Discrete Structures

# Lecture Notes, 10 Sep 2010

 Last Update: 13 September 2010 Note: or material is highlighted

1. "Recursive" Definition of ("Well-Formed" or Grammatically Correct) Proposition:

1. Base Case:
Atomic propositions (represented by p, q, r, etc.) are (well-formed (wf), grammatically correct) propositions.

2. "Recursive" Cases:
If A, B are (wf, grammatically correct) propositions (either atomic or molecular),
then:

1. ¬A
2. (AB)
3. (AB)
4. (AB)
5. (AB)
6. (AB)

are (wf, grammatically correct) "molecular" (or "compound") propositions.

3. "Closure" clause: Nothing else is a (wf, grammatically correct) proposition.

2. Computing truth tables for molecular propositions:

1. Base case: The truth value of an atomic proposition is either T or else it is F.

2. Recursive case:

1. The recursive case is based on the Principle of Compositionality:

The truth value of a molecular proposition depends on (i.e., is a function of) (i.e., is computed from)

1. the truth values of its "constituent parts"
2. and its "principal connective"

Let * be any of the logical connectives.
Then:
• The (constituent) parts of (A * B) are A and B.
• The principal connective of (A * B) is *.

2. So, to compute the t-val of a molecular proposition of the form (A * B),
first compute the t-val of A,
then compute the t-val of B,
then use the t-val for * to combine these t-vals into the t-val for (A * B).

3. How do you compute the t-vals of A and of B?
If they are molecular, follow rule B
(but aren't we in the middle of applying rule B?
Yes!
This is called a "recursive" application of the rule,
because the same rule "recurs" (= re-occurs) inside of its own application)

If they are atomic, then apply the Base Case, rule A:

the t-val is either T or else F.

3. In the t-table for ((p ∨ q) ∧ ¬(p ∧ q)) from last lecture,
the "input" columns are the tvals of the atomic propositions,
the "intermediate columns" are the tvals of the constituent parts,
and the "output" column is the final t-val for the molecular proposition.

3. Translating from English to Logic:

1. This is a hard problem, because there are lots of different ways to say the same thing in English;

2. It is equivalent to the Natural-Language Understanding Problem of computational linguistics:
How do you represent English sentences so that computers can understand them?

1. Trivial way: input English sentence S1; output atomic proposition p1
2. For better ways, see websites, esp. Suber's guide

4. Propositional Equivalences (§1.2)

1. Consider the t-table for (p ∨ ¬p)
(I won't type it in here; you should do it!)

Note that, for all possible t-vals of its atomic proposition, the t-val of the molecular proposition is T.

2. Def:
Let A be a proposition.
Then A is a tautology
=def
for all possible t-vals of A's atomic propositions (i.e., for all rows of A's t-table), t-val(A)=T.

3. Consider the t-table for (p ∧ ¬p)
(I won't type it in here; you should do it!)

Note that, for all possible t-vals of its atomic proposition, the t-val of the molecular proposition is F.

4. Def:
Let A be a proposition.
Then A is a contradiction
=def
for all possible t-vals of A's atomic propositions (i.e., for all rows of A's t-table), t-val(A)=F.

5. Consider the t-table for (pq)
(I won't type it in here; you should do it!)

Note that, there are some possible t-vals of its atomic proposition such that the t-val of the molecular proposition is T,
but there are some other possible t-vals of its atomic proposition such that the t-val of the molecular proposition is F.

6. Def:
Let A be a proposition.
Then A is contingent
=def
there are some possible t-vals of A's atomic propositions (i.e., there are some rows of A's t-table) such that t-val(A)=T
(so it's not a contradiction),
but there are some (other) possible t-vals of A's atomic propositions (i.e., there are some other rows of A's t-table) such that t-val(A)=F
(so it's not a tautology).

7. Recall that the input-output columns of the t-table for ((p ∨ q) ∧ ¬(p ∧ q)) matched those of the t-table for (p ⊕ q).

1. Def:
Let A, B be propositions.
Then A is (logically) equivalent to B
=def
for all rows of A's and B's t-tables, t-val(A) = t-val(B)

2. Notation: AB for: A is logically equivalent to B

• So, we can say:
((p ∨ q) ∧ ¬(p ∧ q)) ≡ (p ⊕ q)

• Note that ≡ is not one of our proposition-forming binary connectives
(it's not part of the recursive definition of "proposition").
And "A ≡ B" is not a proposition of propositional logic.
Instead, it's a sentence of "mathematical English" that is about propositional logic.
Next lecture: the relation between ≡ and ↔…

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