What is a direct proof, formally?

Question

In my logic class I took in college, I sometimes would get wrong marks because I proved a statement indirectly instead of directly. Informally, a direct proof of a conditional $P \implies Q$, is one where you assume $P$ and try to deduce $Q$ "directly", without using the contrapositive of that conditional. But no one has ever given me a formal, rigorous definition of direct proof. Can someone give me one, and with one, can someone give examples of statements which can't be proven directly, only indirectly?

Answer 1

As Rob Arthan said in the comments, one way to distinguish between (one kind of) direct proof and indirect proof is simply to distinguish between proofs that use intuitionistically valid deductive rules and proofs that do not.

To show that there are some theorems (using classical logic) that have no intuitionistically valid proofs, it suffices to show that there is a way to interpret intuitionistic logic such that the intuitionistic deductive rules are sound for that interpretation, and also that there is a tautology of classical logic that is not a tautology under that interpretation. A popular one is LEM (i.e. $A∨¬A$), but it may not be convincing. Here is another:

$(A⇒B)∨(B⇒A)$.

You cannot prove this classical tautology in intuitionistic logic. In that precise sense there can be no direct proof of this tautology. This is more convincing than LEM because it is even slightly counter-intuitive and you really need to 'check cases' (relying on LEM) to prove it. Lest you think that 'checking cases' is direct, observe that every intuitionistic proof of ( $¬B⇒¬A$ ) can be translated to a proof that is intuitionistic except for a single LEM case-checking:

  If $A$:
    $B∨¬B$.
    If $B$:
      $B$.
    If $¬B$:
      [Insert proof of ( $¬B⇒¬A$ ) here.]
      $¬A$.
      Contradiction (i.e. $A,¬A$).
      $B$. [by explosion]
    $B$.
  $A⇒B$.

There are many interpretations of intuitionistic logic under which this classical tautology does not hold. One is Kripke frames (see here for a sketch), and another is the BHK interpretation. Understanding either of them will give you a clear idea of how exactly an intuitionistic proof can be considered to be 'direct'.

Answer 2

From wikipedia at https://en.wikipedia.org/wiki/Direct_proof

"In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions.In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations in which the statement p is true. Logical deduction is employed to reason from assumptions to conclusion."

This sentence "In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations in which the statement p is true.", is particularly important because it tells us that a direct proof only examines what happens if the hypothesis is true. A true hypothesis can't yield a false conclusion, so a direct proof doesn't examine false conclusions, either. However, an indirect proof examines what would happen if the conclusion were false for its proof.

I am not sure if there is any theorem that has been proven to have indirect proof, but here's an indirect proof of the irrationality of $\sqrt2$ as an example. https://www.math.utah.edu/~pa/math/q1.html

Answer 3

A formal notion of a proof is:
Given a set of axioms $\Delta$ and a formula $\alpha$, a proof of $\alpha$ is a finite sequence $(x_i)$, $i \in \{1,...,n\}$, such that:

• $x_n = \alpha$;
• Either $x_j$ , $1 \le j \lt n$, is a axiom or a tautology;
• Or $x_j$ , $1 \lt j \lt n$, is deduced through an inference rule from the set of formulas $\{x_k\}$, $k \lt j$.
  
  A standard, and frequently unique, inference rule is the Modus Ponens: $$\{(\alpha \to \beta), \alpha\} \vdash \beta$$
  This notion of proof is from a view of formal systems. As a bibliographic reference, in "Metalogic: An Introduction to the Metatheory of Standard First-Order Logic", Geoffrey Hunter shortly exposes this notion. But this is a wide use notion of proof, far from restricted to this book.
  
  

Now to the question of what is a direct proof.

As indicatet earlier by Some Guy (https://math.stackexchange.com/users/730299/some-guy) from Wikipedia, https://en.wikipedia.org/wiki/Direct_proof, "In order to directly prove a conditional statement of the form 'If p, then q', it suffices to consider the situations in which the statement p is true.". It seems that when someone says that a proof is a indirect proof, it is because it is used reductio ad absurdum or the modus tollens. And that is right, in these cases we use more than just to consider the hypothesis is true, we do negate the conclusion. Lets have a closer look:

• modus tollens: $\{(\alpha \to \beta), \lnot\beta\} \vdash \lnot\alpha$
  
• reductio ad absurdum: $\{(\lnot \alpha \to \lnot \beta), (\lnot \alpha \to \beta\} \vdash \alpha$
  
  Both are, actually, valid inference rules in classical logic. They can be deduced just from tautologies and the modus ponens. Then they use can be reduced to the use of modus ponens, but with the crucial use also from the right tautologies. These tautologies are tautologies because of our definitions of the formal classical operators, e.g. $\lnot, \land, \lor, \to,$ etc.
  Then why do some peoples desire much more a direct proof or do not trust fully in indirect proofs? The answer lies, I believe, in the credibility of the classical logic. The intuitionists don't believe in the excluded middle, therefore don't they don't hold that the tautology that implies in the validation of the reductio ad absurdum as a inference rule. For these mathematicians, without a direct proof that something is true or that something is false, there has nothing to be said.
  I hope this was fruitfull and helpfull, also that there are no mistakens.