Mathematical Logic Part Three
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1 Mathematical Logic Part Three
2 The Aristotelian Forms All As are Bs x. (A(x B(x Some As are Bs x. (A(x B(x No As are Bs x. (A(x B(x Some As aren t Bs x. (A(x B(x It It is is worth worth committing committing these these patterns patterns to to memory. memory. We ll We ll be be using using them them throughout throughout the the day day and and they they form form the the backbone backbone of of many many firstorder firstorder logic logic translations. translations.
3 The Art of Translation
4 Using the predicates - Person(p, which states that p is a person, and - Loves(x, y, which states that x loves y, write a sentence in first-order logic that means everybody loves someone else.
5 How How many many of of the the following following first-order logic logic statements are are correct correct translations of of everyone loves loves someone someone else? else? p. (Person(p q. (Person(q Loves(p, q p. (Person(p q. (Person(q p q Loves(p, q p. (Person(p q. (Person(q p q Loves(p, q p. (Person(p q. (Person(q p q Loves(p, q Answer Answer at at PollEv.com/cs103 PollEv.com/cs103 or or text text CS103 CS103 to to once once to to join, join, then then 0, 0, 1, 1, 2, 2, 3, 3, or or 4. 4.
6 Everybody loves someone else
7 Every person loves some other person
8 Every person p loves some other person
9 Every person p loves some other person All All As As are are Bs Bs x. x. (A(x (A(x B(x B(x
10 p. (Person(p p loves some other person All All As As are are Bs Bs x. x. (A(x (A(x B(x B(x
11 p. (Person(p p loves some other person
12 p. (Person(p there is some other person that p loves
13 p. (Person(p there is a person other than p that p loves
14 p. (Person(p there is a person q, other than p, where p loves q
15 p. (Person(p there is a person q, other than p, where p loves q
16 p. (Person(p there is a person q, other than p, where p loves q Some Some As As are are Bs Bs x. x. (A(x (A(x B(x B(x
17 p. (Person(p q. (Person(q, other than p, where p loves q Some Some As As are are Bs Bs x. x. (A(x (A(x B(x B(x
18 p. (Person(p q. (Person(q, other than p, where p loves q
19 p. (Person(p q. (Person(q p q p loves q
20 p. (Person(p q. (Person(q p q Loves(p, q
21 Using the predicates - Person(p, which states that p is a person, and - Loves(x, y, which states that x loves y, write a sentence in first-order logic that means there is a person that everyone else loves.
22 There is a person that everyone else loves
23 There is a person p where everyone else loves p
24 There is a person p where everyone else loves p Some Some As As are are Bs Bs x. x. (A(x (A(x B(x B(x
25 p. (Person(p everyone else loves p Some Some As As are are Bs Bs x. x. (A(x (A(x B(x B(x
26 p. (Person(p everyone else loves p
27 p. (Person(p every other person q loves p
28 p. (Person(p every person q, other than p, loves p
29 p. (Person(p every person q, other than p, loves p All All As As are are Bs Bs x. x. (A(x (A(x B(x B(x
30 p. (Person(p q. (Person(q p q q loves p All All As As are are Bs Bs x. x. (A(x (A(x B(x B(x
31 p. (Person(p q. (Person(q p q q loves p
32 p. (Person(p q. (Person(q p q Loves(q, p
33 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: Everyone loves someone else.
34 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: Everyone loves someone else. p. (Person(p q. (Person(q p q Loves(p, q
35 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: Everyone loves someone else. p. (Person(p q. (Person(q p q Loves(p, q For every person,
36 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: Everyone loves someone else. p. (Person(p q. (Person(q p q Loves(p, q For every person, there is some person
37 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: Everyone loves someone else. p. (Person(p q. (Person(q p q Loves(p, q For every person, there is some person who isn't them
38 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: Everyone loves someone else. p. (Person(p q. (Person(q p q Loves(p, q For every person, there is some person who isn't them that they love.
39 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: There is someone everyone else loves.
40 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: There is someone everyone else loves. p. (Person(p q. (Person(q p q Loves(q, p
41 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: There is someone everyone else loves. p. (Person(p q. (Person(q p q Loves(q, p There is some person
42 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: There is someone everyone else loves. p. (Person(p q. (Person(q p q Loves(q, p There is some person who everyone
43 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: There is someone everyone else loves. p. (Person(p q. (Person(q p q Loves(q, p There is some person who everyone who isn't them
44 Combining Quantifiers Most interesting statements in first-order logic require a combination of quantifiers. Example: There is someone everyone else loves. p. (Person(p q. (Person(q p q Loves(q, p There is some person who everyone who isn't them loves.
45 For Comparison p. (Person(p q. (Person(q p q Loves(p, q For every person, there is some person who isn't them that they love. p. (Person(p q. (Person(q p q Loves(q, p There is some person who everyone who isn't them loves.
46 Everyone Loves Someone Else
47 Everyone Loves Someone Else No No one one here here is is universally universally loved. loved.
48 There is Someone Everyone Else Loves
49 There is Someone Everyone Else Loves This This person person does does not not love love anyone anyone else. else.
50 Everyone Loves Someone Else and There is Someone Everyone Else Loves
51 p. (Person(p q. (Person(q p q Loves(p, q For every person, there is some person who isn't them that they love. p. (Person(p q. (Person(q p q Loves(q, p There is some person who everyone who isn't them loves.
52 Quantifier Ordering The statement x. y. P(x, y means for any choice of x, there's some choice of y where P(x, y is true. The choice of y can be different every time and can depend on x.
53 Quantifier Ordering The statement x. y. P(x, y means there is some x where for any choice of y, we get that P(x, y is true. Since the inner part has to work for any choice of y, this places a lot of constraints on what x can be.
54 Order matters when mixing existential and universal quantifiers!
55 Time out for a little stretch!
56 B A C D Which Which person person in in this this diagram diagram do do you you most most aspire aspire to to be? be? Answer Answer at at PollEv.com/cs103 PollEv.com/cs103 or or text text CS103 CS103 to to once once to to join, join, then then A, A, B, B, C, C, or or D. D.
57 Back to CS103!
58 Set Translations
59 Using the predicates - Set(S, which states that S is a set, and - x y, which states that x is an element of y, write a sentence in first-order logic that means the empty set exists.
60 Using the predicates - Set(S, which states that S is a set, and - x y, which states that x is an element of y, write a sentence in first-order logic that means the empty set exists. First-order First-order logic logic doesn't doesn't have have set set operators operators or or symbols symbols built built in. in. If If we we only only have have the the predicates predicates given given above, above, how how might might we we describe describe this? this?
61 How How many many of of the the following following first-order logic logic statements are are correct correct translations of of the the empty empty set set exists? exists? S. (Set(S x. x S S. (Set(S x. x S S. (Set(S x. x S S. (Set(S x. x S Answer Answer at at PollEv.com/cs103 PollEv.com/cs103 or or text text CS103 CS103 to to once once to to join, join, then then 0, 0, 1, 1, 2, 2, 3, 3, or or 4. 4.
62 The empty set exists.( ( (
63 There is some set S that is empty.( ( (
64 S. (Set(S S is empty.
65 S. (Set(S there are no elements in S
66 S. (Set(S there is an element in S
67 S. (Set(S there is an element x in S
68 S. (Set(S x. x S
69 S. (Set(S x. x S
70 S. (Set(S x. x S S. (Set(S there are no elements in S(
71 S. (Set(S x. x S S. (Set(S every object does not belong to S(
72 S. (Set(S x. x S S. (Set(S every object x does not belong to S(
73 S. (Set(S x. x S S. (Set(S x. x S
74 S. (Set(S x. x S S. (Set(S x. x S
75 S. (Set(S x. x S S. (Set(S x. x S Both Both of of these these translations translations are are correct. correct. Just Just like like in in propositional propositional logic, logic, there there are are many many different different equivalent equivalent ways ways of of expressing expressing the the same same statement statement in in first-order first-order logic. logic.
76 Mechanics: Negating Statements
77 Which Which of of the the following following is is the the negation negation of of the the statement x. x. y. y. Loves(x, Loves(x, y? y? A. A. x. x. y. y. Loves(x, y y B. B. x. x. y. y. Loves(x, y y C. C. x. x. y. y. Loves(x, y y D. D. x. x. y. y. Loves(x, y y E. E. None None of of these. these. F. F. Two Two or or more more of of these. these. Answer Answer at at PollEv.com/cs103 PollEv.com/cs103 or or text text CS103 CS103 to to once once to to join, join, then then A, A, B, B, C, C, D, D, E, E, or or F. F.
78 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x
79 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x
80 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x
81 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x
82 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x
83 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x x. P(x For some choice of x, P(x For any choice of x, P(x
84 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x x. P(x For some choice of x, P(x For any choice of x, P(x
85 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x x. P(x For some choice of x, P(x For any choice of x, P(x
86 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x x. P(x x. P(x For any choice of x, P(x
87 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x x. P(x x. P(x For any choice of x, P(x
88 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x x. P(x x. P(x For any choice of x, P(x
89 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x x. P(x x. P(x x. P(x
90 An Extremely Important Table x. P(x x. P(x x. P(x x. P(x When is this true? For any choice of x, P(x For some choice of x, P(x For any choice of x, P(x For some choice of x, P(x When is this false? x. P(x x. P(x x. P(x x. P(x
91 Negating First-Order Statements Use the equivalences to negate quantifiers. Mechanically: x. A x. A x. A x. A Push the negation across the quantifier. Change the quantifier from to or vice-versa. Use techniques from propositional logic to negate connectives.
92 Taking a Negation x. y. Loves(x, y ( Everyone loves someone. x. y. Loves(x, y x. y. Loves(x, y x. y. Loves(x, y ( There's someone who doesn't love anyone.
93 Two Useful Equivalences The following equivalences are useful when negating statements in first-order logic: (p q p q (p q p q These identities are useful when negating statements involving quantifiers. is used in existentially-quantified statements. is used in universally-quantified statements. When pushing negations across quantifiers, we strongly recommend using the above equivalences to keep with and with.
94 Negating Quantifiers What is the negation of the following statement, which says there is a cute puppy? We can obtain it as follows: x. (Puppy(x Cute(x x. (Puppy(x Cute(x x. (Puppy(x Cute(x x. (Puppy(x Cute(x This says no puppy is cute. Do you see why this is the negation of the original statement from both an intuitive and formal perspective?
95 S. (Set(S x. (x S ( There is a set with no elements. S. (Set(S x. (x S S. (Set(S x. (x S S. (Set(S x. (x S S. (Set(S x. (x S S. (Set(S x. x S ( Every set contains at least one element.
96 These two statements are not negations of one another. Can you explain why? S. (Set(S x. (x S ( There is a set that doesn't contain anything S. (Set(S x. (x S ( Everything is a set that contains something Remember: usually goes goes with with,, not not
97 Restricted Quantifiers
98 Quantifying Over Sets The notation x S. P(x means for any element x of set S, P(x holds. (It s vacuously true if S is empty. The notation x S. P(x means there is an element x of set S where P(x holds. (It s false if S is empty.
99 Quantifying Over Sets The syntax x S. φ x S. φ is allowed for quantifying over sets. In CS103, feel free to use these restricted quantifiers, but please do not use variants of this syntax. For example, don't do things like this: x with P(x. Q(x y such that P(y Q(y. R(y. P(x. Q(x
100 Expressing Uniqueness
101 Using the predicate - Level(l, which states that l is a level, write a sentence in first-order logic that means there is only one level. A A fun fun diversion: diversion:
102 There is only one level.
103 Something is a level, and nothing else is.
104 Some thing l is a level, and nothing else is.
105 Some thing l is a level, and nothing besides l is a level
106 l. (Level(l nothing besides l is a level.
107 l. (Level(l anything that isn't l isn't a level
108 l. (Level(l any thing x that isn't l isn't a level
109 l. (Level(l x. (x l x isn't a level
110 l. (Level(l x. (x l Level(x
111 l. (Level(l x. (x l Level(x
112 l. (Level(l x. (Level(x x = l
113 Expressing Uniqueness To express the idea that there is exactly one object with some property, we write that there exists at least one object with that property, and that there are no other objects with that property. You sometimes see a special uniqueness quantifier used to express this:!x. P(x For the purposes of CS103, please do not use this quantifier. We want to give you more practice using the regular and quantifiers.
114 Using the predicates - Set(S, which states that S is a set, and - x y, which states that x is an element of y, write a sentence in first-order logic that means two sets are equal if and only if they contain the same elements.
115 Two sets are equal if and only if they have the same elements.
116 Any two sets are equal if and only if they have the same elements.
117 Any two sets S and T are equal if and only if they have the same elements.
118 S. (Set(S T. (Set(T S and T are equal if and only if they have the same elements.
119 S. (Set(S T. (Set(T (S = T if and only if they have the same elements.
120 S. (Set(S T. (Set(T (S = T they have the same elements.
121 S. (Set(S T. (Set(T (S = T S and T have the same elements.
122 S. (Set(S T. (Set(T (S = T every element of S is an element of T and vice-versa
123 S. (Set(S T. (Set(T (S = T x is an element of S if and only if x is an element of T
124 S. (Set(S T. (Set(T (S = T x. (x S x T
125 S. (Set(S T. (Set(T (S = T x. (x S x T
126 S. (Set(S T. (Set(T (S = T x. (x S x T You You sometimes sometimes see see the the universal universal quantifier quantifier pair pair with with the the connective. connective. This This is is especially especially common common when when talking talking about about sets sets because because two two sets sets are are equal equal when when they they have have precisely precisely the the same same elements. elements.
127 S. (Set(S T. (Set(T (S = T x. (x S x T
128 Next Time Binary Relations How do we model connections between objects? Equivalence Relations How do we model the idea that objects can be grouped into clusters? First-Order Definitions Where does first-order logic come into all of this? Proofs with Definitions How does first-order logic interact with proofs?
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