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Subsection 7.6.7 So Where Does That Leave Us?

By now, you’ve seen that, in the predicate logic framework that we’ve described, we can:

  • Naturally represent and reason with mathematical facts. (By the way, there are many more examples of this in our follow-on course, Sets, Relations and Functions.)

    x (Prime(x)  ((x = 1)  y (Div(x, y)  ((x = y)  (y = 1)))))

  • Naturally represent and reason with formal specifications for programs.

    x (Disk(x)  On(x, Pole1, time1))

  • Naturally represent and reason with constraints and policies that are stated in terms of objects in databases.

    x ((Loan(x)  Approved(x))  (y (z (Super(y)  Signed(y, x)  Signed(z, x)  (y = z)))))

  • Sometimes represent and reason with our knowledge about the everyday world.

    x (y (Friends(x, y)  Friends(y, x)))

Of course, we wish that reasoning about the everyday world were easier. But even if that’s what you care primarily about, the logical foundation that we’ve built will serve you well. There exist formal extensions to predicate logic that can solve many of the problems that we’ve encountered. And sound everyday reasoning, while not sticking to our formalism, rests on its foundation.

[1] Everyone who’s invited to the party works at MegaLogicLand.

[2] Every couple who’s coming to the party has to bring something.

[3]  Jamie is planning to attend the party, but she knows that her partner is bringing cookies, so she’s not worried about getting something.

[4]  Since Jordan works at MegaLogicLand, he assumes he’s invited to the party.

Jamie’s reasoning is sound. In its essence, [2] says:

[2] Party  Bring something

Jamie knows that her partner’s bringing cookies makes that claim true for them. So she need take no further action.

But Jordon’s reasoning is flawed. In its essence, [1] says:

[1] Invited  MegaLogicLand

Jordan has used Modus Ponens backwards to attempt to reason:

 MegaLogicLand  Invited

And that, we know, does not preserve truth.

Big Idea

The logical tools that we have developed form the basis for sound reasoning in mathematics, in many areas of computer science, and in our everyday world.

Exercises Exercises

1.

Consider the following claims:

[1] Everyone who thinks that it’s going to rain tomorrow will bring an umbrella.

[2] It’s going to rain tomorrow.

[3] The weather forecast is very accurate if you ask for just one day in the future.

[4] Bailey just checked the weather forecast.

[5] Dana never checks the weather forecast.

[6] Bailey will bring an umbrella tomorrow.

[7] Dana won’t bring an umbrella tomorrow.

Take [1] – [5] as premises. Which of the following statements is correct:

  1. [6] follows from the premises but [7] does not.

  2. [7] follows from the premises but [6] does not.

  3. Both [6] and [7] follow from the premises.

  4. Neither [6] nor [7] follows from the premises.

Answer.

Correct answer is A

Solution.

Explanation: [6] follows: It’s going to rain. Bailey checked the weather forecast, which is accurate, so Bailey knows that. Thus Bailey will bring an umbrella.

But [7] doesn’t follow. [1] says that thinking it’s going to rain implies bringing an umbrella. We can’t conclude from [1] that not thinking it’s going to rain implies no umbrella. It’s possible for someone, say Dana, to bring an umbrella all the time. It’s also possible that Dana knows it’s going to rain by some other means than checking the weather forecast. Bottom line: we don’t know whether Dana will bring an umbrella.