Symbolization and Derivation Cheat Sheet
PHL245
Allen Yao
Unit 3 - Symbolization
Conditionals and Biconditionals
English
Logic
“If A then B”
A⇒B
“Only if A then B”
B⇒A
“A is necessary for B”
B⇒A
“A is sufficient for B”
A⇒B
“A is necessary and sufficient for B”
A⇔B
“If and only if A then B”
A⇔B
Conjunctions and Disjunctions
English
Logic
“A and B”
A∧B
“A or B”
A∨B
“Neither A nor B”
¬(A ∨ B)
“Not both A and B”
¬(A ∧ B)
“A or B, but not both”
A⇔B
“A unless B”
A ∨ B or ¬A ⇒ B
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Numbers
English
Logic
“All of A, B, and C”
A∧B∧C
“None of A, B, and C”
¬(A ∨ B ∨ C)
“At least one of A, B, and C”
A∨B∨C
“At least two of A, B, and C”
(A ∧ B) ∨ (A ∧ C) ∨ (B ∧ C)
“Exactly one of A, B, and C”
(A ∧ ¬B ∧ ¬C) ∨ (¬A ∧ B ∧ ¬C) ∨ (¬A ∧ ¬B ∧ C)
“Exactly two of A, B, and C”
(A ∧ B ∧ ¬C) ∨ (¬A ∧ B ∧ C) ∨ (A ∧ ¬B ∧ C)
“At most one of A, B, and C”
None ∨ Exactly One
“At most two of A, B, and C”
¬(A ∧ B ∧ C)
Restrictive Clause (only commas matter!)
English
Logic
“A, which B, are C”
(A ⇒ C) ∧ B
“A that B are C”
A∧B ⇒C
Cases
English
Logic
“If A then B and in that case C”
(A ⇒ B) ∧ (B ⇒ C)
“A is necessary for B and in that case C”
(B ⇒ A) ∧ (B ⇒ C)
“If A1 . . . An . . . . . . Ak and in that case B”
(A1 . . . A2 . . . . . . An ) ∧ (Ak ⇒ B)
“If A1 . . . A2 . . . . . . An and in the former case B”
(A1 . . . A2 . . . . . . An ) ∧ (A1 ⇒ B)
“If A1 . . . A2 . . . . . . An and in the latter case B”
(A1 . . . A2 . . . . . . An ) ∧ (An ⇒ B)
“If A1 . . . A2 . . . . . . An and in the kth case B”
(A1 . . . A2 . . . . . . An ) ∧ (Ak ⇒ B)
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Unit 4 - Derivation
Basic Rules
Name
Abbrv
Logic
Comments
Modus Ponens
MP
A ⇒ B, A ∴ B
Modus Tollens
MT
A ⇒ B, ¬B ∴ ¬A
Double Negation
DN
¬¬A ∴ A
Repetition
R
A∴A
Simplification
S
A∧B ∴A
Adjuncation
ADJ
A, B ∴ A ∧ B
Modus Tellendo Ponens
MTP
A ∨ B, ¬A ∴ B
Addition
ADD
A∴A∨B
Biconditional-Conditional
BC
A⇔B∴A⇒B
Conditional-Biconditional
CB
(A ⇒ B) ∧ (B ⇒ A) ∴ A ⇔ B
Works the other way too.
Also works for B.
Also works for ¬B ∴ A.
Also works for B ⇒ A.
Derived Rules
Name
Abbrv
Logic
Comments
Negation of Conditionals
NC
¬(A ⇒ B) ∴ A ∧ ¬B
Negation of Biconditionals
NB
¬(A ⇔ B) ∴ A ⇔ ¬B
Demorgan’s Laws
DM
Factor/distribute the ¬ out of /
into the brackets
and switch the ∨/∧ to ∧/∨
Separation of Cases
SC
A ∨ B, A ⇒ C, B ⇒ C ∴ C
A ⇒ C, ¬A ⇒ C ∴ C
Conditional as Disjunction
CDJ
A ⇒ B ∴ ¬A ∨ B
¬A ⇒ B ∴ A ∨ B
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