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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 \begin{array}{c|c|c} p & q & p \Rightarrow q \ \hline T & T & T \ T & F & F \ F & T & T \ F & F & T \end{array}

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 do you write "P if Q" in propositional logic? 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 do you write "P if Q" using "necessary"? Back: P is necessary for 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 do you write "P if Q" using "sufficient"? Back: Q is sufficient for 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 do you write "P only if Q" in propositional logic? 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 do you write "P only if Q" using "necessary"? Back: Q is necessary for 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 do you write "P only if Q" using "sufficient"? Back: P is sufficient for 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 do you write "P is necessary for Q" in propositional logic? 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 do you write "P is necessary for Q" using "if"? Back: P if 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 do you write "P is necessary for Q" using "only if"? Back: Q only if 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 do you write "P is sufficient for Q" in propositional logic? 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 do you write "P is sufficient for Q" using "if"? Back: Q if 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 do you write "P is sufficient for Q" using "only if"? Back: P only if 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 do you write "P if Q" using "only if"? Back: Q only if 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 do you write "P is sufficient for Q" using "necessary"? Back: Q is necessary for 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 do you write "P only if Q" using "if"? Back: Q if 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 do you write "P is necessary for Q" using "sufficient"? Back: Q is sufficient for P. 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%%

Laws

Commutativity

For propositions E1 and E2:

  • (E1 \land E2) = (E2 \land E1)
  • (E1 \lor E2) = (E2 \lor E1)
  • (E1 = E2) = (E2 = E1)

%%ANKI Basic Which of the basic logical operators do the commutative laws apply to? Back: \land, \lor, and = Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the commutative laws allow us to do? Back: Reorder operands. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the commutative law of e.g. \land? Back: E1 \land E2 = E2 \land E1

END%%

Associativity

For propositions E1, E2, and E3:

  • E1 \land (E2 \land E3) = (E1 \land E2) \land E3
  • E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3

%%ANKI Basic Which of the basic logical operators do the associative laws apply to? Back: \land and \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the associative laws allow us to do? Back: Remove parentheses. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the associative law of e.g. \land? Back: E1 \land (E2 \land E3) = (E1 \land E2) \land E3 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Distributivity

For propositions E1, E2, and E3:

  • E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)
  • E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)

%%ANKI Basic Which of the basic logical operators do the distributive laws apply to? Back: \land and \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the distributive laws allow us to do? Back: "Factor" propositions. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the distributive law of e.g. \land over \lor? Back: E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

De Morgan's

For propositions E1 and E2:

  • \neg (E1 \land E2) = \neg E1 \lor \neg E2
  • \neg (E1 \lor E2) = \neg E1 \land \neg E2

%%ANKI Basic Which of the basic logical operators do De Morgan's laws involve? Back: \neg, \land, and \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is De Morgan's law (distributing \land) expressed as an equivalence? Back: \neg (E1 \land E2) = \neg E1 \lor \neg E2 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: programming::equiv-trans

END%%

Law of Negation

For any proposition E1, it follows that \neg (\neg E1) = E1.

%%ANKI Basic How is the law of negation expressed as an equivalence? Back: \neg (\neg E1) = E1 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: programming::equiv-trans

END%%

Law of Excluded Middle

For any proposition E1, it follows that E1 \lor \neg E1 = T.

%%ANKI Basic Which of the basic logical operators does the law of excluded middle involve? Back: \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is the law of excluded middle expressed as an equivalence? Back: E1 \lor \neg E1 = T Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: programming::equiv-trans

END%%

%%ANKI Basic Which equivalence schema is "refuted" by sentence, "This sentence is false." Back: The law of excluded middle Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Law of Contradiction

For any proposition E1, it follows that E1 \land \neg E1 = F.

%%ANKI Basic Which of the basic logical operators does the law of contradiction involve? Back: \land Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is the law of contradiction expressed as an equivalence? Back: E1 \land \neg E1 = F Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: programming::equiv-trans

END%%

%%ANKI Cloze The law of {1:excluded middle} is to {2:\lor} whereas the law of {2:contradiction} 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 What does the principle of explosion state? Back: That any statement can be proven from a contradiction. Reference: “Principle of Explosion,” in Wikipedia, July 3, 2024, https://en.wikipedia.org/w/index.php?title=Principle_of_explosion.

END%%

%%ANKI Basic How is the principle of explosion stated in first-order logic? Back: \forall P, F \Rightarrow P Reference: “Principle of Explosion,” in Wikipedia, July 3, 2024, https://en.wikipedia.org/w/index.php?title=Principle_of_explosion.

END%%

%%ANKI Basic What does the law of contradiction say? Back: For any proposition P, it holds that \neg (P \land \neg P). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How does the principle of explosion relate to the law of contradiction? Back: If a contradiction could be proven, then anything can be proven. Reference: “Principle of Explosion,” in Wikipedia, July 3, 2024, https://en.wikipedia.org/w/index.php?title=Principle_of_explosion.

END%%

%%ANKI Basic Suppose P and \neg P. Show schematically how to use the principle of explosion to prove Q. Back: \begin{align*} P \ \neg P \ P \lor Q \ \hline Q \end{align*}$$Reference: “Principle of Explosion,” in Wikipedia, July 3, 2024, https://en.wikipedia.org/w/index.php?title=Principle_of_explosion.

END%%

%%ANKI Cloze The law of {contradiction} and law of {excluded middle} create a dichotomy in "logical space". Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

%%ANKI Basic Which property of partitions is analagous to the law of contradiction on "logical space"? Back: Disjointedness. Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

%%ANKI Basic Which property of partitions is analagous to the law of excluded middle on "logical space"? Back: Exhaustiveness. Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

%%ANKI Cloze The law of {1:contradiction} is to "{2:mutually exclusive}" whereas the law of {2:excluded middle} is "{1:jointly exhaustive}". Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

%%ANKI Basic Which logical law proves equivalence of the law of contradiction and excluded middle? Back: De Morgan's law. Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

As 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 \{\langle b, T \rangle, \langle c, F \rangle\}? 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 \{\langle b, T \rangle, \langle c, F \rangle\}? 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: \{\{\langle a, T \rangle, \langle b, T \rangle\}\} 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: \{\{\langle a, T \rangle, \langle b, T \rangle\}, \{\langle a, T \rangle, \langle b, F \rangle\}, \{\langle a, F \rangle, \langle b, T \rangle\}\} 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