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title TARGET DECK FILE TAGS tags
Propositional Logic Obsidian::STEM logic::propositional
logic
propositional

Overview

A branch of logic derived from negation (\neg), conjunction (\land), disjunction (\lor), implication (\Rightarrow), and biconditionals (\Leftrightarrow). A proposition is a sentence that can be assigned a truth or false value.

%%ANKI Cloze {Propositional} logic is also known as {zeroth}-order logic. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What are the basic propositional logical operators? Back: \neg, \land, \lor, \Rightarrow, and \Leftrightarrow Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is a proposition? Back: A declarative sentence which is either true or false. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What two categories do propositions fall within? Back: Atomic and molecular propositions. 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 an atomic proposition? Back: One that cannot be broken up into smaller propositions. 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 molecular proposition? Back: One that can be broken up into smaller propositions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Cloze A {molecular} proposition can be broken up into {atomic} propositions. 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 sentence from a proposition? Back: The latter has an associated truth value. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What are constant propositions? Back: Propositions that contain only constants as operands. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How does Lean define propositional equality? Back: Expressions a and b are propositionally equal iff a = b is true. Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d. Tags: lean

END%%

%%ANKI Basic How does Lean define propext? Back:

axiom propext {a b : Prop} : (a ↔ b) → (a = b)

Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d. Tags: lean

END%%

Implication

Implication is denoted as \Rightarrow. It has truth table

p q p \Rightarrow q
T T T
T F F
F T T
F F T

Implication has a few "equivalent" English expressions that are commonly used. Given propositions P and Q, we have the following equivalences:

  • P if Q
  • P only if Q
  • P is necessary for Q
  • P is sufficient for Q

%%ANKI Basic What name is given to operand a in a \Rightarrow b? Back: The antecedent Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What name is given to operand b in a \Rightarrow b? Back: The consequent Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How does "P if Q" translate with \Rightarrow? Back: Q \Rightarrow P Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How does "P only if Q" translate with \Rightarrow? Back: P \Rightarrow Q Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How does "P is necessary for Q" translate with \Rightarrow? Back: Q \Rightarrow P Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How does "P is sufficient for Q" translate with \Rightarrow? Back: P \Rightarrow Q Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Which of if or only if map to necessary? Back: if Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Which of if or only if map to sufficient? Back: only if Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Which logical operator maps to "if and only if"? Back: \Leftrightarrow Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Which logical operator maps to "necessary and sufficient"? Back: \Leftrightarrow 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 converse of P \Rightarrow Q? Back: Q \Rightarrow P Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic When is implication equivalent to its converse? Back: It's indeterminate. 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 contrapositive of P \Rightarrow Q? Back: \neg Q \Rightarrow \neg P Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic When is implication equivalent to its contrapositive? Back: They are always equivalent. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Given propositions p and q, p \Leftrightarrow q is equivalent to the conjunction of what two expressions? Back: p \Rightarrow q and q \Rightarrow p. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Sets

A state is a function that maps identifiers to T or F. A proposition can be equivalently seen as a representation of the set of states in which it is true.

%%ANKI Basic What is a state? Back: A function mapping identifiers to values. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Is (b \land c) well-defined in \{(b, T), (c, F)\}? Back: Yes. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Is (b \lor d) well-defined in \{(b, T), (c, F)\}? Back: No. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic A proposition is well-defined with respect to what? Back: A state to evaluate against. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What proposition represents states \{(b, T), (c, T)\} and \{(b, F), (c, F)\}? Back: (b \land c) \lor (\neg b \land \neg c) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does proposition a \land b represent? Back: \{\{(a, T), (b, T)\}\} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does proposition a \lor b represent? Back: \{\{(a, T), (b, T)\}, \{(a, T), (b, F)\}, \{(a, F), (b, T)\}\} 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 b \lor \neg c"? Back: b \lor \neg c 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%%

%%ANKI Basic What is the weakest proposition? Back: T Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does T represent? Back: The set of all states. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the strongest proposition? Back: F Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does F represent? Back: The set of no states. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What does a proposition represent? Back: The set of states in which it is true. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is p stronger than q? Back: When p \Rightarrow q. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic If p \Rightarrow q, which of p or q is considered stronger? Back: p Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is p weaker than q? Back: When q \Rightarrow p. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic If p \Rightarrow q, which of p or q is considered weaker? Back: q Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Why is b \land c stronger than b \lor c? Back: The former represents a subset of the states the latter represents. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given sets a and b, a = b is equivalent to the conjunction of what two expressions? Back: a \subseteq b and b \subseteq a. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Cloze {a \Rightarrow b} is to propositional logic as {a \subseteq b} is to sets. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Cloze {a \Leftrightarrow b} is to propositional logic as {a = b} is to sets. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Bibliography