5. Logic And Proof

In the previous chapters we observed several procedures that exert a particularly strong pressure on conviction. Counting, pairing, and geometrical construction share certain features. Their steps are simple, inspectable, and repeatable. Different people who follow the same procedure usually arrive at the same result. When disagreements arise, the procedure can be performed again and the error often becomes visible. The convictional force of the result is still provisional, but the procedure exerts such pressure that conviction usually settles almost at once.

A natural next step is to make the steps of reasoning themselves more explicit and more tightly constrained. Each move of thought should be visible and checkable, and hidden assumptions should be minimized as far as possible.

This demand contributed to the development of formal logic and mathematical proof.

Engineering Conviction

From the perspective of conviction formation, logic introduces a distinctive strategy.

Instead of relying on naturally convincing procedures such as counting or diagrammatic construction, logic deliberately engineers a system in which reasoning becomes controllable.

This process has two stages.

First, a system of rules is constructed that specifies exactly how statements may be combined and transformed. The goal is to make every step of reasoning explicit and inspectable.

Second, the system must demonstrate that it works. It is applied to problems, compared with accepted reasoning, and tested for contradictions or counterexamples. Over time, successful use stabilizes conviction that the system captures important patterns of valid reasoning.

The convincing force of logic therefore does not arise simply from symbolic notation. It arises from a two-stage process: first, the construction of a controllable system, and second, the demonstration that this system reliably tracks patterns of reasoning we already experience as compelling.

Logical Form

To see how such a system works, consider a simple example of reasoning.

If it rains, the ground becomes wet.
It rains.
Therefore the ground becomes wet.

The conclusion appears convincing because the structure of the reasoning is clear. The second statement satisfies the condition stated in the first, and the conclusion follows.

Now consider a different example.

If a number is divisible by four, it is even.
Twelve is divisible by four.
Therefore twelve is even.

Although the subject matter is completely different, the structure of the reasoning is the same.

Logic makes this structure explicit by replacing the concrete statements with symbols.

If A then B.
A.
Therefore B.

Once written in this form, the reasoning no longer depends on rain, numbers, or any other specific topic. The letters stand only for statements. What matters is the pattern connecting them.

By representing arguments in this way, logic turns reasoning into something that can be inspected step by step.

Controlled And Inspectable Reasoning

Formal logic then specifies rules for how such statements may be combined.

For example, a rule may state that from the statements

If $A\ then\ B$ and $A$ the statement $B$ may be derived.

The rules function much like the procedures we encountered earlier in counting or geometry. They define exactly which moves are allowed and which are not.

Because the rules are explicit, disagreements can be settled by examining the steps. A derivation can be written down and checked line by line. If an error occurs, it usually appears at a specific point in the sequence.

Logic therefore attempts to make reasoning itself behave like the procedures we previously examined: transparent, repeatable, and open to public verification.

One of the simplest such systems is propositional logic. A version of it was already known to the Stoic philosophers in the 3rd century BC, and it remains in use today.

In it, each statement is treated as either true or false. More complex statements are built from simpler ones using a small number of logical connectors such as $not$, $and$, $or$, and $if - then$.

The behavior of these connectors can be defined by simple rule tables.

Negation (not)

Disjunction (or)

Conjunction (and)

Implication (if - then)

The arrow "→" symbolizes "if ... then ...".

Using these rule tables, the truth of more complex statements can be calculated or inferred from the truth of simpler statements, if enough information is given.

We can now return to the initial example.
The logical structure of the argument can be written as:

The table for implication shows that whenever A is true and A → B is true, the only possible case is the first row of the table.

So the value of B must also be True:

The conclusion follows mechanically from the rules. If we only knew that A is false and A → B is true, then we could not conclude anything concrete about B. According to the last two rows of the implication table B can be either true or false.

In this form, checking the reasoning becomes procedural. One simply verifies that the rules have been applied correctly. Different persons who apply the same tables will arrive at the same result.

Propositional logic presented in this way resembles a structured search through a small space of possibilities. Other logical systems organize reasoning differently, but the underlying goal remains the same: to make the steps of reasoning explicit enough that they can be inspected and verified by anyone.

From the perspective of conviction formation, this is the essential feature of logical reasoning. It amounts to searching through and applying a finite, well-behaved set of rules that has proven reliable in practice and is based on forms of reasoning we already find compelling.

Representations of Reasoning

Formal logic isolates the structure of reasoning by representing statements with symbols such as A and B. In doing so it treats reasoning as a sequence of statements connected by certain key words: and, or, not, if - then.

This representation implicitly assumes that all reasoning can be captured through such linguistic structures. In other words, it treats thinking primarily as a manipulation of symbolic statements.

The earlier examples suggest that this is not the only way conviction does form.

In counting we manipulate objects directly.
In pairing we establish correspondences between collections.
In geometry we reason with diagrams and constructions. Shapes can be drawn, divided, moved, and recombined. These operations themselves reveal relations that can be inspected.

These procedures can exert strong convictional force even before they are translated into explicit formal symbolic statements.

Greek geometry illustrates this particularly clearly. Diagrams and constructions were not merely illustrations of arguments expressed elsewhere. They were themselves part of the reasoning process. A construction or rearrangement could reveal a relation that became immediately visible.

Symbolic reasoning does not possess this immediacy. The elements of symbolic logic cannot simply be seen, cut apart, or rearranged in the way geometrical figures can. The representation must first be carefully designed before reasoning becomes controllable.

Finding suitable symbolic structures for reasoning took a considerable time in intellectual history. The development from Aristotle's syllogistic logic to the predicate logic of Frege illustrates how much refinement was required before symbolic reasoning became a powerful and reliable tool.

This historical development reflects the second stage of the engineering process discussed earlier. A formal system must not only be constructed. It must also prove itself through sustained use, correction, and refinement.

Proof

In mathematics the ideas of engineering systems to capture valid discursive reasoning take a particularly systematic form in the notion of proof.

A proof is a sequence of statements in which each step follows from earlier ones according to accepted rules. The final statement of the sequence is the result to be established.

Because each step must be justified, the chain of reasoning becomes explicit. Hidden moves are not allowed. The reader is invited to inspect every step.

In practice proofs rely on definitions, previously established results, and simple rules of inference. When these elements are combined carefully, complex conclusions can be derived from relatively simple beginnings.

The tool of proof is logic.

From the perspective of conviction formation, proof is a remarkable device. It organizes reasoning in such a way that agreement does not depend on the authority of the person presenting the argument. Anyone who understands the rules can in principle verify whether the conclusion follows.

Conviction stabilizes through the structure of the procedure itself.

Following a proof consists in checking that each step either applies an accepted rule of inference or follows from earlier results whose proofs are already known. Shortcuts are often taken in practice, but in principle every step could be expanded into a sequence of elementary rule applications.

In this way conviction travels through the proof: the inference rules have acquired credibility through systematization and long use, and the reader verifies that those rules have been applied correctly in the particular argument.

Contradiction

An important proof device is the proof by contradiction. It plays a large role in classical mathematics, though how far it can be used has been challenged by intuitionism.

Its convincing force is indirect. Instead of establishing a claim directly, it removes conviction from its alternative, so that conviction must settle on the remaining option.

Suppose we want to establish $not\ A$, but no direct proof is available. Assume we already know that $A\ or\ not\ A$ holds. We then assume $A$ and combine it with the other premises. From this assumption a consequence is derived of the form $B\ and\ not\ B$.

Such a result cannot stabilize conviction. The same statement is both affirmed and denied, and no coherent mapping to statements can sustain that. If the contradiction does have any convincing force, and the rules are supposed to preserve convincing force, then A cannot have had any to begin with.

Since $A\ or\ not\ A$ was granted from the start, rejecting $A$ leaves only $not\ A$. In this way conviction is produced not by supporting the target claim directly, but by showing that its alternative collapses.

The proof of the irrationality of $\sqrt{2}$ in the earlier chapter is of that kind.

Proof And Logic As Objects

Once proofs are written down explicitly, they acquire a curious status. They are no longer only instruments used to obtain results. They also become objects that can themselves be studied.

Mathematicians compare proofs, shorten them, generalize them, or discover alternative arguments for the same result. In doing so, the tools of reasoning are applied to reasoning itself. This often leads to clearer arguments, new proof techniques, and occasionally entirely new areas of investigation.

In modern mathematics proofs can even be checked or constructed by computer programs. The reasoning process becomes so explicit that machines can verify whether each step follows from the previous ones, turning proofs themselves into objects of systematic analysis.

This development illustrates how strongly conviction in mathematics is tied to the transparency of the reasoning process.

The same development applies to logic itself. Logical systems can be studied, compared, and refined using mathematical methods. Logic is thus used to analyze and improve its own formal structure.

Algorithms

Formal logic and proof show how reasoning can be organized into explicit rules whose steps can be checked one by one.

A related development appears in the notion of an algorithm. An algorithm is a finite sequence of precise instructions that transforms given inputs into a result. Each step is simple enough to be carried out mechanically, without requiring further judgment.

A familiar example appears in elementary arithmetic. Long division follows a fixed sequence of steps that can be executed reliably by anyone who has learned the procedure.

Here conviction stabilizes differently than in proof. In a proof, conviction depends on checking that each inferential step follows from the previous ones. In an algorithm, conviction depends on the correctness of the procedure as a whole and on the faithful execution of each step.

Once such procedures are stated with sufficient precision, they can be executed repeatedly and reliably, not only by humans but also by machines. Modern computers are devices for carrying out such procedures at enormous speed.

Algorithms therefore extend the move toward explicit control already seen in logic. They make certain procedures executable in a way that minimizes the need for further judgment during execution.

Limits Of Formal Control

The development of logic and proof represents a powerful attempt to bring reasoning under strict control. Within such systems each step can be inspected, and errors can often be located precisely.

But this control always operates within a framework: a set of definitions, axioms, and rules of inference.

The reliability of conclusions therefore depends on the stability of that framework itself.

As earlier chapters have shown, even highly successful frameworks can later reveal unexpected limitations. The discovery of incommensurable magnitudes did not undermine geometric reasoning. It revealed that the number system used to interpret geometrical magnitudes was incomplete.

Results of a similar character appear much later in the history of logic. Gödel's incompleteness theorem shows that sufficiently expressive formal systems cannot capture all truths about the structures they describe.

This reveals a limitation of formal systems, not a failure of their method. Their real power lies in something different: they create environments in which reasoning can be made explicit, inspected, and corrected with exceptional precision.

The success of this strategy, however, also depends on the objects to which it is applied. Mathematics deals with deliberately simplified and carefully defined objects, and under these conditions formal control has proven extraordinarily powerful. In other domains the situation is sometimes less cooperative. Attempts to impose the same degree of formal structure have often produced more modest results, for example in philosophy.

The Layer Beneath Logic

Scenes like the following are familiar from crime novels or detective films.

Suspect: "I did not do it. But I feel sorry for the poor man. No one deserves to be stabbed with a knife to death like that." Inspector: "But we never said the victim was stabbed with a knife."

The suspect is then arrested.

What gives this scene its convincing force and relief?

One possible answer would be logic. Another might be probability, which we will get to in the next chapter. But many people immediately find the conclusion persuasive without consciously applying logic or probability.

A possible answer is that such judgments rely on unconscious applications of logic or probability. But this explanation is unlikely to carry much weight. Most viewers have never studied formal logic or statistical reasoning in any depth, and few could apply such methods even deliberately, let alone implicitly while reacting to a short scene.

This explanation also presupposes too much of what it is meant to explain. Logic provides a language for describing such inferences, yet it does not follow that the convincing force of the scene arises from the formal system of logic itself. Logic was built over centuries to capture and discipline patterns of immediate conviction. That logic can later formalize and discipline such inferences does not show that it is their original source.

The examples considered so far already make it difficult to argue that conviction formation can be reduced to logic, and they point in the same direction for probability. Both presuppose forms of convincing force that they later formalize and discipline.

By the same token, if reading this paragraph makes it more doubtful that conviction formation can be reduced to logic or probability: What exactly produced that shift in conviction?