Title: CS1502 Formal Methods in Computer Science
1CS1502 Formal Methods in Computer Science
- Lecture Notes 4
- Tautologies and Logical Truth
2Constructing a Truth Table
- Write down sentence
- Create the reference columns
- Until you are done
- Pick the next connective to work on
- Identify the columns to consider
- Fill in truth values in the column
- EG (A (A v (B C))) v B
- (in Boole and on board)
3Tautology
- A sentence S is a tautology if and only if every
row of its truth table assigns true to S.
4Example
- Is ?(A ? (?A ? (B ? C))) ? B a tautology?
5Example
6Logical Possibility
- A sentence S is logically possible if it could be
true (i.e., it is true in some world) - It is TW-possible if it is true in some world
that can be built using the program
7Examples
- Cube(b) ? Large(b)
- ?(Tet(c) ? Cube(c) ? Dodec(c))
- e ? e
Logically possible
TW-possible
Not TW-possible
Logically possible
Not Logically possible
8Spurious Rows
- A spurious row in a truth table is a row whose
reference columns describe a situation or
circumstance that is impossible to realize on
logical grounds.
9Example
Spurious!
Spurious!
10Logical Necessity
- A sentence S is a logical necessity (logical
truth) if and only if S is true in every logical
circumstance. - A sentence S is a logical necessity (logical
truth) if and only if S is true in every
non-spurious row of its truth table.
11Example
Logical Necessity
TW-Necessity
Not a tautology
12Example
Not a TW-Necessity
Not a Logical Necessity
Not a tautology
According to the book, the first row is spurious,
because a cannot be both larger and smaller than
b. Technically, though, Larger and Smaller
might mean any relation between objects. So, the
first row is really only TW-spurious. This
issue wont come up with any exam questions based
on this part of the book. (The book refines
this later.)
13Tet(b) ? Cube(b) ? Dodec(b)
Tet(b) ??Tet(b)
aa
Cube(a) v Cube(b)
Cube(a) ? Small(a)
14Tautological Equivalence
- Two sentences S and S are tautologically
equivalent if and only if every row of their
joint truth table assigns the same values to S
and S.
15Example
S and S are Tautologically Equivalent
16Logical Equivalence
- Two sentences S and S are logically equivalent
if and only if every non-spurious row of their
joint truth table assigns the same values to S
and S.
17Example
S
S
Not Tautologically equivalent
Logically Equivalent
18Tautological Consequence
- Sentence Q is a tautological consequence of P1,
P2, , Pn if and only if every row that assigns
true to all of the premises also assigns true to
Q. - Remind you of anything?
- P1,P2,,Pn Q is also a valid argument!
- A Con Rule Tautological Consequence
19Example
premises
conclusion
Tautological consequence
20Logical Consequence
- Sentence Q is a logical consequence of P1, P2, ,
Pn if and only if every non-spurious row that
assigns true to all of the premises also assigns
true to Q.
21premise
conclusion
Not a tautological consequence
Is a logical consequence
22Summary
23Summary
Logical-Consequences of P1Pn
- Every tautological consequence of a set of
premises is a logical consequence of these
premises. - Not every logical consequence of a set of
premises is a tautological consequence of these
premises.
Tautological- Consequences of P1Pn
24Summary
- Every tautological equivalence is a logical
equivalence. - Not every logical equivalence is a tautological
equivalence.
25Summary
- Every tautology is a logical necessity.
- Not every logical necessity is a tautology.