In propositional logic, hypothetical syllogism is the name of a valid rule of inference (often abbreviated HS and sometimes also called the chain argument, chain rule, or the principle of transitivity of implication). The rule may be stated:
where the rule is that whenever instances of "", and "" appear on lines of a proof, "" can be placed on a subsequent line.
Hypothetical syllogism is closely related and similar to disjunctive syllogism, in that it is also a type of syllogism, and also the name of a rule of inference.
If Jones wins the election, Smith will retire after the election.
If Smith dies before the election, Jones will win the election.
If Smith dies before the election, Smith will retire after the election.
Clearly, (3) does not follow from (1) and (2). (1) is true by default, but fails to hold in the exceptional circumstances of Smith dying. In practice, real-world conditionals always tend to involve default assumptions or contexts, and it may be infeasible or even impossible to specify all the exceptional circumstances in which they might fail to be true. For similar reasons, the rule of hypothetical syllogism does not hold for counterfactual conditionals.
Formal notation
The hypothetical syllogism inference rule may be written in sequent notation, which amounts to a specialization of the cut rule:
An alternative form of hypothetical syllogism, more useful for classical propositional calculus systems with implication and negation (i.e. without the conjunction symbol), is the following:
(HS1)
Yet another form is:
(HS2)
Proof
An example of the proofs of these theorems in such systems is given below. We use two of the three axioms used in one of the popular systems described by Jan Łukasiewicz. The proofs relies on two out of the three axioms of this system:
^Adams, Ernest W. (1975). The Logic of Conditionals. Dordrecht: Reidel. p. 22.
External links
Philosophy Index: Hypothetical Syllogism
January 06, 2023
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In classical logic a hypothetical syllogism is a valid argument form a syllogism with a conditional statement for one or both of its premises Hypothetical syllogismTypeSyllogismFieldPropositional calculus Classical logic Intuitionistic logic Most systems of relevance logicStatementWhenever instances of P Q displaystyle P to Q and Q R displaystyle Q to R appear on lines of a proof P R displaystyle P to R can be placed on a subsequent line Symbolic statementP Q Q R P R displaystyle frac P to Q Q to R therefore P to R An example in English 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 The term originated with Theophrastus 1 Contents 1 Propositional logic 2 Applicability 3 Formal notation 4 Proof 5 Alternative forms 5 1 Proof 5 2 As a metatheorem 6 See also 7 References 8 External linksPropositional logic EditIn propositional logic hypothetical syllogism is the name of a valid rule of inference often abbreviated HS and sometimes also called the chain argument chain rule or the principle of transitivity of implication The rule may be stated P Q Q R P R displaystyle frac P to Q Q to R therefore P to R where the rule is that whenever instances of P Q displaystyle P to Q and Q R displaystyle Q to R appear on lines of a proof P R displaystyle P to R can be placed on a subsequent line Hypothetical syllogism is closely related and similar to disjunctive syllogism in that it is also a type of syllogism and also the name of a rule of inference Applicability EditThe rule of hypothetical syllogism holds in classical logic intuitionistic logic most systems of relevance logic and many other systems of logic However it does not hold in all logics including for example non monotonic logic probabilistic logic and default logic The reason for this is that these logics describe defeasible reasoning and conditionals that appear in real world contexts typically allow for exceptions default assumptions ceteris paribus conditions or just simple uncertainty An example derived from Ernest W Adams 2 If Jones wins the election Smith will retire after the election If Smith dies before the election Jones will win the election If Smith dies before the election Smith will retire after the election Clearly 3 does not follow from 1 and 2 1 is true by default but fails to hold in the exceptional circumstances of Smith dying In practice real world conditionals always tend to involve default assumptions or contexts and it may be infeasible or even impossible to specify all the exceptional circumstances in which they might fail to be true For similar reasons the rule of hypothetical syllogism does not hold for counterfactual conditionals Formal notation EditThe hypothetical syllogism inference rule may be written in sequent notation which amounts to a specialization of the cut rule P Q Q R P R displaystyle frac P vdash Q quad Q vdash R P vdash R where displaystyle vdash is a metalogical symbol and A B displaystyle A vdash B meaning that B displaystyle B is a syntactic consequence of A displaystyle A in some logical system and expressed as a truth functional tautology or theorem of propositional logic P Q Q R P R displaystyle P to Q land Q to R to P to R where P displaystyle P Q displaystyle Q and R displaystyle R are propositions expressed in some formal system Proof EditStep Proposition Derivation1 P Q displaystyle P to Q Given2 Q R displaystyle Q to R Given3 P displaystyle P Conditional proof assumption4 Q displaystyle Q Modus ponens 1 3 5 R displaystyle R Modus ponens 2 4 6 P R displaystyle P to R Conditional Proof 3 5 Alternative forms EditAn alternative form of hypothetical syllogism more useful for classical propositional calculus systems with implication and negation i e without the conjunction symbol is the following HS1 Q R P Q P R displaystyle Q to R to P to Q to P to R Yet another form is HS2 P Q Q R P R displaystyle P to Q to Q to R to P to R Proof Edit An example of the proofs of these theorems in such systems is given below We use two of the three axioms used in one of the popular systems described by Jan Lukasiewicz The proofs relies on two out of the three axioms of this system A1 ϕ ps ϕ displaystyle phi to left psi to phi right A2 ϕ ps 3 ϕ ps ϕ 3 displaystyle left phi to left psi rightarrow xi right right to left left phi to psi right to left phi to xi right right The proof of the HS1 is as follows 1 p q r p q p r q r p q r p q p r displaystyle p to q to r to p to q to p to r to q to r to p to q to r to p to q to p to r instance of A1 2 p q r p q p r displaystyle p to q to r to p to q to p to r instance of A2 3 q r p q r p q p r displaystyle q to r to p to q to r to p to q to p to r from 1 and 2 by modus ponens 4 q r p q r p q p r q r p q r q r p q p r displaystyle q to r to p to q to r to p to q to p to r to q to r to p to q to r to q to r to p to q to p to r instance of A2 5 q r p q r q r p q p r displaystyle q to r to p to q to r to q to r to p to q to p to r from 3 and 4 by modus ponens 6 q r p q r displaystyle q to r to p to q to r instance of A1 7 q r p q p r displaystyle q to r to p to q to p to r from 5 and 6 by modus ponens The proof of the HS2 is given here As a metatheorem Edit Whenever we have two theorems of the form T 1 Q R displaystyle T 1 Q to R and T 2 P Q displaystyle T 2 P to Q we can prove P R displaystyle P to R by the following steps 1 Q R P Q P R displaystyle Q to R to P to Q to P to R instance of the theorem proved above 2 Q R displaystyle Q to R instance of T1 3 P Q P R displaystyle P to Q to P to R from 1 and 2 by modus ponens 4 P Q displaystyle P to Q instance of T2 5 P R displaystyle P to R from 3 and 4 by modus ponens See also EditModus ponens Modus tollens Affirming the consequent Denying the antecedent Transitive relationReferences Edit History of Logic Theophrastus of Eresus in Encyclopaedia Britannica Online Adams Ernest W 1975 The Logic of Conditionals Dordrecht Reidel p 22 External links EditPhilosophy Index Hypothetical Syllogism Retrieved from https en wikipedia org w index php title Hypothetical syllogism amp oldid 1125821653, wikipedia, wiki, book, books, library,