====== (Valid) Inferences ======

This category contains articles with valid logical inferences, in particular from propositional logic.

===== 1. (Material) Conditionals =====

| ^  <i :la>[[logic:inferences:modus_ponens|Modus ponens]]</i>  ^  <i :la>[[logic:inferences:modus_tollens|Modus tollens]]</i>  ^
^ Premise 1 |  <span "If A, then B">A → B</span> \\ <small>(If A, then B)</small>  |  <span "if A, then B">A → B</span> \\ <small>(If A, then B)</small>  |
^ Premise 2 |  A  |  <span "not B">⌐B</span> <small>(not B)</small>  |
^ Conclusion |  B  |  <span "not A">⌐A</span> <small>(not A)</small>  |
| ||
^ Type |  [[glossary:conditional|Conditional]]  |  [[glossary:conditional|Conditional]]  | 

There are also more complex forms that are derived from these two moduses, for example:
  * [[logic:inferences:constructive_dilemma|Constructive dilemma]]
  * [[logic:inferences:destructive_dilemma|Destructive dilemma]]
  * [[logic:inferences:contraposition|Contraposition]]
  * [[logic:inferences:sorites|Sorites]] (polysyllogism)
  * [[logic:inferences:hypothetical_syllogism|Hypothetical syllogism]]

===== 2. Conjunctions and Disjunctions =====
 
| ^  <i :la>[[logic:inferences:modus_ponendo_tollens|Modus ponendo tollens]]</i>  ^^ ^  <i :la>[[logic:inferences:modus_tollendo_ponens|Modus tollendo ponens]]</i>  ^^
^Premise 1 |  <span "not both, A and B">⌐(A ∧ B)</span> \\ <small>(not both, A and B)</small>  || |  <span "A or [incl.] B">A ∨ B</span> \\ <small>(A or [incl.] B)</small>  ||
^Premise 2 |  A  |  B  | |  <span "not A">⌐A</span> <small>(not A)</small>  |  <span "not B">⌐B</span> <small>(not B)</small>  |
^Conclusion |  <span "not B">⌐B</span> <small>(not B)</small>  |  <span "not A">⌐A</span> <small>(not A)</small>  | |  B  |  A  |
| |||
^Type |  [[glossary:conjunction|Conjunction]]  || |  [[glossary:disjunction|Disjunction]]  ||

===== 3. Syllogisms =====

[[glossary:syllogism|Syllogisms]] are forms of logical inferences that infer a new proposition from //two// premises containing a total of //three// terms. There are 24 //valid// syllogisms which can be grouped into four categories by their basic forms: 

  * **[[logic:inferences:modus_barbara:index|Modus Barbara]]**, with variants: [[logic:inferences:modus_barbara:modus_baroco|Baroco]], [[logic:inferences:modus_barbara:modus_bocardo|Bocardo]], [[logic:inferences:modus_barbara:modus_barbari|Barbari]] and [[logic:inferences:modus_barbara:modus_bamalip|Bamalip]].
  * **[[logic:inferences:modus_celarent:index|Modus Celarent]]**, with variants: [[logic:inferences:modus_celarent:modus_calemes|Calemes]], [[logic:inferences:modus_celarent:modus_calemos|Calemos]], [[logic:inferences:modus_celarent:modus_camestres|Camestres]], [[logic:inferences:modus_celarent:modus_camestros|Camestros]], [[logic:inferences:modus_celarent:modus_celaront|Celaront]], [[logic:inferences:modus_celarent:modus_cesare|Cesare]] and [[logic:inferences:modus_celarent:modus_cesaro|Cesaro]].
  * **[[logic:inferences:modus_darii:index|Modus Darii]]**, with variants: [[logic:inferences:modus_darii:modus_dimatis|Dimatis]], [[logic:inferences:modus_darii:modus_datisi|Datisi]], [[logic:inferences:modus_darii:modus_disamis|Disamis]] and [[logic:inferences:modus_darii:modus_darapti|Darapti]].
  * **[[logic:inferences:modus_ferio:index|Modus Ferio]]**, with variants:  [[logic:inferences:modus_ferio:modus_festino|Festino]], [[logic:inferences:modus_ferio:modus_ferison|Ferison]], [[logic:inferences:modus_ferio:modus_felapton|Felapton]], [[logic:inferences:modus_ferio:modus_fesapo|Fesapo]] and [[logic:inferences:modus_ferio:modus_fresison|Fresison]].

In addition, the [[logic:inferences:sorites|Sorites]] (polysyllogism) can also be seen as a more general form of the [[logic:inferences:modus_barbara:index|Modus Barbara]].

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