L2W1

Today: We'll go over some more logic and think about how we'll use our courseware.

Administrative Details:
Participation counts: Includes attendance, forum, POP quizzes
Homework: resultsTrack OR print.
HW#1 next time (informal proofs may be hard; you may feel the x-word’s not worth it)
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Forum, email

Review:

  1. Arguments
  2. Conclusion and Premise Indicators
  3. Valid
  4. Sound
  5. Strong
  6. Cogent???

 

Deductive and Inductive Arguments
Idea?

 

 

 

 

How about this one:

All whales are mammals, so no whales are fish. (Assuming fish are all non-mammals.)

 

Exercises on Validity

Exercises on Deductive/Inductive

 

 

 

(Logically) Possible = what could have been true.

(Logically) Impossible = what couldn't have been true.

 

 

 

 

Examples something that IS POSSIBLE: No Bush ever becomes president of the US. (It’s settled that this is false; but it could have been true if certain elections had gone differently.)

But it's NOT possible that a square be round.

 

Informal Proofs:

 

 

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(*) If an argument is invalid, then you can show it’s unsound.

Proof:
Suppose A is any arbitrary invalid argument.
Then by the definition of soundness, A is sound if and only if (iff)
A is both (1) valid and (2) has all true premises.
By our supposition, A is not valid so does not meet part (1) of the defintion of soundness.
So, we’ve seen that if A is invalid then it’s unsound.
Q.E.D.

(**) Any valid argument with a false conclusion has at least one false premise.

Proof:
Suppose that A is a valid argument with false conclusion.
Then A can’t have all true premises, o/w it would have to have a true conclusion.
So, A must have at least one false premise.
Q.E.D.

Second proof of ** by the indirect method:
Suppose that A is a valid argument with false conclusion.
Assume for contradiction that the statement is wrong, that A has no false premise.
Then all A’s premises are true.
And because A is valid, it follows that it’s conclusion has to be true.
But this contradicts the supposition.
Hence if the supposition is true, the premises can’t all be true so at least one must be false.
Q.E.D.

(***) If an argument has a false premise, then it’s unsound.
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More:

A. Suppose an argument is invalid, then you can show it’s unsound.
B. If an argument has a false premise, then it’s unsound.
C. If an argument is sound, then it has a true conclusion.
D. Any valid argument with a false conclusion has at least one false premise.
E. A valid argument CAN have a false premise but a true conclusion.

 

 

 

More Deductive Logic

Definitions: 1,2,3

A sentence if logically true iff it's impossible for it to be false.

A sentence if logically false iff it's impossible for it to be true.

A sentence if logically true iff it's is possibly true and possibly false.

 

Examples

 

Informal Proof: If a sentence S is logically true, then it's negation not-S is logically false.

 

 

Turn next to pairs of sentences. For example,

All whales are mammals.
No whales are non-mammals.

Intuitively these two sentences mean the same thing. We will call them logically equivalent. To give a definition that is a little more precise than talk about "meaning" allows, we utilize the notion of possibility yet again:

The two members of a pair of sentences are logically equivalent if and only if it is not possible for one of the pair to be true while the other is false.

Also, we say that one sentence logically entails another when the argument from the first as premise to the second as conclusion is valid. In other words,

One sentence logically entails a second sentence just in case the first could not possibly be true while the second is false.

Finally, sets of sentences can tell coherent stories or, on the other hand, their members can conflict with one another. Roughly, a set of sentences is logically consistent when there is no contradiction between its members.

A set of sentences is logically consistent if and only if it is possible for all members of the set to be true together.
A set of sentences is logically inconsistent if and only it it is not logically consistent, i.e., it is not possible for all members of the set to be true together.

 

 

 

 

 

1. Little boy/pool example (I didn’t go in the Berger’s pool, my hair’s not wet. Anyway, Billy made me do it!)
2. Informal Proof: If \ contains a logically false sentence, then \ is logically inconsistent.