rule add
Addition: from any sentence you may infer its disjunctionwith any other sentence.
add:
2
(2 ∨ #)(# ∨ 2)
3 / 18
rule mtp
Modus tollendo ponens: from a disjunction and the negationof one of its disjuncts you may infer the other disjunct.
mtp:
(2 ∨ #)¬#
2
5 / 18
rule s
Simplification: if you have a conjunction, you may infer eitherconjunct.
s:
(2 ∧ #)
2#
7 / 18
God is made of spaghetti. God owns a velociraptor.∴ God is made of spaghetti and owns a velociraptor.
8 / 18
rule adj
Adjunction: if you have any two sentences, you may infertheir conjunction, in either order.
adj:
2#
(2 ∧ #)(# ∧ 2)
9 / 18
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)
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 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)
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 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)
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 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)
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 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)
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 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)
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 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)
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 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)
10 / 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
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
11 / 18
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
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
11 / 18
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
11 / 18
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
11 / 18
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
11 / 18
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
11 / 18
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
11 / 18
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
12 / 18
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
12 / 18
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
12 / 18
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
12 / 18
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
12 / 18
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
12 / 18
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
12 / 18
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
12 / 18
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
12 / 18
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?
14 / 18
rule bc
Biconditional-to-conditional: from a biconditional you mayinfer either of the corresponding conditionals.
bc:
(2 ↔ #)
(2 → #)(# → 2)
16 / 18
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
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