truth tables & truth trees

truth table: A & B

A | B | A & B

T | T |    T

T | F |    F

T | T |    F

F | F |    F

truth table: A ∨ B

A | B | A ∨ B

T | T |    T

T | F |    T

F | T |    T

F | F |    F

truth table: A ↔ B

A | B | A ↔ B

T | T |    T

T | F |    F

F | T |    F

F | F |    T

truth table: ¬A

A | ¬A

T | F

F | T

truth table: A → B

A | B | A → B

T | T |    T

T | F |    F

F | T |    T

F | F |    T

not, necessarily, possibly, should, may

translate to ¬

and, both, although, though, but

translate to &

iff, when and only when, just in case 

translate to ↔

if… then, only if, if

translate to →

formula is logically contingent

iff the final column has both T & F

formula is a logical truth

iff final column has only T

formula is unsatisfiable 

iff final column has only F

formula is satisfiable

iff final column has at least on T

tautology

lotical truth

formulas are equivalent

iff the columns are identical (same truth value in every row)

argument is valid

iff in every row where every premise is T, the conclusion is also T

truth tree: A & B

A & B

A

B

truth tree: ¬¬A

¬¬A

A

truth tree: A ∨ B

A ∨ B

branch 1: A

branch 2: B

truth tree: ¬(A & B)

¬(A & B)

branch 1: ¬A

branch 2: ¬B

truth tree: ¬(A ∨ B)

¬(A ∨ B)

    ¬A

    ¬B

truth tree: A → B

A → B

branch 1: ¬A

branch 2: B

truth tree: ¬(A → B)

¬(A → B)

     A

   ¬B

truth tree: A ↔ B

A ↔ B

branch 1:

     A

     B

branch 2:

   ¬A

   ¬B

truth tree: ¬(A ↔ B)

¬(A ↔ B)

branch 1:

     A

    ¬B

branch 2:

    ¬A

      B

using truth trees to determine: logical truth

negate the given formula

ex: is (p ∨ ¬p) a logical truth?

solution:

¬(p ∨ ¬p)

    ¬p

  ¬¬p

    x

yes → (p ∨ ¬p) is a logical truth

using truth trees to determine: unsatisfiability

double negate the given formula

ex: is (p ∨ ¬p) unsatisfiable?

solution:

¬¬(p ∨ ¬p)

 (p ∨ ¬p)

     p

   ¬p

     x

yes → (p ∨ ¬p) is unsatisfiable

using truth trees to determine: equivalence

test both directions of implication