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| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Propositional Logic | Obsidian::STEM | 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:
PifQPonly ifQPis necessary forQPis sufficient forQ
%%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%%
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
- 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.