Deductive reasoning Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! === 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: # <code><math>P \rightarrow Q</math></code> (First premise is a conditional statement) # <math>P</math> (Second premise is the antecedent) # <math>Q</math> (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 |last1=Morreau |first1=Michael |title=The Hypothetical Syllogism |journal=Journal of Philosophical Logic |date=2009 |volume=38 |issue=4 |pages=447β464 |doi=10.1007/s10992-008-9098-y |jstor=40344073 |s2cid=34804481 |url=https://www.jstor.org/stable/40344073 |issn=0022-3611}}</ref> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page