Editing Deductive reasoning (section)

=== Prominent rules of inference ===
==== Modus ponens ====
{{Main|Modus ponens|selfref = None}}

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 [[Material conditional|conditional statement]] (<math>P \rightarrow Q</math>) and as second premise the antecedent (<math>P</math>) of the conditional statement. It obtains the consequent (<math>Q</math>) of the conditional statement as its conclusion. The argument form is listed below:
# <math>P \rightarrow Q</math>&nbsp; (First premise is a conditional statement)
# <math>P</math>&nbsp; (Second premise is the antecedent)
# <math>Q</math>&nbsp; (Conclusion deduced is the consequent)

In this form of deductive reasoning, the consequent (<math>Q</math>) obtains as the conclusion from the premises of a conditional statement (<math>P \rightarrow Q</math>) and its antecedent (<math>P</math>). However, the antecedent (<math>P</math>) cannot be similarly obtained as the conclusion from the premises of the conditional statement (<math>P \rightarrow Q</math>) and the consequent (<math>Q</math>). Such an argument commits the [[logical fallacy]] of [[affirming the consequent]].

The following is an example of an argument using modus ponens:
# If it is raining, then there are clouds in the sky.
# It is raining.
# Thus, there are clouds in the sky.

==== Modus tollens ====
{{Main|Modus tollens}}
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 (<math>\lnot Q</math>) and as conclusion the negation of the antecedent (<math>\lnot P</math>). 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:

# <math>P \rightarrow Q</math>. (First premise is a conditional statement)
# <math>\lnot Q</math>. (Second premise is the negation of the consequent)
# <math>\lnot P</math>. (Conclusion deduced is the negation of the antecedent)

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

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

==== Hypothetical syllogism ====
{{main|hypothetical syllogism}}
A ''hypothetical [[syllogism]]'' is an inference that 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:
# <math>P \rightarrow Q</math>
# <math>Q \rightarrow R</math>
# Therefore, <math>P \rightarrow R</math>.

In there being a subformula in common between the two premises that does not occur in the consequence, this resembles syllogisms in [[term logic]], although it differs in that this subformula is a proposition whereas in Aristotelian logic, this common element is a term and not a proposition.

The following is an example of an argument using a hypothetical syllogism:

# If there had been a thunderstorm, it would have rained.
# If it had rained, things would have gotten wet.
# Thus, if there had been a thunderstorm, things would have gotten wet.<ref>{{Cite journal |last=Morreau |first=Michael |year=2009 |title=The Hypothetical Syllogism |journal=Journal of Philosophical Logic |volume=38 |issue=4 |pages=447–464 |doi=10.1007/s10992-008-9098-y |issn=0022-3611 |jstor=40344073 |s2cid=34804481}}</ref>