4. Sources Of Logical Constraint

The small systems in the previous chapters were deliberately limited. Their purpose was to keep enough of the relevant machinery visible that we could inspect where some of the constraint came from.

One source of constraint was representation.

In the propositional system, replacing ordinary claims with letters was useful because the internal structure of those claims did not matter for the alarm inference.

For the Socrates inference, the same move lost too much.

We needed to preserve the fact that the same object and the same properties appeared in different propositions. Formalization preserves only what the representation makes explicit.

The second source was fixed structure.

Once the truth-value mappings of the connectives had been defined, the truth value of a complex proposition was no longer settled independently of the truth values of its parts. Once universal quantification had been defined over a complete finite collection, the same condition applied to every member of that collection.

The third source was the treatment of possibilities.

For two Boolean variables, we could list every assignment. That gave us a complete space in which to ask whether the premises could remain true while the conclusion became false.

For an invalid inference, one surviving case was enough.

The quantified example worked similarly. We did not need to list every possible interpretation to disprove the inference.

One interpretation in which both premises were true and the conclusion false gave us a counterexample.

The fourth source was compression.

The expression $\forall x(M(x)\to D(x))$ was shorter than writing the same condition separately for every object.

But in our finite system the compressed expression could still be unfolded. The variable and quantifier preserved the repeated structure in a form that could be reused.

These sources interact.

A representation makes some structure available. Definitions and mappings constrain what can happen within it. Formal devices compress repeated structures. Possibilities can then be tested, and conclusions fail when a counterexample remains.

The constraint comes from the relations made explicit and fixed within the system, not from the logical symbols by themselves.

Where These Small Systems Sit

We can now use some of the standard terminology we mostly avoided while constructing the systems.

The system in the second chapter contained familiar parts of classical propositional logic. The letters $A$, $B$, and $C$ were propositional variables, and our construction rules defined a small language of well-formed formulas.

The assignments of $\text{true}$ and $\text{false}$ were Boolean valuations. The symbols $\neg, \land, \lor, \to$ were truth-functional connectives, with $\to$ interpreted as material implication.

The tables we used were truth tables.

When we asked whether there was an assignment in which the premises were true and the conclusion false, we were testing semantic validity. A surviving assignment of that kind can be called a countervaluation.

The third chapter added machinery familiar from first-order logic.

The symbol $s$ functioned as an individual constant naming Socrates. $M$ and $D$ were unary predicate symbols. Expressions such as $M(s)$ were atomic formulas. The symbol $x$ was a variable. Our three-object collection served as a finite domain of discourse.

$\forall x(M(x)\to D(x))$ used universal quantification. The table assigning the properties "man" and "mortal" to the objects gave us a small interpretation. Together with the domain, this is close to what is called a finite structure.

When such a structure makes the relevant premises true, it is a model of those premises. When it also makes the proposed conclusion false, it is a countermodel to the inference.

Our third chapter should not be mistaken for a reconstruction of first-order logic. It used only a tiny fragment of first-order language, interpreted over a finite domain.

Full first-order logic distinguishes more carefully between the symbols of the language and the objects they denote.

For example, the symbol $s$ and Socrates himself are not the same thing. We mostly collapsed that distinction because nothing in our investigation depended on keeping it visible.

The standard terminology also gives us a useful distinction between syntax and semantics.

Syntax concerns which expressions belong to the formal language and how they can be constructed. Semantics concerns how those expressions receive truth conditions.

An interpretation assigns meanings to the nonlogical parts of the language.

Semantic validity concerns what happens across the relevant interpretations: an inference is valid if there is no interpretation in which all its premises are true and its conclusion false. Those are more precise names for much of the machinery we constructed from the other direction.

We began with convincing and unconvincing inferences, and asked what had to become explicit before their difference could be inspected.

What Modern Logic Adds

The small systems worked partly because they were small. That advantage disappears quickly.

With two propositional variables, there are four Boolean valuations. With ten variables, there are already $1024$. With twenty, there are more than a million.

The possibility space is still finite, but direct inspection is no longer attractive.

First-order logic creates a larger problem. Its domains need not be finite at all.

A universal statement may concern every natural number, every point in a space, or every object in some other infinite structure.

The method we used in the Socrates example, writing out the condition separately for every object, is then unavailable.

Modern logic develops machinery for working beyond those limits.

One part of that machinery is recursive syntax. Our simple formation rules already pointed in that direction.

A complex expression was built from simpler expressions according to fixed rules, and its structure could be recovered afterward. Modern formal languages develop this systematically. They can generate expressions of arbitrary finite complexity while preserving how those expressions were constructed. That extends one source of constraint we already encountered: traceable structure.

Semantics is extended in a similar way.

Instead of defining truth only for a few hand-built examples, modern logic defines systematically how formulas are evaluated in interpretations or structures.

Universal quantification no longer depends on being able to write every member of the domain on the page. Its semantic definition can range over every object in the domain whether that domain contains three objects or infinitely many.

This increases reach, but it also changes what carries the convincing force.

In our small systems, much of the construction could be inspected directly. In larger systems, more of the work is carried by mathematical definitions and the theory built around them.

Another important addition is proof.

So far, we have approached validity mainly through semantics. We asked whether a counterexample existed.

For small propositional systems, a truth table can answer that directly. For larger systems, an exhaustive search may become impractical or impossible.

Formal proof systems offer another route. Natural deduction, sequent calculi, tableaux, and axiomatic calculi differ in how they organize proofs, but for our purposes they share an important feature.

They allow reasoning to proceed through local formal steps.

Instead of inspecting every interpretation, we transform formulas according to specified rules. A large space of possibilities can be replaced by a sequence of smaller operations. That is another form of compression. It also creates another question. Why should the permitted proof steps convince us?

A formal rule does not become trustworthy merely because it has been given a name. This is one place where metatheory enters.

A proof system can itself become an object of mathematical investigation. A soundness result connects syntactic derivation back to semantics. Roughly, if a proof system is sound, then anything derivable from the premises by its rules is semantically valid. The local transformations do not lead us to a conclusion for which a semantic counterexample remains.

Completeness establishes a converse relation for systems where it holds.

Roughly, if a conclusion is semantically valid, then an appropriate complete proof system can derive it. For classical propositional logic and classical first-order logic, sound and complete proof systems are available.

The proof system can be examined in turn.

We can investigate not only whether a particular inference succeeds. We can also investigate whether the procedures we use for producing proofs correspond to the semantic relation we wanted them to track. That gives us another level at which the system can itself be examined.

The metatheoretical proofs themselves use mathematics and logic. But this gives us additional ways to expose dependencies and test them.

Modern logic also does not consist of a single final formal system.

Classical propositional and first-order logic preserve some distinctions and ignore others. Other systems change the available machinery.

Modal logic introduces formal treatment of notions such as necessity and possibility. Temporal logics represent temporal structure. Many-valued logics allow more than two truth values. Relevance and paraconsistent logics alter other parts of the inferential setting.

The details do not matter here.

What matters is that increasing formal precision does not by itself choose one unique formal system for every inquiry.

The question of what structure needs to be represented remains.

From Inspection To Practice

There is another difference between our investigation and ordinary logical practice.

A trained logician usually does not reconstruct the truth table for:

$$ A $$

and:

$$ A\to B $$

before accepting:

$$ B. $$

The form is familiar.

A proof system may contain a rule corresponding to the move. The user may also know that the relevant proof system is sound.

Notation becomes familiar. Common forms become recognizable. Proof rules become ordinary working tools. Training makes some transformations available without deliberate reconstruction. Books, teachers, colleagues, and mathematical practice contribute further layers of reliance.

Mature logical practice depends on more than one source of convincing force at once.

Recursive examination becomes relevant when something no longer convinces, when two standards conflict, when an unfamiliar inference appears, or when the limits of a system matter.

We can then reopen parts of what had been compressed. We may reconstruct a truth table, examine a proof rule semantically, or challenge a formalization. A metatheoretical result can also become an object of inquiry.

Logical practice fits the same active and passive structure encountered elsewhere in conviction formation.

We can construct examples, search for countermodels, learn calculi, check proofs, alter representations, and choose which formal systems to investigate.

We cannot simply decide what the result of those activities will convince us of.

Observations

The investigation also leaves a few observations that reach beyond what the logic example alone can establish.

They are worth keeping because the following investigations of probability and causality give us another chance to see whether the same patterns recur.

Formalization can make more of what a conclusion depends on available for inquiry. Once premises, mappings, alternatives, and dependencies have been made explicit, questions become available that may have been difficult to formulate before.

We can ask which premise would have to change, whether the representation is adequate, whether a counterexample can be found, or whether a rule was applied correctly.

Part I gave us a reason why this may matter for conviction formation. We can deliberately ask a question without being able to decide whether it will become genuinely puzzling. Trying to answer it may reveal that an expected answer is unavailable, that an assumption is harder to defend than we thought, or that something else we wanted to retain has to move.

The formal system does not determine what this will convince us of. It can help create the conditions under which a question acquires force.

This gives recursive examination a more specific mechanism. Formalization can expose dependencies. Exposed dependencies make further questions available. And trying to answer those questions can disturb convictions without the result being ours to choose.

Something similar may happen whenever a distinction or criterion itself becomes convincing enough to enter inquiry.

Once the difference between waking perception and dreaming or hallucination matters to us, a conviction can come under pressure from the question whether what seems to be there is actually there. Once a kind of correspondence between a conviction and what it is about becomes a convincing possibility, the question whether the conviction is true in that sense becomes available.

Other criteria can acquire a similar role.

Consistency, empirical adequacy, reliability, causal adequacy, and many others do not determine what we must believe.

But once they have become convincing, convictions no longer encounter the same landscape without them. Once a criterion becomes convincing enough to guide inquiry, a conviction may have to withstand questions it did not face before.

Logic gives us an especially explicit case. Formalization can expose where the pressure falls and where disagreement can continue.

The wider pattern seems to be the same: once a criterion itself becomes convincing enough to guide inquiry, it changes what later convictions have to withstand.

The logic case does not establish that this happens in exactly the same way for every criterion. But it shows how such pressure can arise without the criterion standing outside conviction formation or determining the result in advance.

The path we followed resembles something mathematicians often call getting an intuition.

We began with very small examples. Then we stripped away structures that did not matter for the inference at hand, and made the relevant possibilities visible.

Counterexamples showed where an apparent inference failed.

When repeated structure appeared, we represented it explicitly.

And once the structure became familiar, it could be compressed into notation and used without reconstructing every earlier step.

Getting an intuition can mean many things in mathematics, and this investigation does not offer a general theory of it. But at least one familiar form seems close to what happened here. We find a representation in which the constraint becomes visible enough to inspect. Repeated use can then stabilize it until the compressed formal move itself becomes convincing without the whole path being rebuilt each time. That possibility also fits the transition from our small constructions to mature logical practice.

These observations are not established generally by the logic case. But they give us things to watch for when we turn to probability and causality.

What Logic Adds, And What It Does Not

We can now separate three questions that are easy to run together.

Validity concerns a relation between premises and conclusion.

Within the semantics of a formal system, an inference is valid when the premises cannot all be true while the conclusion is false.

Truth concerns whether a proposition is true.

A valid inference can begin from false premises. Validity does not make them true.

Convincingness concerns what a finite person or group comes to rely on.

A valid inference may fail to convince someone who does not accept its representation, its premises, its semantics, or the formal system in which it is expressed. Conversely, invalid inferences can convince people.

Logic does not remove those differences. Nor does validity establish that a conclusion is relevant, important, or a sufficient basis for action.

Those limits are not surprising once we see where the constraint came from. The small systems always depended on inputs. Modern systems move some of those choices into larger bodies of formal theory, but they do not abolish them.

But logical constraint is not arbitrary. Once enough of the relevant conditions remain in place, some possibilities disappear.

If $A$ and $A\to B$ remain true under the mappings we fixed, then rejecting $B$ requires something else to change.

If $\forall x(M(x)\to D(x))$ and $M(s)$ remain true under the interpretation we fixed, then rejecting $D(s)$ requires something else to move.

The premise may be abandoned. The interpretation may be challenged. Or the representation may turn out to have lost something important. A different formal system may become relevant. But the conclusion is not independently free while everything else stays where it is.

That is the logical constraint our small constructions made visible.

Modern logic extends many of these sources through formal languages, general semantics, proof systems, and metatheory. These allow structures to remain stable and usable far beyond the cases a finite person can inspect all at once. They also provide further objects for examination when the sources of constraint themselves become doubtful.

Logic does not stand outside conviction formation and decide what we must believe. It changes some of the conditions under which convictions form by making structures explicit, excluding some possibilities, and carrying constraints across longer sequences of reasoning.

When enough of those conditions remain convincing, some conclusions become difficult to reject without giving something else up.