L2 inference rulesrule mtp Modus tollendo ponens: from a disjunction and the negation of one of its...
Transcript of L2 inference rulesrule mtp Modus tollendo ponens: from a disjunction and the negation of one of its...
L2 inference rules| University of Edinburgh | PHIL08004 |
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Alfred reads books ∴ Alfred reads books or God is an alien.
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rule add
Addition: from any sentence you may infer its disjunctionwith any other sentence.
add:
2
(2 ∨ #)(# ∨ 2)
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Either Alfred does yoga or Elle does yoga. But Alfred doesn’tdo yoga ∴ Elle does yoga.
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rule mtp
Modus tollendo ponens: from a disjunction and the negationof one of its disjuncts you may infer the other disjunct.
mtp:
(2 ∨ #)¬#
2
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Alfred reads books and Alfred does yoga. ∴ Alfred does yoga.
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rule s
Simplification: if you have a conjunction, you may infer eitherconjunct.
s:
(2 ∧ #)
2#
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God is made of spaghetti. God owns a velociraptor.
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God is made of spaghetti. God owns a velociraptor.∴ God is made of spaghetti and owns a velociraptor.
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rule adj
Adjunction: if you have any two sentences, you may infertheir conjunction, in either order.
adj:
2#
(2 ∧ #)(# ∧ 2)
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Biconditional
I “James is angry if and only if Ann is happy”
I ‘if and only if’ is a connective used to combine twopropositions
I A biconditional says that the truth of either propositionrequires the truth of the other, i.e., either both propositionsare true, or both are false
I P iff QI (P → Q) and (Q → P)
I (P ↔ Q)
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Biconditional
I “James is angry if and only if Ann is happy”
I ‘if and only if’ is a connective used to combine twopropositions
I A biconditional says that the truth of either propositionrequires the truth of the other, i.e., either both propositionsare true, or both are false
I P iff QI (P → Q) and (Q → P)
I (P ↔ Q)
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Biconditional
I “James is angry if and only if Ann is happy”
I ‘if and only if’ is a connective used to combine twopropositions
I A biconditional says that the truth of either propositionrequires the truth of the other, i.e., either both propositionsare true, or both are false
I P iff QI (P → Q) and (Q → P)
I (P ↔ Q)
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Biconditional
I “James is angry if and only if Ann is happy”
I ‘if and only if’ is a connective used to combine twopropositions
I A biconditional says that the truth of either propositionrequires the truth of the other, i.e., either both propositionsare true, or both are false
I P iff QI (P → Q) and (Q → P)
I (P ↔ Q)
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Biconditional
I “James is angry if and only if Ann is happy”
I ‘if and only if’ is a connective used to combine twopropositions
I A biconditional says that the truth of either propositionrequires the truth of the other, i.e., either both propositionsare true, or both are false
I P iff QI (P → Q) and (Q → P)
I (P ↔ Q)
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Biconditional
I “James is angry if and only if Ann is happy”
I ‘if and only if’ is a connective used to combine twopropositions
I A biconditional says that the truth of either propositionrequires the truth of the other, i.e., either both propositionsare true, or both are false
I P iff QI (P → Q) and (Q → P)
I (P ↔ Q)
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Biconditional
I “James is angry if and only if Ann is happy”
I ‘if and only if’ is a connective used to combine twopropositions
I A biconditional says that the truth of either propositionrequires the truth of the other, i.e., either both propositionsare true, or both are false
I P iff QI (P → Q) and (Q → P)
I (P ↔ Q)
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Biconditional
I “James is angry if and only if Ann is happy”
I ‘if and only if’ is a connective used to combine twopropositions
I A biconditional says that the truth of either propositionrequires the truth of the other, i.e., either both propositionsare true, or both are false
I P iff QI (P → Q) and (Q → P)
I (P ↔ Q)
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iff
I x is a rectangle iff:
I x is a polygon
necessary but not sufficient
I x has four sides
necessary but not sufficient
I x is a quadrilateral with right angles
necessary and sufficient
I x is a square
sufficient but not necessary
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iff
I x is a rectangle iff:
I x is a polygon
necessary but not sufficient
I x has four sides
necessary but not sufficient
I x is a quadrilateral with right angles
necessary and sufficient
I x is a square
sufficient but not necessary
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iff
I x is a rectangle iff:I x is a polygon
necessary but not sufficientI x has four sides
necessary but not sufficient
I x is a quadrilateral with right angles
necessary and sufficient
I x is a square
sufficient but not necessary
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iff
I x is a rectangle iff:I x is a polygon necessary but not sufficient
I x has four sides
necessary but not sufficient
I x is a quadrilateral with right angles
necessary and sufficient
I x is a square
sufficient but not necessary
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iff
I x is a rectangle iff:I x is a polygon necessary but not sufficientI x has four sides
necessary but not sufficientI x is a quadrilateral with right angles
necessary and sufficient
I x is a square
sufficient but not necessary
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iff
I x is a rectangle iff:I x is a polygon necessary but not sufficientI x has four sides necessary but not sufficientI x is a quadrilateral with right angles
necessary and sufficientI x is a square
sufficient but not necessary
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iff
I x is a rectangle iff:I x is a polygon necessary but not sufficientI x has four sides necessary but not sufficientI x is a quadrilateral with right angles necessary and sufficient
I x is a square
sufficient but not necessary
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iff
I x is a rectangle iff:I x is a polygon necessary but not sufficientI x has four sides necessary but not sufficientI x is a quadrilateral with right angles necessary and sufficientI x is a square
sufficient but not necessary
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iff
I x is a rectangle iff:I x is a polygon necessary but not sufficientI x has four sides necessary but not sufficientI x is a quadrilateral with right angles necessary and sufficientI x is a square sufficient but not necessary
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Puzzle. Two of the inhabitants—A and B—are standing togetherunder a tree. A makes the following statement: “I am a knave iffB is a knave.”
What is B?
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rule bc
Biconditional-to-conditional: from a biconditional you mayinfer either of the corresponding conditionals.
bc:
(2 ↔ #)
(2 → #)(# → 2)
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rule cb
Conditionals-to-biconditional: from two conditionals wherethe antecedent of one is the consequent of the other, andvice versa, you may infer a biconditional containing the partsof the conditionals.
cb:
(2 → #)(# → 2)
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Practice exercises
2:1 (P∨R). (¬R∧T). (P∨(Q∧T))→S ∴ S
2:3 (P↔¬Q)→R ∴ (¬R∧P)→Q
2:4 ¬(¬R→(Q∧T)). ¬S∨R ∴ ¬S
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