 Hypothetical syllogism

Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B)
Modus tollens (A→B, ¬B ⊢ ¬A)
Modus ponendo tollens (¬(A∧B), A ⊢ ¬B)
Conjunction introduction (A, B ⊢ A∧B)
Simplification (A∧B ⊢ A)
Disjunction introduction (A ⊢ A∨B)
Disjunction elimination (A∨B, A→C, B→C ⊢ C)
Disjunctive syllogism (A∨B, ¬A ⊢ B)
Hypothetical syllogism (A→B, B→C ⊢ A→C)
Constructive dilemma (A→P, B→Q, A∨B ⊢ P∨Q)
Destructive dilemma (A→P, B→Q, ¬P∨¬Q ⊢ ¬A∨¬B)
Biconditional introduction (A→B, B→A ⊢ A↔B)
Biconditional elimination (A↔B ⊢ A→B)Predicate calculus Universal generalization
Universal instantiation
Existential generalization
Existential instantiationIn logic, a hypothetical syllogism has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic, it is a shorthand for the theory of consequence.
Propositional logic
Hypothetical syllogism is one of the proof rules in classical logic that may or may not be available in a nonclassical logic. The hypothetical syllogism (abbr. H.S.) is a valid argument of the following form:
 If P → Q.
 If Q → R.
____________________
 Then P → R.
Symbolically, this is expressed:
Example of use:
 If I do not wake up, then I cannot go to work.
 If I cannot go to work, then I will not get paid.
 Therefore, if I do not wake up, then I will not get paid.
See also
 Modus ponens
 Modus tollens
 Modus tollendo ponens
 Affirming the consequent
 Denying the antecedent
 Disjunctive syllogism
 Inference rule
 Transitive relation
References
Categories: Rules of inference
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Look at other dictionaries:
hypothetical syllogism — noun Etymology: translation of Late Latin hypotheticus syllogismus 1. : a syllogism consisting wholly of hypothetical propositions called also pure hypothetical syllogism 2. : a syllogism consisting partly of hypothetical propositions called also … Useful english dictionary
mixed hypothetical syllogism — noun see hypothetical syllogism … Useful english dictionary
pure hypothetical syllogism — noun see hypothetical syllogism … Useful english dictionary
Syllogism — A syllogism (Greek: συλλογισμός – syllogismos – conclusion, inference ) is a kind of logical argument in which one proposition (the conclusion) is inferred from two or more others (the premises) of a certain form. In antiquity, there were… … Wikipedia
syllogism — /ˈsɪlədʒɪzəm / (say siluhjizuhm) noun 1. Logic an argument with two premises and a conclusion. Both the premises of a categorical syllogism are categorical propositions, containing just three distinct terms between them, e.g. all men are mortal… … Australian English dictionary
syllogism — /sil euh jiz euhm/, n. 1. Logic. an argument the conclusion of which is supported by two premises, of which one (major premise) contains the term (major term) that is the predicate of the conclusion, and the other (minor premise) contains the… … Universalium
hypothetical — /haɪpəˈθɛtɪkəl / (say huypuh thetikuhl) adjective Also, hypothetic. 1. assumed by hypothesis; supposed: a hypothetical case. 2. relating to, involving, or of the nature of hypothesis: hypothetical reasoning. 3. given to making hypotheses: a… … Australian English dictionary
hypothetical — hypothetically, adv. /huy peuh thet i keuhl/, adj. 1. assumed by hypothesis; supposed: a hypothetical case. 2. of, pertaining to, involving, or characterized by hypothesis: hypothetical reasoning. 3. given to making hypotheses. 4. Logic. a. (of a … Universalium
Disjunctive syllogism — Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) … Wikipedia
logic — logicless, adj. /loj ik/, n. 1. the science that investigates the principles governing correct or reliable inference. 2. a particular method of reasoning or argumentation: We were unable to follow his logic. 3. the system or principles of… … Universalium