3. The rule of transitivity of conditionals
This is an inference where you can infer a new conditional from the two given conditionals if they have this form:
This is an inference where you can infer a new conditional from the two given conditionals if they have this form:
Note: we identify rules of inference by their form, that is, by the form of their premises and the conclusion.
Notice the form of this rule. It has two conditionals as premises, and one conditional as a conclusion. The second clause of one premise (see the blue triangle below) is the same as the first clause of the other premise.
Note also that the order of premises is not important, just their form.
The transitivity is an inference you can draw (like a Contrapositive). Here you can combine two statements to make a new one.
If A → B, and B → C; then A → C.
Valid Inference (Transitivity):
A → C
If A, then B: If I press the off button (A), then the generator will turn off (B).
If B, then C: If the generator is off (B), then the website will shut down (C).
If A, then C: If I press the off button (A), then the website will shut down (C).
This video consolidates everything we’ve covered. Watch how he contraposes conditionals joined with “and.“
If Jean went to the concert, then Ricardo also went to the concert. If Ricardo went to the concert, then Adrian also went to the concert. Jean went to the concert.
If the above statements are true, which of the following must also be true?
(A) If Jean did not go to the concert, then Adrian also did not go to the concert.
(B) Adrian also went to the concert.
(B) Adrian also went to the concert.
Statement |
Symbols |
Valid/Invalid |
Description |
1. If Jean went to the concert, then Ricardo also went to the concert. |
J → R |
Given |
Premise 1 |
2. If Ricardo did not go to the concert, then Jean did not go to the concert. |
~R → ~J |
Valid |
Contrapositive of premise 1 |
3. If Ricardo went to the concert, then Adrian also went to the concert. |
R → A |
Given |
Premise 2 |
4. If Adrian did not go to the, then Ricardo did not go to the concert. |
~A → ~R |
Valid |
Contrapositive of premise 2 |
5. If Jean went to the concert, then Adrian also went to the concert. |
J → A |
Valid |
Inference from premise 1 and 2 |
6. If Jean did not go to the concert, then Adrian also did not go to the concert. |
~J → ~A |
Invalid |
Inverse of inference from premise 1 and 2 |
Increase in the country’s GDP means the country’s economy is improving. If the country’s agriculture is not doing good, then the country’s economy is not improving.
If the above statements are true, which of the following must also be true?
(A) If the country’s agriculture is doing good, then the country’s economy is improving.
(B) If the country’s agriculture is doing good, then the country’s GDP is increasing.
(C) If the country’s agriculture is not doing good, then the country’s GDP is not increasing.
(C) If the country’s agriculture is not doing good, then the country’s GDP is not increasing.
Statement |
Symbols |
Valid?/ |
|
1. Increase in the country’s GDP means the country’s economy is improving. |
GDP → economy |
Given, Premise 1 |
|
2. If the country’s agriculture is not doing good, then the country’s economy is not improving. |
~agriculture → ~economy |
Given, Premise 2 |
|
3. If the country’s agriculture is not doing good, then the country’s GDP is not increasing. |
~agriculture → ~GDP |
Valid, Inference from Premise 1 and 2 |
|
4. If the country’s agriculture is doing good, then the country’s economy is improving. |
agriculture → economy |
Invalid, Inverse of Premise 1 |
|
5. If the country’s agriculture is doing good, then the country’s GDP is increasing. |
agriculture → GDP |
Invalid, Inverse of Inference |
|
Due to a conflict in schedule, I can watch movie 1 only if I do not watch movie 2. If I don’t watch movie 3, then I can watch movie 2.
If the above statements are true, which of the following must also be true?
(A) If I watch movie 3 then I can watch movie 1.
(B) If I can watch movie 1 then I can also watch movie 3.
(C) If I watch movie 3 then I can’t watch movie 2.
(B) If I can watch movie 1 then I can also watch movie 3.
Statement |
Symbols |
Valid/Invalid |
Description |
1. I can watch movie 1 only if I do not watch movie 2. |
M1 → ~M2 |
Given |
Premise 1 |
2. If I don’t watch movie 3, then I can watch movie 2. |
~M3 → M2 |
Given |
Premise 2 |
3. If I can watch movie 1 then I can also watch movie 3. |
M1 → M3 |
Valid |
Inference from Premise 1 and 2 |
4. If I watch movie 3 then I can watch movie 1. |
M3 → M1 |
Invalid |
Converse of Inference |
5. If I watch movie 3 then I can’t watch movie 2. |
M3 → ~M2 |
Invalid |
Inverse of Premise 2 |
Next LSAT: Sep 08/ Sep 09
Every problem is absurdly simple when it is explained to you.
-Sherlock Holmes