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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). There exists a hierarchy of terms used to describe a string of English:

  • A sentence is any grammatical string of words.
  • A predicate is a sentence with free variables.
  • A statement is a sentence that can be assigned a truth or false value.
    • A predicate with free variables "plugged in" is a statement.

%%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 propositional statement? 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 propositional statements fall within? Back: Atomic and molecular statements. 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 statement? Back: One that cannot be broken up into smaller statements. 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 statement? Back: One that can be broken up into smaller statements. 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} statement can be broken up into {atomic} statements. 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 statement? Back: The latter is a sentence that can be derived a 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 distinguishes a predicate from a statement? Back: A statement 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 statements defined in terms of predicates? Back: A statement 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 statement? 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%%

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%%

Bibliography