3. A Small Quantified System

The propositional system from the previous chapter could inspect some kinds of conclusions already.

15 is even or 15 is divisible by 3.
15 is not even.

Thus: 15 is divisible by 3.

It could do that because the inferentially relevant structure was already visible in the propositions and their combination.

Our second example is different:

All men are mortal.
Socrates is a man.

Thus: Socrates is mortal.

If we represent these three propositions simply as $A$, $B$ and $C$, then our small propositional system sees three unrelated propositions:

$A$
$B$

Thus: $C$

But this representation has lost some of the structure that is important to what happens.

"Man" occurs in the first premise and again in the second.

"Mortal" occurs in the first premise and again in the conclusion.

Socrates is the same person in the second premise and the conclusion.

And "all men are mortal" does not concern Socrates alone.

None of that survives in $A$, $B$ and $C$.

If we want to inspect the constraint in this inference, we need a representation that preserves more of what happens inside the propositions.

Things, Properties, And Propositions

Consider "Socrates is a man."

This proposition has parts that play different roles.

"Socrates" is not true or false.

Neither is "is a man" by itself.

But the complete claim "Socrates is a man" can be true or false.

For our small system, we will distinguish between things we talk about, properties we attribute to them, and propositions produced by attributing a property to a thing.

Let $s$ stand for Socrates.

Let $M$ stand for the property "is a man".

Then $M(s)$ shall stand for "Socrates is a man."

Similarly, let $D$ stand for the property "is mortal".

Then $D(s)$ shall stand for "Socrates is mortal."

The notation now preserves something our earlier propositional letters did not. The same object $s$ can occur in several propositions. And the same property can be attributed to several objects.

That gives us a way to preserve some of the internal structure that matters to the inference.

It does not tell us whether Socrates is in fact a man or mortal. Those truth values still have to come from the context of the application of the formal system.

What the notation gives us is a way to keep track of what is being said about what.

A Small Collection

The first premise causes another problem.

All men are mortal.

This does not concern only Socrates.

To inspect the word "all", we need to know what objects are under consideration.

For the present construction, suppose there are exactly three:

$$ {s,a,b}. $$

The letters stand for three objects. One of them is Socrates.

The collection is complete for this small system. There is no fourth object hiding outside it that also has to be checked.

This is obviously not a claim about the world.

Ordinary universal claims can range over far larger collections, including ones we cannot simply list. We restrict the present system because a finite collection lets us inspect what universal coverage amounts to.

That is the same advantage we used in the previous chapter. There, two truth values for two variables gave us four possible assignments. Here, a finite collection lets us make every object under consideration explicit.

Saying It For Every Object

We can now try to represent "All men are mortal."

Without introducing any new symbol, we can write the relevant condition separately for every object in our collection:

$$ M(s)\to D(s) $$

$$ M(a)\to D(a) $$

$$ M(b)\to D(b). $$

We already defined the implication in the previous chapter. For each object, the expression excludes one combination: the object being a man while not being mortal.

In our finite model, the representation of "all men are mortal" requires this condition to hold for every object we have admitted.

All three cases can be inspected directly.

But the construction is repetitive. The same pattern appears three times. Only the object changes.

A Repeated Place

The first position inside both properties is occupied once by $s$, once by $a$, and once by $b.$ The property however does not change.

We can mark that repeated place with another symbol: $x$.

Then the common form becomes $M(x)\to D(x)$.

Here, $x$ is not another object in our collection. It marks a place into which each of those objects can be inserted. It is a variable.

If we put $s$ there, we get:

$$ M(s)\to D(s). $$

If we put $a$ there:

$$ M(a)\to D(a). $$

And for $b$:

$$ M(b)\to D(b). $$

The new notation compresses the repeated structure. But we can reconstruct each individual case.

Taming The Variable

We now have a common form:

$$ M(x)\to D(x). $$

But the variable $x$ is still only marking a place. The two occurrences of $x$ tell us to put the same object into both places, but we have not yet said which objects are to be considered, or how far that use of $x$ extends.

For that we introduce $\forall$.

It does two jobs for us. It says that we are considering every object in our collection, and it marks the scope within which $x$ is governed by that requirement.

That matters because a later use of $x$ need not always belong to the same quantifier. Parentheses let us mark where the reach of a quantifier ends and where another use could begin. We may also need other quantifiers for different kinds of claims.

For our present claim, we want the requirement to cover the whole formula:

$$ \forall x(M(x)\to D(x)). $$

The quantifier tells us that throughout the formula in its scope, $x$ ranges over every object in our collection.

For the small system we have constructed, this means that all of these must be true:

$$ M(s)\to D(s), $$

$$ M(a)\to D(a), $$

$$ M(b)\to D(b). $$

Because the collection is finite and complete, we can unfold the quantified formula again and inspect every case.

What We Have Constructed

We can now distinguish objects from properties, preserve the same object across several propositions and the same property across several objects, and express one condition that has to hold for every object in a fixed finite collection.

The variable and the universal quantifier compress that repeated structure, but we can still unfold it whenever we want to inspect the individual cases.

The interpretation has to be supplied from outside of the system.

What the system gives us is a way to make more of the structure of the inference explicit.

Is it enough to recover the constraint we started with?

Testing The Inference

Our original inference was:

All men are mortal.
Socrates is a man.

Thus: Socrates is mortal.

In the system we have constructed, this becomes:

$$ \forall x (M(x)\to D(x)) $$

$$ M(s) $$

with the proposed conclusion:

$$ D(s). $$

The notation alone does not show why the conclusion follows.

So we unfold the universal premise.

Our complete collection is:

$$ {s,a,b}. $$

That means the first premise requires:

$$ M(s)\to D(s), $$

$$ M(a)\to D(a), $$

and:

$$ M(b)\to D(b). $$

The first of these concerns Socrates.

So among the things we have taken as true are:

$$ M(s)\to D(s) $$

and:

$$ M(s). $$

The truth-value mapping for implication from the previous chapter already gives us what we need.

It is the same structure as:

$$ A\to B $$

$$ A $$

Thus:

$$ B. $$

There was no assignment in which both premises were true while the conclusion was false.

The same propositional mapping still applies even though the letters now occur inside a more structured notation.

If:

$$ M(s)\to D(s) $$

is true, and:

$$ M(s) $$

is true, then:

$$ D(s) $$

cannot be false under the mapping we already fixed.

The quantified premise gave us the Socrates instance because Socrates belongs to the complete collection over which the condition ranges.

The second premise tells us that Socrates satisfies the first part of that condition.

Together they leave no case in which Socrates is a man but not mortal.

The inference is valid in the small system.

If we add more objects, even as many as there are human beings, we get a much larger collection, but still a finite one. In principle, the quantified expression can still be unfolded into one condition for each object.

That is as far as our present construction carries us. But it can fully accommodate the real example. For an infinite collection, we could no longer inspect universal coverage by unfolding every case into a finite list.

Quantified logic can handle such structures, but doing so requires machinery we have not constructed here.

An Inference That Fails

We can now test the other inference:

All men are mortal.
Socrates is mortal.

Thus: Socrates is a man.

Its formal representation is:

$$ \forall x (M(x)\to D(x)) $$

$$ D(s) $$

with the proposed conclusion:

$$ M(s). $$

To show that the conclusion does not follow, we need one case in which both premises are true while the conclusion is false.

We can try to construct one systematically.

The proposed conclusion is:

$$ M(s). $$

So we begin by making it false:

$$ M(s)=\text{false}. $$

The second premise has to remain true:

$$ D(s)=\text{true}. $$

Now consider the part of the universal premise that concerns Socrates:

$$ M(s)\to D(s). $$

With the values we just chose, this becomes:

$$ \text{false}\to\text{true}, $$

which is true under the mapping we already fixed.

So making the conclusion false has not yet broken either premise.

We still need the universal condition to hold for $a$ and $b$. We can choose any values for them that do not make $M(x)$ true while $D(x)$ is false.

For example:

Now we can inspect the universal premise object by object.

For Socrates $M(s)\to D(s)$ is true.

Its first part is false and its second part true.

For $a$ $M(a)\to D(a)$ is also true.

Both parts are true.

For $b$ $M(b)\to D(b)$ is true as well.

Its first part is false. Remember that is how we defined the implication.

So every object in the collection satisfies $M(x)\to D(x)$.

That means $\forall x (M(x)\to D(x))$ is true.

The second premise $D(s)$ is also true.

But $M(s)$ is false.

The premises can be true while the proposed conclusion is false.

Translated back into ordinary language, the premises do not exclude this situation:

All men are mortal. Socrates is mortal. Socrates is not a man.

There is a counterexample.

What This Construction Shows

The system has reproduced the inferential difference we started with.

To do that, it had to preserve more structure than the propositional system from the previous chapter.

It had to keep track of objects and properties, preserve the same object across several propositions, and make explicit what it means for one condition to hold for every object in the collection.

The universal quantifier compresses that repeated structure. In our finite model, we can unfold it again into one case for each object.

The convincing force of this additional constraint seems to come from preserving these relations explicitly: the same objects across propositions, the properties attributed to them, and the requirement that a universal claim hold for every object in the collection.

Once the universal premise is fixed, Socrates belongs to the collection, and $M(s)$ is true, the condition that applies to every object also applies to Socrates. The propositional constraint constructed in the previous chapter can then operate on that particular case.

But the application still depends on what we supplied.

We chose the objects under consideration, decided which object $s$ represents, chose the properties represented by $M$ and $D$, treated the collection as complete, and translated "all men are mortal" as a universally quantified conditional. We also relied on the truth-value mappings constructed in the previous chapter.

Any of those choices can be questioned.

If they remain convincing for the problem at hand, then the formal system makes the relevant structure explicit enough that a counterexample can either be constructed or excluded.

What seems to make the resulting constraint convincing is that we can inspect how it arises from the representation we supplied.

An Aside On Hidden Structure

The quantified system also helps us see why we used the arithmetic example in the previous chapter.

Consider:

If the alarm light is on, the machine has stopped.
The alarm light is on.

Thus: the machine has stopped.

Written propositionally, this looks simple enough:

$$ A\to B $$

$$ A $$

Thus:

$$ B. $$

It is natural to read the first premise as a general rule. But propositional logic evaluates a complex proposition from a particular assignment of truth values to its atomic propositions. That requirement is easy to overlook in ordinary-language examples.

"The alarm light is on" and "the machine has stopped" do not have fixed truth values unless a particular machine, time, or situation has been settled. Without that, there is no single valuation for $A$ and $B$ to begin with. Sometimes the machine has stopped and sometimes it has not, and similar for the alarm light.

The truth value of $A\rightarrow B$ is determined from the truth values of $A$ and $B$ by the mappings we fixed earlier. If the atomic propositions have not yet received definite truth values, neither has the complex proposition built from them.

Read as a general rule, this example does not yield a propositional interpretation at all.

We can fix that by referring to a specific situation where the truth values of $A$ and $B$ are indeed determined, but then we lose the rule character. Or we can represent the changing circumstances explicitly with a richer system, for example a quantified system.

The arithmetic example avoids much of this problem.

"15 is even" and "15 is divisible by 3" already concern one fixed object. Their truth values do not change from one occasion to another. We can therefore treat them as complete propositions without first fixing a surrounding situation.

The quantified system is a better fit when the truth of a claim depends on which object, time, or situation is under consideration. Those circumstances can become part of the representation instead of being silently supplied by the reader.

In that sense, the two systems also illuminate each other. The quantified system makes it easier to see where a propositional representation has hidden a dependency, and to say more precisely what has been left implicit.