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| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Predicate Logic | Obsidian::STEM | formal-system::predicate |
|
Overview
Predicate logic is a logical system 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%%
Existentials
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%%
Uniqueness
We can also denote existence and uniqueness using \exists!. For example, \exists! x, P(x) indicates there exists a unique x satisfying P(x), i.e. there is exactly one x such that P(x) holds: (\exists! x, P(x)) = (\exists x, P(x)) \land (\forall x, \forall y, (P(x) \land P(y)) \Rightarrow (x = y)))
The first conjunct denotes existence while the second denotes uniqueness.
%%ANKI
Basic
What non-counting quantifer denotes unique existential quantification?
Back: \exists!
Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
END%%
%%ANKI
Basic
Unique existential quantification can be expressed using what counting quantification?
Back: \exists^{=1}
Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
END%%
%%ANKI
Basic
How is \exists! x, P(x) expanded using the basic existential and universal quantifiers?
Back: (\exists x, P(x)) \land (\forall x, \forall y, (P(x) \land P(y)) \Rightarrow (x = y))
Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
END%%
%%ANKI
Basic
How do we write the equivalent existence (not uniqueness) assertion made by \exists! x, P(x)?
Back: \exists x, P(x))
Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
END%%
%%ANKI
Basic
How do we write the equivalent uniqueness (not existence) assertion made by \exists! x, P(x)?
Back: \forall x, \forall y, (P(x) \land P(y)) \Rightarrow (x = y)
Reference: Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.
END%%
Counting
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%%
Universals
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%%
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%%
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.
- Patrick Keef and David Guichard, “An Introduction to Higher Mathematics,” n.d.