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title TARGET DECK FILE TAGS tags
Predicate Logic Obsidian::STEM formal-system::predicate
logic
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%%

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%%

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%%

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