Question: Using the rules of validity, show what syllogisms are valid, when the conclusion is O.
This is the correct answer of the same question, but with the conclusion being I (it is preferred to have the answer in this format, showing the rules that applied to each step that got you to the correct answer):
Suppose the conclusion is I. Then at least one premise is particular (Rule 2). No premise is negative (Rule 3). One premise is universal (Rule 2). Therefore, the premises are either AI or IA. If AI, M must be first in the major premise. Therefore, if AI, the figure can either be 1 or 3, All-1 or All-3. If the premise is IA, then M must still be distributed (Rule 1). Therefore, M is first in the second premise. Therefore, IAI is valid in figure 4 or 3.
Here is relevant information to help solve the problem:
E & O are negative
A & E are universal
I & O are particulars
A & I are affirmative
A: All _____ are ______
E: No _____ are ______
I: Some _____ are _____
O: Some _____ are not _____
A term has distribution or is said to distribute if it is the 1st term of the universal or the 2nd term of the negative.
If the premise is true and the conclusion is false it is invalid.
The first term is the major premise. The second term is the minor premise. The third term is the conclusion. And the number reflects which Figure you used to make it a valid syllogism. (ie. AAA-1 or EIE-3).
Here they the Figures:
Premise 1: M P P M M P P M
Premise: 2: S M S M M S M S
Conclusion: S P S P S P S P
Figure 1 Figure 2 Figure 3 Figure 4
The mood refers to the string of the two premises and the conclusion regardless of the figure and is written using the A, E, I, O format.
Rules to determine validity: A Standard Form Categorical Syllogism (SFCS) is valid if and only if there are no substitutions producing true premises and false conclusion.
There are four Rules for determining the validity of a Standard Form Categorical Syllogism (SFCS):
1. The Middle term must be distributed once and only once.
2. The number of particular conclusions must EQUAL the number of particular premises (0 or 1).
3. The number of negative conclusions must EQUAL the number of negative premises (0 or 1).
4. If a term is distributed in the conclusion, it must be distributed in the premises.
* I am told that it is easier to find validity if you start with Rules 2 and 3. *
(In proofing my posting I see that the way I typed each item out actually gets mushed together. I hope you are used to reading it and understanding that it is line by line. For example, for the four Figure possibilities, each figure has 2 letters in premise 1's, then 2's, then the conclusion. Figure ones premise 1 is: M P. It's premise 2 is: S M and conclusion: S P. Hopefully you can understand by the way it gets posted.
Thank you! Good luck :-)