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itional Logic
6
Propositional Logic
We use the
propositional variables
𝑃
and
𝑄. Consider the
formula
𝐴
given by
(𝑃∨𝑃⇒𝑄)⇒¬(𝑃∧𝑄)
1.Give the
value function
ℐ𝜑(𝐴)
for every possible
variable assignment
𝜑, e.g., via a
truth table.
Solution:
There are four
variable assignments:
𝜑(𝑃) 𝜑(𝑄) ℐ𝜑(𝐴) 0 0 10 1 11 0 11 1 0
Note that the left side of the
implication
is always true and
𝐴
is equivalent to
¬(𝑃∧𝑄).
Grading:
0.5
points for each assignment.
AC1misunderstood what an assignment is
AC2wrong set of assignments
AC3individual miscalculations
2.Give some
formula
in
CNF
that is equivalent to
¬𝐴.
Solution:
E.g.,
𝑃∧𝑄.
Grading:
1
point if the
formula
is well-formed and in CNF but not equivalent to
¬𝐴. If equivalent to an incorrect truth table, points are given anyway.
AC1correct CNF relative to false truth table (inherited error)
AC2(𝑃∨¬𝑃∨𝑄)∧𝑃∧𝑄
(correct but unsimplified)
AC3𝑄∧𝑃∧𝑄
(correct but partially simplified)
AC4not well-formed
AC5not equivalent
AC6not CNF
AC7not expanding
𝐴
3.Assume we have constructed a
saturated
tableau
with root
𝐴𝖥
that has a single
open
branch. What useful information about
𝐴
can we read off that open branch?
Solution:
The literals on the branch determine an assignment that falsifies
𝐴. In particular,
𝐴
is not a theorem.
Grading:
1
point for the falsifying assignment.
1
point for the non-theorem-hood.
1
point for giving some parts of the correct answer or given a partially wrong answer.
AC1A is a theorem
AC2anything about how to close the branch
AC3anything about applying a substitution
AC4partial answer
AC5trying to prove
𝐴
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