Q\in\mathrm{Cons}.$$

Our propositional world is therefore enriched from

$$W=(C,P)$$

to

$$\boxed{W=(C,P,\mathrm{Cons}).}$$

Consistency is no longer identified with realization in $C$; it is represented by its own family $\mathrm{Cons}$.

6. The Realizability Family

Although realization in $C$ is not enough to characterize all consistency, it still gives us an important source of consistent sets.

Collect all sets of propositions that have a common realization in $C$:

$$\mathcal R_C:=\left\{P'\subseteq P:\exists x\in C,\ p(x)=1\text{ for all }p\in P'\right\}.$$

This family has the same downward property. Indeed, take $P'\in\mathcal R_C$ and $Q\subseteq P'$. By definition, there exists $x\in C$ such that every proposition in $P'$ is true at $x$. Since $Q\subseteq P'$, the same $x$ also makes every proposition in $Q$ true. Therefore

$$Q\in\mathcal R_C.$$

Thus

$$P'\in\mathcal R_C,\quad Q\subseteq P'\quad\Longrightarrow\quad Q\in\mathcal R_C.$$

Moreover, every set with a common realization is certainly consistent, so

$$\boxed{\mathcal R_C\subseteq\mathrm{Cons}.}$$

The inclusion may be strict: some sets of propositions may be consistent even though nothing currently contained in $C$ realizes them.

Realization generates consistency, but does not necessarily exhaust it.