Logic Cheat Sheet
Truth Table Notes

P, Q  :   P  /\ Q   |   P  v  Q   |   P ->  Q   |   P <-> Q
T  T  :   T  T  T   |   T  T  T   |   T  T  T   |   T  T  T
T  F  :   T  F  F   |   T  T  F   |   T  F  F   |   T  F  F
F  T  :   F  F  T   |   F  T  T   |   F  T  T   |   F  F  T
F  F  :   F  F  F   |   F  F  F   |   F  T  F   |   F  T  F

Interpret P/\Q .: P     as (P/\Q)->P
Interpret P, P/\Q .: P  as (P/\(P->Q))->P

/\  can be interpreted as 'and', 'but', 'although', 'even though', and 'despite the fact that'.
v   can be interpreted as 'or' or 'unless'.
->  can be interpreted as 'if ... then', 'given that', and 'only if'.
<-> can be interpreted as 'if then and only then', 'just in case', 'exactly on the condition that', 'if and only if'.

For this dictionary:

A = Annie dances.
B = Brian sings.

Interpret "If Brian sings, then Annie dances." as:  B->A
Interpret "Annie dances if Brian sings." as:        B->A
Interpret "Annie dances only if Brian sings." as:   A->B

Universal quantifiers almost always involve an implication; e.g. Ax( F(x) -> G(x) )

That is, "Everything that has the property of being a frog also has the property of being green".

Existential quantifiers almost always involve a conjunction; e.g. Ex( F(x) /\ G(x) )

That is, "Some things are frogs and green."

Basic Proof Strategies

When what is to be shown is:

  a) A single premise/statement like .: P
  b) A negation
  c) A disjunction
  d) A tautological derivation

Think about using an indirect derivation before anything else.
REMEMBER THAT ALL INDIRECT DERIVATIONS MUST BE A NEGATION, even if it equates to an affirmative statement such as SHOW: ~~P. Remember too that most often, your assumption (if any) should have the opposite truth value than that of the SHOW line, for instance: 1. SHOW: ~~P 2. ~P :AS 3. ... For conditionals, just assume the antecedent by default; maybe discover later that you didn't need to. 1. SHOW: P->Q 2. P :AS 3. ... If proving a biconditional, show that each side independently implies the other, and more than likely, you'll be using CB as part of the proof. 1. SHOW: P<->Q 2. SHOW: P->Q 3. P :PR 4. Q :PR 5. :CD 4 6. SHOW: Q->P 7. Q :PR 8. P :PR 9. :CD 8 10. P<->Q :CB 2 6 11. :DD 10 If the proof involves a disjuction, think of using CASES if ID isn't working. 1. SHOW: P 2. Q->P :PR 3. R->P :PR 4. Q v R :PR 5. P :CASES 2 3 4 6. :DD 5 For Predicate Proofs Existential derivations don't have to relate to what is being shown, aside from the shown being derivable from the assumption made from a previous existential premise. Remember that a single universal instanciation can be called many times, each with different variable names. See LOGICAL RULES AND NOTATION for further reference. During Open Book Exams Use the PROPOSITIONAL AND PREDICATE RULE BUILDERS to derive proofs. Your proof will be valid when the input box at the very top displays all premises used in your proof and the resulting derivation. If this doesn't make immediate sense to you, then copy a previous proof you've completed correctly and paste it into the rule builder. Notice the header is all filled in. Notice that if you modify any bits of it, say even just adding a new line and a space within the proof, the header blanks itself out. Only when the header above the rule builder is filled in with all, not just some, of the premises used, and what is being proved by them, you know your proof is perfectly valid. When asked to translate an English sentence into logical notation, look for similar problems in carnap.io and input and check your answer there, adjusting any variables and properties accordingly. If it checks out, readjust any variables and properties and you have your answer for the exam.