Lesson LOGIC PROBLEM INVOLVING IF THE DOG WAGS ITS TAIL THEN THE DOG IS NOT CALM WITH SOLUTION AND EXPLANATION

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PROBLEM

determine which if any of the three statements are equivalent. give a reason for your conclusion, and show complete work.
i)if the dog wags its tail, then the dog is not calm.
ii)either the dog does not wag its tail or the dog is not calm
iii) if the dog is not calm, then the dog wags its tail.

SOLUTION

problem statements are:

i)if the dog wags its tail, then the dog is not calm.
ii)either the dog does not wag its tail or the dog is not calm
iii) if the dog is not calm, then the dog wags its tail.

let:

p = the dog wags its tail.
q = the dog is not calm.

~p = the dog does not wag its tail.
~q = the dog is calm.

statement 1 translates to p -> q
statement 2 translates to ~p v q
statement 3 translates to q -> p

statement 1 and 3 are not equivalent by reason of the fallacy of the converse.

with an implied statement you can have 3 possible additional statements associated with it.

they are:

implied statement:
p -> q
inverse of the implied statement:
~p -> ~q
converse of the implied statement:
q -> p
contra-positive of the implied statement:
~q -> ~!p

only the contra-positive of the implied statement is equivalent to the implied statement.

the inverse of the implied statement is not equivalent to the implied statement.  any assumption that it is leads to the fallacy of the inverse.

the converse of the implied statement is not equivalent to implied statement.  any assumption that it is leads to the fallacy of the converse.

we need to test with statement 2 to see if it's equivalent to either one of the other 2.

when in doubt, you should do a truth table test.
it tells you whether the statements are equivalent or not.

when you do the truth table equivalency test, you are working with demorgan's law.  
that law states that if two statements have the same truth table entries for all possible conditions in the truth table, then those two statements are equivalent.  if you make a column in the truth table that tests for the equivalency of those two statement, then that column will be true for all possible conditions in the truth table.  if it's not true for all conditions then the statements are not equivalent.

the truth table for these 3 statements is shown in the following picture.

in this picture:

statement 1 has been called statement A.
statement 2 has been called statement B.
statement 3 has been called statement C.

The truth table says that A is equivalent to B which means that (p -> q) is equivalent to (~q v p).

That means that statement 1 is equivalent to statement 2.

It would be difficult to see this without use of the truth table.

Logically, trying to make sense of it takes a little stretch, but it can be done.

i believe the answer lies in the law of disjunction.

The law of disjunction states:

if A or B is true, then:
If A is false, then B must be true.
If B is false, than A must be true.

we have two statements that we think are equivalent because the truth table test of equivalency says that they are.

The two statements are:

i)if the dog wags its tail, then the dog is not calm.
ii)either the dog does not wag its tail or the dog is not calm

the contra-positive of the first statement and the first statement are equivalent.
this is already given and has been proven elsewhere.  i provide you with a link to the proof down further.

The first statement is:
if the dog wags its tail, then the dog is not calm.

The contra-positive to the first statement is:
if the dog is calm, then the dog does not wag its tail.

The second statement is:
either the dog does not wag its tail or the dog is not calm.

If we assume that the dog does not wag its tail is false, then the law of disjunction says that the dog is not calm must be true.

this is because the law of disjunction says that it's either one or the either.  if it is not one, then it must be the other.

we can write this another way to say:
if the dog wags its tail, then the dog is not calm.
that looks very much like statement 1.

if we assume that the dog is not calm is false, then the dog does not wag its tail must be true by the same law of disjunction.

we can write this another way to say:
if the dog is calm, then the dog does not wag its tail.
that looks very much like the contra-positive to the statement 1.

i'm satisfied that the truth table is telling the truth and that statements 1 and 2 are equivalent even though anybody would be hard pressed to determine that they are without using the rules of logic to confirm that.

the truth table test is the way to determine the equivalency of two statements.  the fact that you can do that is determined by demorgan's law.

here's a reference that can be helpful to understand the different statement laws and fallacies.

<a href = "http://www.algebra.com/algebra/homework/Conjunction/THEO-061210.lesson" target = "_blank">STATEMENT LAWS AND FALLACIES</a>