This document provides an introduction to propositional and predicate logic, including examples of how they can be used to represent linguistic statements in a formal, computable way. Some key points covered include:
- Propositional logic uses simple statements connected with logical operators like "if-then", while predicate logic introduces predicates and variables to represent internal structures.
- Examples show how predicate logic can represent statements about individuals and their properties or relationships using predicates, variables, and quantifiers.
- While logic provides a useful formalism for linguistics and computer science, some argue it may not accurately capture how humans naturally understand and use language. Modern linguistics also explores how language relates to other aspects of human cognition beyond logical representations.
10. Peter fell over
• Predicate?
• Fell_over
• How many arguments?
• One – peter
• Fell_over (peter)
11. Jim donated $100 to the city hospital
• Donated (x,…..n)
• Donated (jim, $100, city_hospital)
• Donated (x, y, z)
• X = Jim
• Y = $100
• Z= the city hospital
33. All linguists are crazy
• ∀x (Linguist (x) Crazy_person (x))
• Is this true?
• For all individuals x, if x is in the set of
Linguists, x is also in the set of Crazy persons.
34. • Set A = the set of Linguists
• Linguist (x)
• Set B = the set of crazy people
• Crazy_person (x)
35. Who is in these sets?
• Linguist (x)
• {evans,
• imai,
• ono}
36. Who is in the set of crazy people?
• Crazy_person (x)
• {ken,
• jim,
• ben,
• mary,
• evans}
37. All linguists are crazy
• ∀x (Linguist (x) Crazy_person (x))
• Is Untrue
• Because two members of the set of linguists
are not in the set of crazy people.
40. Some linguists are crazy
• ∃x (Linguist (x) & Crazy_person (x))
• Backward E
• Existential Quantifier
• There is at least one individual x
• x is a linguist
• And x is crazy
41. Who is in the set of crazy people?
• Crazy_person (x)
• {ken,
• jim,
• ben,
• mary,
• evans}
42. Evans is in the set of crazy people
• So this is true
• ∃x (Linguist (x) & Crazy_person (x))
• Backward E
• Existential Quantifier
• There is at least one individual x
• x is a linguist
• And x is crazy
60. • We sometimes say things that are not true
• “My brain exploded”
• And do we really think in Logical Form?
• ¬∃x (Girl (x) & Lives_in_fujiyoshida (x))
61. • And if I say “A girl lives in Fujiyoshida” …
• … is it really a statement about existence of an
individual?
• Or am I more concerned with number
• Or the fact that we’re talking about a girl
rather than a boy?
• Or a girl rather than a woman?
• Or something else related to CONTEXT?
69. But Logic is VERY important in
Linguistics!
• The nurse kissed every child on his birthday.
• [The nurse] kissed [every child] on his birthday.
• Kissed (nurse, every_child)
70. • ∀x (Child (x) Kissed (nurse, x)) on x’s
birthday
• The nurse kissed every child on his birthday
83. p → q
• p = Jim is dead
• q = Mary is sad
• p → q =
• If Jim is dead, Mary is sad
84. p V q
• p = My parrot is clever
• q = Taro fell over
• p V q =
• (either) my parrot is clever or Taro fell over
85. p → q
• p = I study hard
• q = I will pass the test
• p → q =
• If I study hard, I will pass the test
86. ¬p → ¬q
• P = I study hard
• Q = I will pass the test
• ¬p → ¬q = If I do not study hard, I will not
pass the test
• If NOT(I study hard) then NOT (I will pass the
test)
87. p → q
• Normal everyday values for p and q
• p = a bear has warm blood
• q = a bear is a mammal
88. ¬p → q
• Normal everyday values for p and q
• p = you study hard
• q = you will fail
• If you DON’T study hard, you will fail
89. Modern Linguistics
• Studying language helps us understand the
human brain
• Do you think the human brain works like
computer code?
• Do you think studying language tells us
something about the brain?
