Adding Sentences Shrinks Models and Grows Consequences

Adding a sentence to a theory sounds like adding information. Syntactically, the theory becomes larger. Semantically, however, the space of possibilities becomes smaller.

This reversal is the key to the monotonicity of classical logic: more assumptions leave fewer models, and fewer models support more consequences.

A sentence removes possible worlds

Let $\mathrm{Mod}(\Gamma )$ be the models of $\Gamma$. Adding a sentence $\phi$ restricts this set by intersection:

$$\mathrm{Mod}(\Gamma \cup \{\phi \})=\mathrm{Mod}(\Gamma )\cap \mathrm{Mod}(\phi ).$$

Every surviving structure must satisfy everything it satisfied before and also satisfy $\phi$. Therefore

$$\Gamma \subseteq \Delta ⟹\mathrm{Mod}(\Delta )\subseteq \mathrm{Mod}(\Gamma ).$$

The map $\Gamma \mapsto \mathrm{Mod}(\Gamma )$ reverses inclusion. A stronger theory describes a smaller region of semantic possibility.

The inclusion need not be strict. If $\Gamma \models \phi$, every model of $\Gamma$ already satisfies $\phi$, so

$$\mathrm{Mod}(\Gamma \cup \{\phi \})=\mathrm{Mod}(\Gamma ).$$

The new sentence then records information already present rather than removing another possibility.

Fewer worlds make agreement easier

The consequences of a theory are the sentences true in every one of its models:

$$\mathrm{Cons}(\Gamma )=\{\psi \mid \mathcal{M}\models \psi \text{ for every }\mathcal{M}\in \mathrm{Mod}(\Gamma )\}.$$

If models are removed, there are fewer structures that must agree. A sentence that was previously rejected by one of the discarded structures may now be true in every survivor. Hence

$$\Gamma \subseteq \Delta ⟹\mathrm{Cons}(\Gamma )\subseteq \mathrm{Cons}(\Delta ).$$

This is monotonicity of entailment:

$$\Gamma \models \psi \text{ and }\Gamma \subseteq \Delta ⟹\Delta \models \psi .$$

Adding premises cannot destroy an existing classical consequence. It can only preserve it or add further ones. The two movements are therefore coupled:

$$\boxed{\Gamma \uparrow ⟹\mathrm{Mod}(\Gamma )\downarrow ⟹\mathrm{Cons}(\Gamma )\uparrow }$$

Inclusion matters more than cardinality

The inclusions imply the weak cardinal inequalities

$$|\mathrm{Mod}(\Delta )|\le |\mathrm{Mod}(\Gamma )|,|\mathrm{Cons}(\Gamma )|\le |\mathrm{Cons}(\Delta )|.$$

But these numbers do not always change.

• If the added sentence was already entailed, both sets remain equal.
• If it was not entailed, at least one old model is removed and the sentence itself is a genuinely new consequence. Both inclusions are then strict.
• Strict inclusion still need not change cardinality when the sets are infinite. For example, a countable set can have a proper countable subset.

This makes inclusion the robust notion of logical growth. Raw cardinality may hide a real increase in information.

There is also a first-order set-theoretic caveat: the collection of all structures may be a proper class. Cardinalities of model collections are meaningful only after fixing a set of candidate structures, such as structures of a bounded size. The inclusion law itself does not require this restriction.

A finite propositional microscope

The cardinal relationship becomes exact when there are $n$ propositional atoms and formulas are counted up to logical equivalence. There are $2^n$ valuations. Suppose $\Gamma$ has

$$m=|\mathrm{Mod}(\Gamma )|$$

models. A consequence may have any truth set containing all $m$ models of $\Gamma$. Each of the remaining $2^n-m$ valuations may be included or excluded independently. Thus

$$|\mathrm{Cons}(\Gamma )/\equiv |=2^{2^n-m}.$$

The fewer models remain, the more semantically distinct consequences become available.

For two atoms $p,q$, begin with $\Gamma =\{p\to q\}$ and then add sentences:

Theory	Number of models $m$	Consequence classes $2^{4-m}$
$\{p\to q\}$	$3$	$2$
$\{p\to q,p\}$	$1$	$8$
$\{p\to q,p,\neg q\}$	$0$	$16$

Adding $p$ removes the two models in which $p$ is false. The sole survivor has $p=q=1$, so many more claims become unavoidable. Adding $\neg q$ removes the last model.

Penguins expose the limit

Not all useful reasoning is monotonic. Consider the familiar argument:

$$\begin{array}{rl}& \forall x(\mathrm{Bird}(x)\to \mathrm{Fly}(x)),\\ & \forall x(\mathrm{Penguin}(x)\to \mathrm{Bird}(x))\\ & \therefore \forall x(\mathrm{Penguin}(x)\to \mathrm{Fly}(x)).\end{array}$$

As a classical argument, this is valid. “All birds fly” is an exceptionless universal statement. Once the conclusion follows, adding more premises cannot retract it.

Everyday reasoning usually intends something weaker: birds normally fly. From $\mathrm{Bird}(t)$ we tentatively infer $\mathrm{Fly}(t)$. If we later learn $\mathrm{Penguin}(t)$, the more specific exception defeats that conclusion and we withdraw it.

Classical logic cannot express this revision merely by adding

$$\forall x(\mathrm{Penguin}(x)\to \neg \mathrm{Fly}(x)).$$

If the theory also names a penguin, the old universal and the new exception remove every model. Classical logic keeps the old consequence and, from the resulting inconsistency, entails every sentence. It never interprets the later premise as an instruction to discard the earlier default.

The penguin example therefore does not refute classical monotonicity. It shows that ordinary words such as “all birds fly” often conceal a defeasible rule. Reasoning with defaults needs a non-monotonic consequence relation in which new information may invalidate an earlier conclusion.

The empty and inconsistent extremes

The empty theory imposes no restrictions:

$$\mathrm{Mod}(\varnothing )=\text{all candidate structures}.$$

Only sentences true in every structure follow from it, so its consequences are precisely the logically valid sentences.

At the opposite extreme, a theory that is not consistent has no models:

$$\mathrm{Mod}(\Gamma )=\varnothing .$$

Every sentence is then true in every model of $\Gamma$ vacuously, because there is no countermodel:

$$\mathrm{Cons}(\Gamma )=\text{all sentences}.$$

This is why the growth of classical consequence does not mean that the theory is becoming better. It means that the theory is becoming more restrictive. At the limit, it rules out every possible world and thereby entails everything.

The corrected mental model is a three-step motion: adding sentences strengthens the description, strengthening the description removes models, and removing models makes more sentences unavoidable. Default reasoning changes a different thing: it permits the consequence relation itself to revise earlier commitments.