va ri eties o f
n e c e ssit y
LECTURE 10
t a uto l ogic al
e q ui valen ce
t a uto l ogic al
c o n s eque nc e
LOGICAL TRUTH AND TARSKI’S WORLD
NECESSITIES
 Let’s call a sentence Tarski’s World-necessary (T W -necessary)
just in case it is true in all the worlds that can possibly be
constructed in Tarski’s World.
 What’s the relationship between logical truth and T W -necessity?
 All logical truths are T W -necessar y.
 Some T W-necessities are not logical truths, e.g.,
‘Tet(a) ∨ Cube(a) ∨ Dodec(a).’
 So the logical necessities are a strict subset of the T W necessities.
EXERCISE 4.8
 For each of the following sentences, say where it goes in the
following Euler circle diagram
5. Larger(a,b) ∨ ¬Larger(a,b)
6. Larger(a,b) ∨ Smaller (a,b)
I>CLICKER QUESTION
 ‘¬[¬Tet(a) ∧ ¬Cube(a) ∧ ¬Dodec(a)]’ is
A. A tautology
B. A T W-necessity
C. A and B
D. None of the above
LOGICAL AND
TAUTOLOGICAL
EQUIVALENCE
EQUIVALENCE
 Our informal definition of equivalence, like our informal
definition of logical truth, also referred to possible worlds:
 P and Q are equivalent if and only if they are true in all and only the
same possible worlds.
 With our dif ferent conceptions of possibility and necessity, we
can distinguish dif ferent kinds of equivalence:
 tautological equivalence
 logical equivalence, and
 TW-equivalence.
TAUTOLOGICAL EQUIVALENCE
 To give a more precise definition of tautological equivalence,
we will use the notion of sentences’ joint truth tables.
 The joint truth table for A and B is the truth table with both
sentences to the right of the reference columns.
A
T
F
A ∨ ¬A
T F
T T
A ∧ ¬A
F F
F T
 Sentences A and B are tautologically equivalent if and only if
at each row, the truth value under A’s main connective is the
same as the truth value under B’s main connective.
SOME IMPORTANT LOGICAL
EQUIVALENCES
 DeMorgan’s Laws
 Double Negation
TAUTOLOGICAL EQUIVALENCE AND
LOGICAL EQUIVALENCE
 All tautological equivalences are logical equivalences.
 But remember how not all logical truths were tautologies?
 Similarly, not all tautological equivalences are logical
equivalences.
 Example:
 ‘a = b’ is logically but not tautologically equivalent to ‘¬ FrontOf(a,b).’