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 twopropositionsI A biconditional says that the truth of either proposition

requires 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 twopropositionsI A biconditional says that the truth of either proposition

requires 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 twopropositionsI A biconditional says that the truth of either proposition

requires 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 twopropositionsI A biconditional says that the truth of either proposition

requires 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 twopropositionsI A biconditional says that the truth of either proposition

requires 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 twopropositionsI A biconditional says that the truth of either proposition

requires 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 twopropositionsI A biconditional says that the truth of either proposition

requires 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 twopropositionsI A biconditional says that the truth of either proposition

requires 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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I If someone believes that p, then they know that p

I belief is not sufficient for knowledge

I If someone knows that p, then they believe that p

I belief is necessary for knowledge

I If someone knows that p, then p

I truth is necessary for knowledge

I If p, then someone knows that p

I truth is not sufficient for knowledge

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I If someone believes that p, then they know that p

I belief is not sufficient for knowledge

I If someone knows that p, then they believe that p

I belief is necessary for knowledge

I If someone knows that p, then p

I truth is necessary for knowledge

I If p, then someone knows that p

I truth is not sufficient for knowledge

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I If someone believes that p, then they know that pI belief is not sufficient for knowledge

I If someone knows that p, then they believe that p

I belief is necessary for knowledge

I If someone knows that p, then p

I truth is necessary for knowledge

I If p, then someone knows that p

I truth is not sufficient for knowledge

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I If someone believes that p, then they know that pI belief is not sufficient for knowledge

I If someone knows that p, then they believe that p

I belief is necessary for knowledge

I If someone knows that p, then p

I truth is necessary for knowledge

I If p, then someone knows that p

I truth is not sufficient for knowledge

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I If someone believes that p, then they know that pI belief is not sufficient for knowledge

I If someone knows that p, then they believe that pI belief is necessary for knowledge

I If someone knows that p, then p

I truth is necessary for knowledge

I If p, then someone knows that p

I truth is not sufficient for knowledge

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I If someone believes that p, then they know that pI belief is not sufficient for knowledge

I If someone knows that p, then they believe that pI belief is necessary for knowledge

I If someone knows that p, then p

I truth is necessary for knowledge

I If p, then someone knows that p

I truth is not sufficient for knowledge

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I If someone believes that p, then they know that pI belief is not sufficient for knowledge

I If someone knows that p, then they believe that pI belief is necessary for knowledge

I If someone knows that p, then pI truth is necessary for knowledge

I If p, then someone knows that p

I truth is not sufficient for knowledge

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I If someone believes that p, then they know that pI belief is not sufficient for knowledge

I If someone knows that p, then they believe that pI belief is necessary for knowledge

I If someone knows that p, then pI truth is necessary for knowledge

I If p, then someone knows that p

I truth is not sufficient for knowledge

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I If someone believes that p, then they know that pI belief is not sufficient for knowledge

I If someone knows that p, then they believe that pI belief is necessary for knowledge

I If someone knows that p, then pI truth is necessary for knowledge

I If p, then someone knows that pI truth is not sufficient for knowledge

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x knows that p iff x has a justified true belief that p

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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)

(2 ↔ #)17 / 18

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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