Section 2: Logical Statements

Now that you’ve seen the symbols that we’re going to use, let’s see a few larger examples of how they can be incorporated into realistic situations. Logic is so pervasive in everything that we do that you can find examples of the ideas in almost any area (although the formal notation is very seldom, if ever, used outside of a propositional logic class).

To start, we need to make some statements. Essentially we’re creating “variables” that represent a boolean statement (meaning it can be either true or false).

p: I speed past a cop.
q: I get a ticket.

Based on that, and the symbols you saw in the previous section, what do you think is meant by the following statement?

p → q
If I speed past a cop, then I get a ticket.

Think about what different situations are possible (i.e. true) given this statement.

  • Speed past the cop and get a ticket?true
  • Speed past the cop and NOT get a ticket?false
  • Not speed past the cop but still get a ticket?true
  • Not speed past the cop and NOT get a ticket?true

If you disagree with the answers shown, think about if there are other kinds of tickets you might receive. (I’m betting that’s where the issue is.)

Let’s try another one.

p: Patty attends the party.
q: Quinn attends the party.
r: Robert attends the party.

What is the practical meaning of each logical statement?

p ∧ r
Patty and Robert attend the party.
p ⊕ q
Patty or Quinn attend the party, but not both.
p ∨ q ∨ r
At least one (but maybe more) of Patty or Quinn or Robert attend the party.
p → (q ∧ r)
If Patty attends the party, Quinn and Robert will both attend the party as well.
p ↔ (q ∧ r)
If Patty attends the party, Quinn and Robert will both attend the party. But if Patty does not attend, Robert and Quinn will not both be at the party. (But one of them might still go! They just won’t both be there.)
(p ⊕ q) ∧ r
Either Patty or Quinn (but not both) will attend the party and Robert will attend the party.
(p ∧ r) ⊕ (q ∧ r)
Either Patty and Robert, or Quinn and Robert will attend the party. (Notice that this is actually the exact same result as the last one! So the two statements are logically equivalent.)



In the next section, you will learn a bit more about how these logical statements can interact with each other.


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