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 the convincingness 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 number inference.

For the Socrates inference, 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.

One surviving counterexample was enough to make an inference invalid.

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 constructions also made those relations open to visual inspection and manipulation. Possibilities could be laid out so that several cases remained visible at once, and the spaces were produced by repeatable procedures rather than assembled arbitrarily. Later, truth-value mappings and formal rules replaced some of this direct manipulation with rule-governed operations.

Part of the convincing force seems to come not only from relations being made explicit and fixed within the system, but from being able to inspect how they are arranged and to reproduce the operations by which the constraint appears.

This continues something already encountered in Part I. Under ordinary conditions, perception can strongly constrain what we become convinced of. Externalizing an inquiry can make parts of it persist outside immediate attention and become available for comparison and repeated inspection. Formal logic adds rule-governed constraints to such externalized structures.

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. 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.

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. It concerns what we called well-formed formulas. 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.

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, tableaus, 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?

This is 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.

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.

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.

What The Logic Investigation Found

One question concerns the constraint inside a formal system.

Once enough of the representation, mappings, and premises remain fixed, some conclusions are no longer independently open. A counterexample either remains available or it does not.

The other concerns the application of that system to a problem.

We may accept the formal constraint while questioning whether the problem was represented adequately. A distinction may have been lost, a premise may be doubtful, or another formal system may preserve what matters better.

When an inference or its formalization becomes questionable, recursive examination can proceed in two directions.

We can examine what follows once the formal conditions are in place. This is often comparatively contained: if the formal system and the relevant inputs are fixed, we can inspect whether the conclusion follows within it.

Or we can examine whether those were the right conditions to put in place for the problem.

That can reopen the representation, the interpretation, the premises, or even the choice of formal system itself.

If more of these dependencies have been exposed and the result is still convincing, then we are on firmer ground. We know more clearly what the conclusion depends on and what would have to change for it to move. It is, in that sense, better than before.

But that does not mean it is better than other convictions. It still does not give us a definite criterion for ordering different convictions as better or worse.

Logic And Answerability

There is another result, but it needs to be made visible.

Suppose a counterexample has been constructed, an inconsistency exposed, or a conclusion shown not to follow from the stated premises.

The conclusion may be revised, a premise rejected, the formalization questioned, or another interpretation or logical system preferred. The constraint may also be judged irrelevant to the problem.

The inquiry nevertheless now contains a difficulty that was not there before, or was not yet visible in that form.

If the original conclusion remains convincing, attention can move to the premise, representation, interpretation, or choice of formal system on which the difficulty depends.

A logical constraint can also become part of a disagreement between people.

If enough of the logical machinery is shared, one person can construct a counterexample to another person's inference, expose an inconsistency, or ask whether a conclusion follows from premises they both accept.

The disagreement may then shift. It may concern a premise rather than the conclusion, the representation rather than the formal consequence, or whether the logical system fits the problem at all.

Once a difficulty has been made visible in terms both participants can sufficiently follow, one can ask the other how it is to be handled. Rejecting the counterexample, the formalization, or the relevance of the logical constraint remains possible. Those responses can themselves become parts of the inquiry.

This is one form of answerability. And it need not remain confined to the exchange in which the difficulty first appeared.

The particular challenge may pass, but the form of examination remains available.

Logical procedures can also be learned, reused, criticized, and introduced into later inquiries. Other people can bring them with them. Once such practices become sufficiently shared, inquiry takes place in an environment in which these forms of examination are available.

In that sense, logic becomes a fact of intellectual life.

The logic case suggests that forms of constraint may matter not only because of the conclusions they produce. Once they become sufficiently available and shared, they may also change what participants in an inquiry become answerable for.

There may be something still more basic underneath this.

Logic needs propositions that are true or false. But that formal distinction may depend on an earlier, ordinary distinction between something being the case and not being the case. That distinction may already make new questions and forms of answerability possible in much the same way as logical constraint does.

Once people distinguish between describing what is the case and inventing something, or between reporting what they take to be the case and deliberately saying otherwise, further questions can arise before any formal logic is introduced.

The investigations of probability and causality will give us other cases in which to see whether the same pattern appears.

Next essay: Sources Of Constraint In Probability Theory And Bayesian Updating