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title | TARGET DECK | FILE TAGS | tags | ||
---|---|---|---|---|---|
Predicate Logic | Obsidian::STEM | logic::predicate |
|
Overview
A branch of logic that uses quantified variables over non-logical objects. A predicate is a sentence with some number of free variables. A predicate with free variables "plugged in" is a prop-logic.
%%ANKI Cloze {Predicate} logic is also known as {first}-order logic. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What is a predicate? Back: A sentence with some number of free variables. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What distinguishes a predicate from a proposition? Back: A proposition does not contain free variables. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How are propositions defined in terms of predicates?
Back: A proposition is a predicate with 0
free variables.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Why is "3 + x = 12
" not a proposition?
Back: Because x
is a variable.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Quantification
A quantifier refers to an operator that specifies how many members of a set satisfy some formula. The most common quantifiers are \exists
and \forall
, though others (such as the counting quantifier) are also used.
%%ANKI
Basic
What are the most common first-order logic quantifiers?
Back: \exists
and \forall
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What term refers to operators like \exists
and \forall
?
Back: Quantifiers.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
- Existential quantification (
\exists
) asserts the existence of at least one member in a set satisfying a property.
%%ANKI
Basic
What symbol denotes existential quantification?
Back: \exists
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic How many members in the domain of discourse must satisfy a property in existential quantification? Back: At least one. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
\exists x : S, P(x)
is shorthand for what?
Back: \exists x, x \in S \land P(x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What term refers to S
in \exists x : S, P(x)
?
Back: The domain of discourse.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the identity element of \lor
?
Back: F
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
- Universal quantification (
\forall
) asserts that every member of a set satisfies a property.
%%ANKI
Basic
What symbol denotes universal quantification?
Back: \forall
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic How many members in the domain of discourse must satisfy a property in universal quantification? Back: All of them. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
\forall x : S, P(x)
is shorthand for what?
Back: \forall x, x \in S \Rightarrow P(x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the identity element of \land
?
Back: T
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Cloze
{1:\exists
} is to {2:\lor
} as {2:\forall
} is to {1:\land
}.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is \forall x : S, P(x)
equivalently written in terms of existential quantification?
Back: \neg \exists x : S, \neg P(x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
How is \exists x : S, P(x)
equivalently written in terms of universal quantification?
Back: \neg \forall x : S, \neg P(x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
- Counting quantification (
\exists^{=k}
or\exists^{\geq k}
) asserts that (at least)k
(say) members of a set satisfy a property.
%%ANKI
Basic
What symbol denotes counting quantification (of exactly k
members)?
Back: \exists^{=k}
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What symbol denotes counting quantification (of at least k
members)?
Back: \exists^{\geq k}
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is \exists x : S, P(x)
written in terms of counting quantification?
Back: \exists^{\geq 1}\, x : S, P(x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is \forall x : S, P(x)
written in terms of counting quantification?
Back: Assuming S
has k
members, \exists^{= k}\, x : S, P(x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Cloze
Propositional logical operator: \forall x, \forall y, P(x, y)
{\Leftrightarrow
} \forall y, \forall x, P(x, y)
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Propositional logical operator: \forall x, \exists y, P(x, y)
{\Leftarrow
} {\exists y, \forall x, P(x, y)
}.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Propositional logical operator: \exists x, \forall y, P(x, y)
{\Rightarrow
} \forall y, \exists x, P(x, y)
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
Propositional logical operator: \exists x, \exists y, P(x, y)
{\Leftrightarrow
} \exists y, \exists x, P(x, y)
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
When does \exists x, \forall y, P(x, y) \Rightarrow \forall y, \exists x, P(x, y)
hold true?
Back: Always.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
When does \forall x, \exists y, P(x, y) \Rightarrow \exists y, \forall x, P(x, y)
hold true?
Back: When there exists a y
that P(x, y)
holds for over all quantified x
.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Identifiers
Identifiers are said to be bound if they are parameters to a quantifier. Identifiers that are not bound are said to be free. A first-order logic formula is said to be in prenex normal form (PNF) if written in two parts: the first consisting of quantifiers and bound variables (the prefix), and the second consisting of no quantifiers (the matrix).
%%ANKI Basic Prenex normal form consists of what two parts? Back: The prefix and the matrix. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic How is the prefix of a formula in PNF formatted? Back: As only quantifiers and bound variables. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic How is the matrix of a formula in PNF formatted? Back: Without quantifiers. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Which identifiers in the following are bound? \exists x, P(x) \land P(y)
Back: Just x
.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Which identifiers in the following are free? \exists x, P(x) \land P(y)
Back: Just y
.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is the following rewritten in PNF? (\exists x, P(x)) \land (\exists y, Q(y))
Back: \exists x \;y, P(x) \land Q(y)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is the following rewritten in PNF? (\exists x, P(x)) \land (\forall y, Q(y))
Back: N/A.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
As Sets
A state is a function that maps identifiers to values. A predicate can be equivalently seen as a representation of the set of states in which it is true.
%%ANKI
Basic
Is (i \geq 0)
well-defined in \{(i, -10)\}
?
Back: Yes.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Is (i \geq 0)
well-defined in \{(j, -10)\}
?
Back: No.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What predicate represents states \{(i, 0), (i, 2), (i, 4), \ldots\}
?
Back: i \geq 0
is even.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is sloppy about phrase "the states in i + j = 0
"?
Back: i + j = 0
is not a set but a representation of a set (of states).
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
Bibliography
- Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.