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### Transcript of L2 inference rules - brian.rabern rule mtp Modus tollendo ponens: from a disjunction and the...

• 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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Addition: from any sentence you may infer its disjunction with any other sentence.

2

(2 ∨ #) (# ∨ 2)

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• Either Alfred does yoga or Elle does yoga. But Alfred doesn’t do yoga ∴ Elle does yoga.

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• rule mtp

Modus tollendo ponens: from a disjunction and the negation of 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 either conjunct.

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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Adjunction: if you have any two sentences, you may infer their conjunction, in either order.

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 two propositions I A biconditional says that the truth of either proposition

requires the truth of the other, i.e., either both propositions are true, or both are false

I P iff Q I (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 two propositions I A biconditional says that the truth of either proposition

requires the truth of the other, i.e., either both propositions are true, or both are false

I P iff Q I (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 two propositions I A biconditional says that the truth of either proposition

requires the truth of the other, i.e., either both propositions are true, or both are false

I P iff Q I (P → Q) and (Q → P)

I (P ↔ Q)

10 / 18

• Biconditional

I “James is angry if and only if Ann is happy”

I ‘if and only if’ is a connective used to combine two propositions I A biconditional says that the truth of either proposition

requires the truth of the other, i.e., either both propositions are true, or both are false

I P iff Q I (P → Q) and (Q → P)

I (P ↔ Q)

10 / 18

• Biconditional

I “James is angry if and only if Ann is happy”

I ‘if and only if’ is a connective used to combine two propositions I A biconditional says that the truth of either proposition

requires the truth of the other, i.e., either both propositions are true, or both are false

I P iff Q I (P → Q) and (Q → P)

I (P ↔ Q)

10 / 18

• Biconditional

I “James is angry if and only if Ann is happy”

I ‘if and only if’ is a connective used to combine two propositions I A biconditional says that the truth of either proposition

requires the truth of the other, i.e., either both propositions are true, or both are false

I P iff Q I (P → Q) and (Q → P)

I (P ↔ Q)

10 / 18

• Biconditional

I “James is angry if and only if Ann is happy”

I ‘if and only if’ is a connective used to combine two propositions I A biconditional says that the truth of either proposition

requires the truth of the other, i.e., either both propositions are true, or both are false

I P iff Q I (P → Q) and (Q → P)

I (P ↔ Q)

10 / 18

• Biconditional

I “James is angry if and only if Ann is happy”

I ‘if and only if’ is a connective used to combine two propositions I A biconditional says that the truth of either proposition

requires the truth of the other, i.e., either both propositions are true, or both are false

I P iff Q I (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 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 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 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 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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• 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 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 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