Deductive reasoning

Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive rule of inference. It applies to arguments that have as first premise a conditional statement (${\displaystyle P\rightarrow Q}$) and as second premise the antecedent (${\displaystyle P}$) of the conditional statement. It obtains the consequent (${\displaystyle Q}$) of the conditional statement as its conclusion. The argument form is listed below:

1. ${\displaystyle P\rightarrow Q}$  (First premise is a conditional statement)
2. ${\displaystyle P}$  (Second premise is the antecedent)
3. ${\displaystyle Q}$  (Conclusion deduced is the consequent)

In this form of deductive reasoning, the consequent (${\displaystyle Q}$) obtains as the conclusion from the premises of a conditional statement (${\displaystyle P\rightarrow Q}$) and its antecedent (${\displaystyle P}$). However, the antecedent (${\displaystyle P}$) cannot be similarly obtained as the conclusion from the premises of the conditional statement (${\displaystyle P\rightarrow Q}$) and the consequent (${\displaystyle Q}$). Such an argument commits the logical fallacy of affirming the consequent.

The following is an example of an argument using modus ponens:

1. If an angle satisfies 90° < ${\displaystyle A}$ < 180°, then ${\displaystyle A}$ is an obtuse angle.
2. ${\displaystyle A}$ = 120°.
3. ${\displaystyle A}$ is an obtuse angle.

Since the measurement of angle ${\displaystyle A}$ is greater than 90° and less than 180°, we can deduce from the conditional (if-then) statement that ${\displaystyle A}$ is an obtuse angle. However, if we are given that ${\displaystyle A}$ is an obtuse angle, we cannot deduce from the conditional statement that 90° < ${\displaystyle A}$ < 180°. It might be true that other angles outside this range are also obtuse.

Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (formula) and the negation of the consequent (${\displaystyle \lnot Q}$) and as conclusion the negation of the antecedent (${\displaystyle \lnot P}$). In contrast to modus ponens, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following:

1. ${\displaystyle P\rightarrow Q}$. (First premise is a conditional statement)
2. ${\displaystyle \lnot Q}$. (Second premise is the negation of the consequent)
3. ${\displaystyle \lnot P}$. (Conclusion deduced is the negation of the antecedent)

The following is an example of an argument using modus tollens:

1. If it is raining, then there are clouds in the sky.
2. There are no clouds in the sky.
3. Thus, it is not raining.

In proposition logic the law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:

1. ${\displaystyle P\rightarrow Q}$
2. ${\displaystyle Q\rightarrow R}$
3. Therefore, ${\displaystyle P\rightarrow R}$.

The following is an example:

1. If the animal is a Yorkie, then it's a dog.
2. If the animal is a dog, then it's a mammal.
3. Therefore, if the animal is a Yorkie, then it's a mammal.

We deduced the final statement by combining the hypothesis of the first statement with the conclusion of the second statement. We also allow that this could be a false statement. This is an example of the transitive property in mathematics. Another example is the transitive property of equality which can be stated in this form:

1. ${\displaystyle A=B}$.
2. ${\displaystyle B=C}$.
3. Therefore, ${\displaystyle A=C}$.