Question 1199145: Use one of the 3 methods of syllogistic logic to determine whether the following syllogism is
valid:
No carrots are red fruits.
Some fruits are carrots.
Therefore, some fruits are not red fruits.
Found 3 solutions by Edwin McCravy, RBryant, ikleyn:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website! Use one of the 3 methods of syllogistic logic to determine whether the following syllogism is
valid:
No carrots are red fruits.[Translation: if we have something that fits the category of being a carrot, then
it does not fit both the category of being red and the category of being a fruit.]
C ⊃ ~(R • F) Some fruits are carrots.[Translation: We DO have some things that fit the category of being a fruit and
the category of being a carrot.
F • C Therefore, some fruits are not red fruits.[Translation: We DO have some things that fit the category of being a fruit but (and)
do not fit the category of being red.]
F • ~R
------------------
1. C ⊃ ~(R • F)
2. F • C / F • ~R
------------------------------
3. F 2, simplification
4. C • F 2, commutation
5. C 4, simplification
6. ~(R • F) 1,5 modus ponens
7. ~R v ~F 6, DeMorgan's law
8. ~F v ~R 7, commutation
9. ~~F 3, double negation
10. ~R 8,9, disjunctive syllogism
11. F • ~R 3,10 conjunction
The conclusion has been reached. This shows that the argument is valid.
Edwin
Answer by RBryant(14)  (Show Source):
You can put this solution on YOUR website!
No carrots are red fruits.
Some fruits are carrots.
Therefore, some fruits are not red fruits.
This syllogism is a VALID syllogism. We know this because the mood and figure of the syllogism: the mood is EIO and the figure is the first figure.
Well what if you have no idea what mood and figure are? Well you should learn it! Are there other ways to show Validity of the syllogism? Yes! We can use diagrams such as Venn Diagrams, Euler Diagram or use the Rules of Syllogism to show Validity.
RULES FOR VALIDITY FOR CATEGORICAL SYLLOGISMS:
1. There must be exactly three and only three terms. NOTE: If there are more than three then you must reduce the terms to three such as eliminate synonyms or different words that mean the same thing.
2. The middle term must not occur in the conclusion.
3. The middle term must be distributed at least once.
4. If a term is distributed in the conclusion, then that term must be distributed in the premises it originally occurred.
5. If the two premises are affirmative, the conclusion must also be affirmative.
6. If either premise is negative, the conclusion must be negative.
7. No conclusion can be drawn from two negative premises (simply because it would commit at least one fallacy).
8. No particular conclusion can be directly derived from universal premises. Note: this can be done but not directly; one would have to use a mediate inference rule such as sub-alternation (this may also be called conversion in some texts). See the Square of Opposition if you do not know mediate inferences.
***Each of the above rule is associated with at least one fallacy. This method does not require any diagrams and based on the rules alone. Violating any rule will result in a syllogism being INVALID.
Here is a Venn Diagram of the sylllogism (which shows Validity of the syllogism):
Another way is using the syllogistic method by comparing the KNOWN valid forms to the syllogism you have in question after you determine the mood and figure:
Once you find the mood and figure of any syllogism you search for it on the list above. The names were used to remember what the valid moods & figures were with no diagrams and without using the syllogism rules above.
What do the NAMES mean? Well today we do not need this reference but it is still useful. All you need is the VOWELS to know the mood. You can ignore the constants such as the b, r,t,n. However the letters S M P do have meanings. S stood for the rule of conversion (which allows us to move from universals to particulars such as an A to an I or an E to an O proposition; M stood for the muttatio which means change the order of the Premises; and the C stood for a contradiction where one would change a Premise in the syllogism to derive a contradiction. This is simiar to indirect Proof but for syllogisms.
This is what a syllogistic method looks like:
1. C e R
2. F i C / ∴ F o R
--------------------------------------
3. F o R 1,2 Ferio
QED.
Notice the abbreviations I used. C = carrots; e = E statement; R = red fruits;
F =fruit; i = I statement ; C= carrots (used twice). Listed nicely we get the following:
C e R = No carrots are red fruits.
F i C = Some fruits are carrots.
F o R = some fruits are not red fruits.
We end up with a FERIO syllogism in the first figure. We can show this as valid also using the inference rule method and Venn Diagrams. From a distance we see an artistic form which renders beauty in a proof.
Answer by ikleyn(52777)  (Show Source):
You can put this solution on YOUR website! .
Use one of the 3 methods of syllogistic logic to determine whether the following syllogism is
valid:
(1) No carrots are red fruits.
(2) Some fruits are carrots.
(3) Therefore, some fruits are not red fruits.
~~~~~~~~~~~~~~~~~
Instead of using methods of syllogistic logic, I will try to solve the problem
using my common sense, by the " step by step method ".
For the convenience of references, I numbered the sentences (statements) in the " given " part.
Step 1. Statement (1) is true, because no carrots are fruits, at all (by the definition).
Step 2. Statement (2) means that the mentioned set of "some fruits" is EMPTY set,
because no fruits are carrots (by the definition).
Step 3. Since the set "some fruits" defined in (2), is empty, it satisfies to any condition.
In particular, we can conclude that this set "some fruits" are not red fruits, as stated in (3).
So, the property (3) is proven.
The most important part of this " common sense " solution is to
identify that the set "some fruits" in part (2) and in part (3) is the same set
and to get understanding that this set, the problem is talking about, is empty.
For empty set, each statement about properties of its elements is correct,
independently, whether it looks to be sensical or non-sensical.
+--------------------------------------------+
| As an example, consider THIS statement: |
| "As far as it is known for now, |
| each horse with 99 legs is blue". |
+--------------------------------------------+
For empty set, each statement about properties of its elements is correct,
but empty, at the same time.
By the way, since the set " some fruits " in this problem is empty set,
it possesses not only the property " 3 ", but the opposite property, too.
What I want to say, regarding the meaning of this problem, is that
its meaning is emptiness : it is like dust carried by wind.
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