We have already seen the structure of reasoning and simple examples. Now we will test that knowledge by reasoning with 4 deduction techniques of propositional logic. Through this, we will not only see that these things work, but we will also begin to give a certain richness of procedures that will take the propositional calculus out of the rudimentary state it has been presented in so far.

If \alpha, \beta and \gamma are expressions of propositional calculus, then it is possible to infer the following deduction techniques from the fundamentals:

Presumption Rule (Pre)

The simplest deduction rule of all is the presumption rule. This is obtained directly by applying the reciprocal of the deduction theorem on the theorem \vdash(\alpha\rightarrow\alpha). If this sounded like arcane language to you, everything you need to know is here.

\{\alpha\}\vdash \alpha

Combined with the monotonicity rule, it will allow you to add convenient expressions within your deductions.

Hypothetical Syllogism (HS)

Hypothetical syllogism, or implication transitivity, is a kind of evolution of modus ponens. Its formulation is as follows:

\{(\alpha\rightarrow\beta), (\beta\rightarrow\gamma)\}\vdash (\alpha\rightarrow\gamma)

There are several ways to obtain this deduction rule, we will see a couple of them shortly.

If we reason from expressions, it will be easy to construct the following reasoning:

Therefore \{\alpha,(\alpha\rightarrow\beta),(\beta\rightarrow\gamma)\}\vdash\gamma

Finally, applying the deduction theorem to this last expression, we have:

\{(\alpha\rightarrow\beta),(\beta\rightarrow\gamma)\}\vdash(\alpha\rightarrow \gamma)

Another way to obtain the demonstration of this rule is by reasoning from deductions, constructing through presumption and monotonicity. Observe the following reasoning from deductions:

You should note here that both demonstrations are identical, only developed in different styles. In practice, you can alternate between both styles depending on what you find more comfortable.

Double Negation Equivalences (DN)

Double negation equivalences reproduce the intuitive notion that the double negation of a statement is equivalent to the statement itself. This, written symbolically, will be in the form

\alpha\dashv\vdash\neg\neg\alpha

Now let’s see a demonstration:

Therefore\{\neg\neg \alpha \} \vdash \alpha

To make the demonstration in the other direction, we can use this one we just did by readapting it through a simple substitution, obtaining the following:

\{\neg\neg \neg \alpha \} \vdash \neg \alpha

And from this, we construct the demonstration in the other direction:

Therefore \{\alpha \} \vdash \neg\neg \alpha

Finally, from these two demonstrations, we have that \alpha \dashv\vdash \neg\neg \alpha .

Contrapositive Equivalence of Implication (CpI)

This corresponds to the following equivalences

(\alpha \rightarrow \beta) \dashv\vdash (\neg\beta \rightarrow \neg\alpha)

(\neg\alpha\rightarrow\beta)\dashv\vdash (\neg\beta\rightarrow\alpha)

(\alpha\rightarrow\neg\beta) \dashv\vdash (\beta\rightarrow\neg\alpha)

The demonstration of this first relationship is done as follows:

On one side, it is obtained directly from the third axiom

Therefore \{(\neg\beta\rightarrow \neg\alpha)\}\vdash (\alpha \rightarrow \beta)

And in the other direction, the demonstration can be obtained from the following reasoning:

Therefore \{(\alpha \rightarrow \beta)\}\vdash (\neg \beta \rightarrow \neg \alpha )

Therefore, from the two previous reasonings, we have that

(\alpha \rightarrow \beta) \dashv\vdash (\neg \beta \rightarrow \neg \alpha )

To demonstrate the second one, we can make the following two reasonings:

Therefore \{(\neg\alpha \rightarrow \beta)\} \vdash(\neg\beta \rightarrow \alpha)

Now we need to make the demonstration in the reverse direction. We can do it through the following reasoning:

Therefore \{(\neg\beta\rightarrow\alpha)\}\vdash (\neg\alpha \rightarrow \beta)

Finally, from these two reasonings, it is concluded that (\neg\beta\rightarrow\alpha) \dashv \vdash (\neg\alpha \rightarrow \beta) , which is what we wanted to demonstrate.

The last equivalence will be left as an exercise. To demonstrate it, you can guide yourself with the two demonstrations I have already given. This is the best way to master deduction techniques.