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

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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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  • rule add

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

    add:

    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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  • rule adj

    Adjunction: if you have any two sentences, you may infer their 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 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)

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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)

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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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  • 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

    11 / 18

  • 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