An appeal to authority is a common type of fallacy that arises when one who has the difficulty in confronting or understanding complicated questions will seek refuge to the ideas, concepts, principles or judgments of a person who enjoys a reputation as an expert or an authority of the matter at issue. In other words, an appeal to authority is a fallacious argument in which the testimony of someone believed to be an authority is cited in support of a conclusion. It must be noted that the person being cited here is not, in fact, an expert or an authority on the matter or for some reason should not be relied upon. Thus, the fallacy of appeal to authority occurs when the authority cited is not qualified in the relevant matters or, less typically, is not free from adverse influences. Thus, the arguer is relying upon the assertions of someone who is not truly in a position to know.
Let us consider the following examples:
I know your doctor says you need your appendix removed, but according to the famous herbalist Mar Lopez, people with your symptoms just need a change in their diet, plus a daily intake of MX3 capsule. So, forget about having your appendix removed.
Augustine said there is no salvation outside the Catholic Church. That’s reasonable enough for me.
The underlying idea of such arguments is that some statement p is true because some authority q has said it is true. The argument’s basic structure is this:
Authority p asserts that q. Therefore, q.
Here, we see immediately that such an argument is neither valid nor inductively strong, since the mere fact that someone asserts q neither makes it so nor makes it probable. Typically, however, the arguer believes more than the mere fact that p asserts that q. The arguer very likely is assuming such things as that p is someone who knows what he or she is talking about regarding q, or that p is speaking without bias, or that p is telling the truth. If those or similar assumptions are well founded, then the appeal to authority p may constitute a good argument, that is, non-fallacious reasoning. It must be noted that not all appeals to authority are fallacious. In fact, some appeal to authority may be inductively strong. After all, we should accept the testimony of qualified and unbiased experts, for there are indeed experts in their own right.
Now, to identify an appeal to authority fallacy, we ask two questions: 1) Is the authority, in fact, a qualified authority about matters related to q? and 2) Is there any good reason to believe that the authority may be biased in matters related to q?
Regarding Example #1 above, we should ask whether Mar Lopez is qualified to claim that proper diet and daily intake of MX3 capsule will render the removal of someone’s appendix unnecessary. So, is Mar Lopez an expert in human anatomy? Can he provide a scientific proof that proper diet and daily intake of MX3 capsule will render the removal of someone’s appendix unnecessary?
Regarding Example #2, we should ask whether St. Augustine, although he was a famous Catholic theologian, has proofs that heaven and hell really exist. As a matter of fact, issues about heaven and hell are very complicated ones. In fact, nobody has proven that indeed heaven and hell exist. If this is the case, how can we meaningfully talk about salvation?
A common variation on the appeal to authority is an appeal to a magazine or newspaper article or a radio or TV program. Consider this example:
“They have found a cure for cancer. I read about it in The New York Times.”
In such case, we ought to ask the same question: Is the source cited a reliable one in this matter? Ordinarily, we should be very suspicious of medical breakthroughs reported in The New York Times, though not of such breakthroughs reported in, say, the Journal of the American Medical Association. On the other hand, we would not expect to get reliable advice on the news or current events in a medical journal. Hence, the appeal to authority fallacy occurs when an argument is supported by reference to a publication or program not known for specialization on the subject.
In summary, not all appeals to authority are fallacious. The appeal to authority fallacy only occurs when an arguer appeals to someone who is not an expert in the field for which he or she is cited as support or who is not unbiased.
To recognize the appeal to authority fallacy, we need only look for an argument based primarily on the premises that some person (or some publication) reports that q is true. The fallacy occurs when the person (or publication) is not relevantly qualified or is not speaking without bias.
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A syllogism can be distinguished from other syllogisms by its form, that is, the mood and figure of a syllogism.
A syllogism’s form is determined by the mood and figure of the argument.
On the one hand, a Mood refers to the kinds of propositions that syllogistic arguments contain, whether A, E, I or O.
On the other hand, a Figure refers to the position of the middle term in the premises.
Note that the validity or invalidity of the syllogism depends exclusively upon its form and is completely independent of its specific content or subject matter.
Example:
All Filipinos are humans. All Cebuanos are Filipinos. Therefore, all Cebuanos are humans.
The mood and figure of a syllogism can be easily known if the letter S, P, and M are used to represent minor term, major term, and middle term respectively. Thus, the above syllogism will have this:
S – Cebuanos
P – humans
M – Filipinos
Using now the letters corresponding for each term, the syllogistic skeleton of the syllogism can be represented as:
All M are P. All S are M. Therefore, all S are P.
The mood of the argument is AAA since the premises and conclusion are all universal affirmative.
As previously stated, figure refers to the middle term’s position in the premises. There are four possible positions of the middle term in the premises, namely, diagonal to the right (figure 1), vertical to the right (figure 2), vertical to the left (figure 3), and diagonal to the left (figure 4).
It is interesting to note that with the four basic statement forms (A, E, I, and O) and four ways of positioning the middle term, it is possible to construct 256 different syllogistic arguments.
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Eduction is a form of immediate inference which involves the act of drawing out the implied meaning of a given proposition. There are 4 kinds of eduction, namely:
conversion,
obversion,
contraposition, and
inversion
Conversion
Conversion refers to the formulation of a new proposition by way of interchanging the subject and the predicate terms of an original proposition, while retaining the quality of the original proposition. The original proposition is called the convertend, while the new proposition is called the converse. Let us consider the example below.
No plant is an animal. Hence, no animal is a plant.
As is well known, the original proposition is called the “convertend”, while the new proposition is called the “converse”. And in the example above, it must be noted that the new proposition “No animal is a plant” is the implied meaning of the original proposition, that is, “No plant is an animal”.
There are two types of conversion, namely, simple and partial conversion.
Simple conversion is a type of conversion where the quantity of the convertend is retained in the conversion. It must be remembered that only universal negative (E) and particular affirmative (I) propositions can be converted through simple conversion.
Example 1:
No angels are mortals. (E) Therefore, no mortals are angels. (E)
Example 2:
Some mortals are men. (I) Therefore, some men are mortals. (I)
As already mentioned, only universal negative (E) and particular affirmative (I) propositions can be converted because in universal affirmative (A) propositions, the quantity of the predicate term in the convertend (which is particular) which becomes the subject term in the converse cannot be retained; while in particular negative (O) propositions, the subject term of the convertend, being made the predicate term of a negative proposition, would be changed from particular to universal. Let us consider the examples below:
Example 1:
All dogs are animals. (A) Therefore, all animals are dogs. (A)
As we can see, the quantity of the predicate term “animals” in the original proposition, that is, the convertend, is particular because the proposition is affirmative. As we learned in the previous discussions, the predicate terms of all affirmative propositions are particular (while the predicate terms of all negative propositions are universal). Now, the quantity of the term “animals” which becomes the subject term in the converse is universal because of the universal signifier “all”. Hence, we cannot convert universal affirmative (A) propositions because, again, we cannot retain the quantity of the predicate term.
Example 2:
Some animals are not mammals. (O) Therefore, some mammals are not animals. (O)
As we can see, the subject term of the convertend is particular because it is signified by the particular signifier “some”, but it becomes universal in the converse because it becomes the predicate term of a negative proposition. As mentioned above, the predicate terms of all negative propositions are always universal.
Partial conversion, on the other hand, is a type of conversion where the quantity of the convertend is reduced from universal to particular. Of course, partial conversion can only be applied to universal affirmative (A) and universal negative (E) propositions, where a universal affirmative proposition (A) is changed to particular affirmative (I) and a universal negative (E) proposition is changed to particular negative (O).
Let us consider the examples below.
Example 1:
All computers are gadgets. (A) Therefore, some gadgets are computers. (I)
Example 2:
No computers are robots. (E) Therefore, some robots are not computers. (O)
Obversion
Obversion refers to the formulation of a new proposition by retaining the subject and the quantity of the original proposition; however, the quality of the original proposition is changed and the predicate term is replaced by its contradictory. The original proposition is called the “obvertend”, while the new proposition is called the “obverse”. Please note that obversion is applicable to all types of categorical propositions. Let us consider the examples below.
Examples 1:
All men are mortal. (A) Therefore, no men are immortal. (E)
Examples 2:
No giants are small creatures. (E) Therefore, all giants are big creatures. (A)
Example 3:
Some men are mortal. (I) Therefore, some men are not immortal. (O)
Example 4:
Some politicians are not corrupt individuals. (O) Therefore, some politicians are non-corrupt individuals. (I)
Contraposition
Contraposition is the result of the combination of the principles of conversion and obversion. There are two types of contraposition, namely, partial and complete contraposition.
In partial contraposition, 1) the subject of the contraposit (that is, the new proposition) is the contradictory of the contraponend (that is, the original proposition); 2) the quality of the contraponend is changed in the contraposit; and 3) the predicate term in the contraposit is the subject term in the contraponend. Let us consider the example below.
Example 1:
All whales are mammals. (A) Therefore, no non-mammals are whales. (E)
Example 2:
No police officers are drug addicts. (E) Therefore, some non-drug addicts are police officers. (I)
Example 3:
Some students are not studious individuals. (O) Therefore, some non-studious individuals are students. (I)
It must be noted that particular affirmative (I) propositions have no contraposits. Hence, we cannot apply contraposition to particular affirmative propositions. This is because contraposition involves to steps, namely: first, obversion, and then, second, conversion. Now, as we learned above, since the obverse of an “I” proposition is “O” proposition, then we cannot proceed because an “O” proposition does not have a converse.
In complete contraposition, on the other hand, 1) the subject term in the contraposit is the contradictory of the predicate term in the contraponend; 2) the quality of the contraponend is not changed in the contraposit; and 3) the predicate term in the contraposit is the contradictory of the subject term in the contraponend. Let us consider the examples below.
Example 1:
All whales are mammals. (A) Therefore, all non-mammals are non-whales. (A)
Example 2:
No criminals are good people. (E) Therefore, some evil people are not non-criminals. (O)
Example 3:
Some students are not studious. (O) Therefore, some non-studious are not non-students. (O)
Inversion
Finally, in inversion, the subject and predicate terms of the new proposition are contradictories of the subject and predicate terms of the original proposition. And it must be noted that when doing inversions, we change the quantity of the invertend (that is, the original proposition); hence, inversions involve the changing of universal affirmative (A) propositions to particular affirmative (I) propositions, and universal negative (E) propositions to particular negative (O) propositions. Please note that particular affirmative (I) and particular negative (O) propositions do not have inverses.
There are two types of inversion, namely, partial inversion and complete inversion.
In partial inversion, the subject of the inverse (that is, the new proposition) is the contradictory of the subject of the invertend (that is, the original proposition). Let us consider the example below.
Example 1:
All priests are trustworthy. (A) Therefore, some non-priests are not trustworthy. (O)
Example 2:
No dogs are feline. (E) Therefore, some non-dogs are cats. (I)
In complete inversion, the subject and predicate of the new proposition are the contradictories of the subject and predicate of the original proposition. Let us consider the examples below.
Example 1:
Anything material is destructible. (A) Therefore, some non-material things are indestructible. (I)
Example 2:
No wealthy person is financially insecure. (E) Therefore, some non-wealthy persons are not financially non-insecure. (O)
Other immediate inferences aside from the traditional square of opposition is the conversion of propositions, which involves the following:
1) conversion,
2) obversion, and
3) contraposition.
Conversion
This type of inference is done by simply interchanging the subject and predicate terms of the proposition with reference to the distribution of each term.
Conversion is very much valid on E and I propositions where the totality or partiality of exclusion and inclusion of both S class and P class are identical. Its application is limited or this type of inference is not applicable to all types of propositions. Thus, applying conversion to the four propositions yields the following result:
Take note that the qualities of the propositions above are the same. Also, it is invalid to apply conversion in particular negative (O) propositions because it is tantamount to inferring that something must be true to all members of a class because it is true to some members.
Obversion
Obversion is another immediate inference which can be correctly performed by following the two guidelines:
By replacing the predicate terms of the statement with its class complement. The complement of a class is the class of all things that are not members of that class, that is, the complement of P is non-P, and vice versa; and
By changing the quality of the statement, that is, if the statement is affirmative, then we make it negative, and if the statement is negative, we make it affirmative. Please note that only the quality of the statement is changed; the quantity should be left as is.
Thus, applying obversion to the four propositions yields the following result:
Contraposition
Other immediate inferences are done by combining conversion and obversion. And one of the combinations is called contraposition, which is done by obverting, converting, and then obverting again. So, to get the contrapositive of a universal affirmative (A) proposition “All S are P”, we can have “No S are non-P” by obversion and “No non-P are S” by applying conversion to the obvertend, and finally, “All non-P are non-S” by applying again obversion to the converse of the obverted proposition. Put simply, the process involves the following:
Replacing the subject term by the complement of the predicate term; and
Replacing the predicate terms by the complement of its subject term.
The table below will make this point clearer.
Exercises
Instruction: Give, where possible, the converse, obverse, and contrapositive of each of the propositions below.
Rizal’s mother is a feminist.
Some feminist arguments are not valid.
Some nonatheist people attend church.
All graduates of PMA are commissioned officers of the AFP.
No reptiles are warm-blooded animals.
Some robbers are honest persons who are forced to steal to feed their family.
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These notes discuss in detail the nature and dynamics of the traditional square of opposition in categorical logic.
Two categorical propositions stated in standard form having the same subject and predicate terms may differ in either quantity and quality or both. The term “opposition” is used by logicians to illustrate these differences. However, “opposition” should not be understood as “disagreement” in ordinary language. For example, given two propositions having the same quality but different in quantity, such as “All students are intelligent” and “Some students are intelligent”, the ideas they express do not disagree; they are only opposed.
The relationship of opposing propositions can be schematically presented by placing them on different angles of a square. Please see the illustration below.
Contradictories
Contradictories are the relationship between statements opposing both quality and quantity, that is,
Universal affirmative (A) and particular negative (O) propositions (A-O), and
Universal negative (E) and particular affirmative (I) propositions (E-I).
Their relationship indicates that one member of each pair is denying the counterpart member of the other, and vice versa. So, whatever, the truth value of one proposition, the truth value of the other is automatically its opposite. Hence, contradictories cannot be both true or false at the same time.
Contraries
The relationship between universal affirmative (A) and universal negative (E) propositions are called contraries. They cannot be both true though both can be false at the same time. Thus, to know that one is true, the truth value of its contrary must be false. But to know that either one is false, it does not always follow that its counterpart is also false though that is a possibility; but the other possibility is that it could be true. It means, therefore, that the truth value of its contrary is undetermined.
Subcontraries
Subcontrary is the relationship between two particular propositions opposing in quality. Thus, this is a relationship between particular affirmative (I) and particular negative (O) propositions. Subcontraries cannot be both false though both can be true at the same time. Know that either one is false allows us to infer its subcontrary as true. But knowing that either one as true does not automatically suggest the falsity of its subcontrary, though again that is a possibility. This means that it is undetermined.
Subalternation
The relationship between propositions sharing in quality but not in quantity is called subalternation. Thus, subalternation is a relation between
Universal affirmative (A) and particular affirmative (I) propositions, and
Universal negative (E) and particular negative (O) propositions.
Universal propositions are called superaltern, while particular propositions are called subaltern. If the superaltern is true, its subaltern is true; but if the superaltern is false, the subaltern is undetermined. On the other hand, if the subaltern is false, the superaltern is false; but if the subaltern is true, the superaltern is undetermined.
Exercises
Instruction: Using the truth value of the given referent, determine the truth value of the other statement.
If A is true, what are the truth values of E, I, and O?
If E is true, what are the truth values of A, I, and O?
If I is true, what are the truth values of A, E, and O?
If O is true, what are the truth values of A, E, and I?
If A is false, what are the truth values of E, I, and O?
If E is false, what are the truth values of A, I, and O?
If I is false, what are the truth values of A, E, and O?
If O is false, what are the truth values of A, E, and I?
Exercises
Instruction: Using the Square of Opposition, determine whether the arguments below are valid or invalid.
All successful executives are intelligent people. So, it is false that some successful executives are not intelligent.
Some Sillimanians are Cebuanos. So, it is true that some Sillimanians are not Cebuanos.
3) No metals are conductors. So, it is true that some metal are conductors.
4) It is not the case that some martyrs are not saints. So, it is false that all martyrs are saints.
5) It is false that no wrestlers are weaklings. So, all wrestlers are weaklings.
6) Some priests are not faithful to their vows. So, all priests are faithful to their vows.
7) No priests are faithful to their vows. So, it is false that some priests are not faithful to their vows.
8) Some soldiers are homosexuals. So, it is false that some soldiers are not homosexuals.
9) Some Sillimanians are not activists. So, it is false that no Sillimanians are activists.
10) It is false that all geniuses are abnormal. So, it is false that some geniuses are abnormal.
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In my other notes on terms and propositions used in categorical logic, we learned that there are four (4) types of categorical propositions, namely:
Universal affirmative (A),
Universal negative (E),
Particular affirmative (I), and
Particular negative (O).
Now, the relationship between and among these four types of categorical propositions is what logicians call the “square of opposition”.
There are four types of relations in the square of opposition, namely:
1) Contrary,
2) Subcontrary,
3) Subalternation, and
4) Contradiction.
Please see the two models of a square of opposition below.
Contrary
Contrary is the relationship between universal affirmative (A) and universal negative (E) propositions. Hence, there is only one pair in contrary (that is, A-E), and the pair differs only in quality. As we can see, both are universal propositions, but one is affirmative and the other negative.
Example 1:
All philosophers are deep thinkers. (A) No philosophers are deep thinkers. (E)
Example 2:
No pastors are corrupt. (E) All pastors are corrupt. (A)
Rules in Contrary: If one of the contraries is true, then the other is false. But if one is false, then the other is doubtful, that is, its truth-value cannot be determined; this is because contraries cannot be both true but can be both false. Let us consider the examples above and assign truth-value to them.
If we assume that the proposition “All philosophers are deep thinkers” is true, then obviously its contrary “No philosophers are deep thinkers” is absolutely false. Of course, if it is already assumed that all philosophers are indeed deep thinkers, then it is impossible for philosophers to be not deep thinkers.
However, if we assume that the proposition “No pastors are corrupt” is false, then we cannot absolutely say that its contrary “All pastors are corrupt” is true. For sure, it’s possible for the contrary to be either true or false. Again, since we cannot have an absolute truth-value to the contrary of the proposition “No pastors are corrupt”, then its truth-value is doubtful.
Subcontrary
Subcontrary is the relationship between particular affirmative (I) and particular negative (O) propositions. Hence, there is only one pair in subcontrary (that is, I-O). And as we can see in the image of a square of opposition above, subcontraries differ only in quality.
Example 1:
Some politicians are women. (I) Some politicians are not women. (O)
Example 2:
Some mangoes in the basket are not ripe. (O) Some mangoes in the basket are ripe. (I)
Rules in subcontrary: If one of the subcontraries is false, then the other is true; and if one is true, the other is doubtful. This is because subcontraries cannot be both false, but can be both true. Let us consider the examples above and assign truth-value to them.
If we assume that the proposition “Some politicians are women” is false, then its subcontrary “Some politicians are not women” is absolutely true. Of course, if it is not true (therefore false) that some of the politicians are women, then it is absolutely true that some of the politicians are not women.
However, if we assume that the proposition “Some mangoes in the basket are not ripe” is true, then its subcontrary “Some mangoes in the basket are ripe” is doubtful or cannot be absolutely determined; in other words, it can be true or it can be false. Consider this: Imagine we are facing a basket of mangoes. Now, suppose we see that all the mangoes in the side of the basket facing us are not ripe, then the proposition “Some mangoes in the basket are not ripe” is true. However, we cannot be certain about the truth-value (therefore doubtful) of the proposition “Some mangoes in the basket are ripe”. This is because we see only one side of the basket that is full of mangoes. For sure, it is possible that the rest of the mangoes in the basket are not ripe (therefore, “Some mangoes in the basket are ripe” is false) or there is at least one mango on the other side or in the middle of the basket (that we do not see because we are just facing one side of the basket that contains not ripe mangoes) that is ripe (hence, “Some mangoes in the basket are ripe” is true).
Subalternation
Subalternation is the relation between universal and particular propositions having the same quality. Hence, there are two pairs of subalternation, namely, universal affirmative (A) to particular affirmative (I) propositions, and universal negative (E) and particular negative (O) propositions.
Example 1:
All jasmine flowers are white. (A) Some jasmine flowers are white. (I)
Example 2:
Some students are brilliant. (I) All students are brilliant. (A)
Example 3:
No teachers are lazy. (E) Some teachers are not lazy. (O)
Example 4:
Some fruits are not delicious. (O) No fruits are delicious. (E)
Rules in subalternation: If the universal is true, then the particular is true. If the universal is false, then the particular is doubtful. If the particular is true, then the universal is doubtful. And if the particular is false, then the universal is false. Let us consider the examples above and assign truth-value to them.
If we assume that the proposition “All jasmine flowers are white” is true, then its subaltern “Some jasmine flowers are white” is absolutely true. As we can see, the truth of the universal affects the truth of the particular. Thus, if it is true that all jasmine flowers are white, then it is impossible for at least one of the jasmine flowers to be not white; hence, “Some jasmine flowers are white” is absolutely true.
If we assume that the proposition “No teachers are lazy” is false, then its subaltern “Some teachers are not lazy” is doubtful in the sense that it can either be true or false.
If we assume that the proposition “Some fruits are not delicious” is true, then its superaltern “No fruits are delicious” is doubtful because it can either be true or false, that is, it is possible that all fruits are not delicious is true and it is also possible that all fruits are not delicious is false.
If we assume that the proposition “Some fruits are not delicious” is false, then its superaltern “No fruits are delicious” must be false. As we can see, the falsity of the particular affects the falsity of the universal. Indeed, if the particular is false, then it is impossible for the universal to be true; it should be false.
Contradiction
A contradiction is a relation between universal and particular propositions having different quality. Hence, there are two pairs of contradiction, namely, universal affirmative (A) and particular negative (O) propositions, and universal negative (E) and particular affirmative (I) propositions.
Example 1:
All men are mortal. (A) Some men are not mortal. (O)
Example 2:
No men are mortal. (E) Some men are mortal. (I)
Example 3:
Some drivers are sweet lovers. (I) No drivers are sweet lovers. (E)
Example 4:
Some students are not intelligent. (O) All students are intelligent. (A)
Rule in contradiction: One member of each pair is a denial of the other. In other words, if the universal is true, then particular is false; and if the particular is false, then the universal is true. Let us consider some of the examples above and assign truth-value to them.
If we assume that the proposition “All men are mortal” is true, then its contradictory “Some men are not mortal” is absolutely false. Of course, obviously, if it is assumed that all men are mortal, then it is impossible for at least one man to be immortal.
If we assume that the proposition “Some students are not intelligent” is false, then its contradictory “All students are intelligent” is absolutely true. Of course, the first proposition says that there is not at least one student that is not intelligent; hence, we can logically conclude that all students are intelligent.
Another method of symbolizing categorical propositions is the use of the Venn diagram.
John Venn, who introduced the method (thus the name Venn Diagram) used two overlapping circles to represent the relationship between two classes. Consider the diagram below.
The shaded portion represents a class that has no members.
The area with an “X” signifies that the class has at least one member.
is read as “S but not P” and this represents the class of things that are part of S but are not part of P.
is read as “not S but P” and this represents the class of things that are part of P but are not part of S.
is read as “S but P” and this represents the class of things that are both parts of S and P.
The diagrams below are used to represent the four standard types of categorical propositions.
The shaded area of the Venn diagram above represents a class that has no members. In the Venn diagram for a universal affirmative (A) proposition, the area “S but not P” is shaded to indicate that all members of S are members of P. Thus, we say, “All S are P”.
The shaded area of the Venn diagram above represents a class that has no members. In the Venn diagram for a universal negative proposition (E), the area SP is shaded to indicate that this class has no members. Thus, we say, “No S are P”.
A particular affirmative (I) proposition asserts that there is at least one member of S that is a member of P. This is diagrammed by placing an “X” in the area SP, which is common to the two classes. Thus, in the diagram above, we place an “X” in the area SP.
A particular negative (O) proposition asserts that there is at least one member of S that is not a member of P. Thus, in the diagram above, we place the “X” on the area “S but not P” to indicate that indeed there is at least one member of S that is not a member of P.
Venn Diagram and Existential Import
In traditional or Aristotelian logic, one assumes that universal affirmative (A) and universal negative (E) propositions have existential import. Thus, in the example “All angels are holy”, one assumes that there are angels and that all of them are holy. However, in applying the Venn diagram, one does not make this assumption. In the Venn diagram, all angels are said to be holy only if there are indeed angels. But the Venn diagram for a universal affirmative (A) proposition does not contain an area in which there is a symbol to show that there is an angel. Hence, the propositions “All angels are holy” or “No angels are holy” are non-existential propositions. In the first place, there are no angels in reality. The Venn diagram below will demonstrate this point.
As we can see, both application of traditional rules and use of the Venn diagram presuppose that only particular affirmative (I) and particular negative (O) propositions have existential import. Thus, a Venn diagram for the particular affirmative (I) proposition “Some angels are holy” contains an “X” to show that there is at least one angel that is holy. Please see the diagram below.
Venn Diagram and Test of Validity
A Venn diagram can be used to show the validity of categorical syllogisms. Three intersecting circles are needed to diagram a categorical syllogism, one circle for each class. The following rules will be observed in testing the validity of syllogism using a Venn diagram:
The universal premise should be diagrammed first if the argument also contains a particular premise;
The letter “X” should be placed on the line dissecting an area if the whole area is so designated in the premise;
Only the premises should be diagrammed; and
If the conclusion is self-evident in the diagram, then the argument or syllogism is valid.
Let us consider the example below, which is already in its standard form.
Example 1:
How do we determine the validity of the syllogism above using a Venn diagram?
First, we need to draw three intersecting circles (that is, circles for S, P, and M) and then number the areas by starting at the center, and then clockwise. Please see the Venn diagram of this syllogism below.
Now that we have drawn three intersecting circles, each for S, P, and M, our next task is to diagram the syllogism above.
Let us start with the first premise, that is, “All M are P”. It must be noted that since the premise talks about the circles for M and P only, so we will imagine that the circle for S does not exist; hence, we will diagram M and P only. Now, since the premise says “All M are P”, that is, all members of M are members of P, then we will shade areas 5 and 6 to show that all members of M, which are areas 1 and 4, are part of P. The Venn diagram of the syllogism above now looks like this:
After we diagrammed the first premise, let us proceed to diagram the second premise, which reads “All S are M”. This time, the premise talks about S and M only, so we will imagine that the circle for P does not exist. Now, since the premise says “All S are M”, that is, all members of S are members of M, then we will shade areas 7 and 2 to show that indeed all members of S, which is area 1, are part of M. The diagram now looks like this:
Since the Venn diagram of the above syllogism is now complete, let us proceed to analyze the diagram to determine whether the syllogism is valid or invalid. As rule #3 says, we diagram only the premises; hence, we do not diagram the conclusion. And as rule #4 says, the argument or syllogism is valid if the conclusion is self-evident in the Venn diagram.
Now, the conclusion says “All S are P”. As we can see in the Venn diagram of the syllogism above, the conclusion “All S are P” is perfectly diagrammed, that is, it is self-evident. In fact, since areas 2, 6, and 7 are shaded, then they do not exist anymore. What is left now of the class S is area 1, which all belongs to P. Thus, the above syllogism is valid.
Let us consider another example.
Example 2:
Let us draw three intersecting circles for this syllogism, each for S, M, and P, and then number the areas by starting at the center, and then clockwise. As rule #1 says, we will diagram first the universal premise if the syllogism also contains a particular premise. Since the first premise in the syllogism above is particular, while the second premise is universal, then we will diagram first the second premise, that is, “All M are S”.
The second premise says “All M are S”, so we will shade areas 4 and 5 to show that all members of M, which are areas 1 and 6, are part of S. The Venn diagram of the syllogism above will now look like this:
Let us proceed to diagram the second premise, which says “Some M are P”. Since this is a particular proposition, then we will not use the shading method; instead, we will place an “X” on the designated area. Since the premise says “Some M are P”, and since area 4 is already shaded, then it does not exist anymore. Thus, we will place the “X” on area 1 to show that indeed there is at least one member of M that is a member of P. The Venn diagram of the syllogism above now looks like this:
The Venn diagram of the syllogism above is now complete. Let us proceed to determine the validity of this syllogism.
Again, rule #3 says, we diagram only the premises; hence, we do not diagram the conclusion. And as rule #4 says, the argument or syllogism is valid if the conclusion is self-evident in the Venn diagram. Now, the conclusion of the syllogism above says “Some S are P”, and if we look at the Venn diagram above, there is an “X” on area 1, which indicates that there is at least one member of S that is a member of P. Hence, the conclusion which reads “Some S are P” is perfectly diagrammed in the Venn diagram above; indeed, the conclusion is self-evident. Therefore, the above syllogism is valid.
Let us consider another example.
Example 3:
Let us draw three intersecting circles for this syllogism, each for S, M, and P, and then number the areas by starting at the center, and then clockwise.
As we already know, we will diagram the universal premise first, which is “All S are M”. Thus, the Venn diagram of the syllogism above now looks like this:
Then let us diagram the second premise, which says “Some M are P”. Since areas 1 and 4 are so designated in the premise, then we will place the “X” on the line that dissects areas 1 and 4 to show that the whole area is so designated. Hence, the Venn diagram of the syllogism above now looks like this:
Since the “X” is on the line that dissects areas 1 and 4, this gives us an inconclusive reading of the conclusion. For this reason, the above syllogism is invalid. Indeed, the conclusion is not self-evident; it is not perfectly diagrammed.
Let us analyze one more example:
Example 4:
Let us draw three intersecting circles for this syllogism, each for S, M, and P, and then number the areas by starting at the center, and then clockwise.
Let us first diagram the first premise, which reads “All M are P. The Venn diagram of the above syllogism now looks like this:
Then let us diagram the second premise, which reads “No S are M”. The Venn diagram of the above syllogism now looks like this:
The conclusion of the above syllogism, which reads “No S are P”, asserts that no members of S should be a member of P. But area 2 of the Venn diagram above, which is also an area of SP, is not shaded. Thus, the Venn diagram above does not perfectly diagram the conclusion; indeed, the conclusion is not self-evident. Therefore, the above syllogism is invalid.
Antilogism is another method to test the validity of categorical syllogisms. This test of validity is a type of indirect proof in which the conclusion of the syllogism to be tested is replaced by its contradictory. The antilogism of a valid syllogism must meet the three requirements, namely:
There must be two universal propositions and one particular proposition, or two equations and one inequation.
The two universal propositions (two equations) must have a common term between them which is once negative and once affirmative.
The other two terms must appear unchanged in the particular proposition (inequation).
Let us consider the example below.
Antilogism and the Validity of Categorical Syllogisms
Antilogism is another method to test the validity of categorical syllogisms. This test of validity is a type of indirect proof in which the conclusion of the syllogism to be tested is replaced by its contradictory. The antilogism of a valid syllogism must meet the three requirements, namely:
There must be two universal propositions and one particular proposition, or two equations and one inequation.
The two universal propositions (two equations) must have a common term between them which is once negative and once affirmative.
The other two terms must appear unchanged in the particular proposition (inequation).
Let us consider the example below.
Example 1:
All men are mortal. All Filipinos are men. So, all Filipinos are mortal.
How do we determine the validity of the syllogism above using the antilogism method?
First, let us symbolize the syllogism in the algebraic notation. Let M stand for men and F for Filipinos. The algebraic notation of the above syllogism is as follows:
Next, let us construct its antilogism by replacing the conclusion with its contradictory. The contradictory of a proposition in algebraic form is easily formulated by changing an inequality (particular) to an equality (universal), or an equality (universal) to an inequality (particular). Thus, the antilogism of the example above is:
Now, let us check to see if the antilogism meets the three requirements mentioned above. As we can see:
There are three equations, namely: propositions (that is, premises) 1 and 2, and 1 inequation (that is, the conclusion).
There is a common term between the equations (universal propositions), which is once negative and once affirmative, namely:
The other two terms are unchanged in the inequation (conclusion), namely:
Hence, the above syllogism is valid because it meets the three requirements for antilogism of valid syllogisms.
Let us consider another example.
Example 2:
All professionals are former amateurs. But some former amateurs are wealthy persons. Therefore, some wealthy persons are professionals.
Let us check whether the syllogism is valid or invalid.
The first requirement is met: the first premise and the conclusion are equalities, that is, universal propositions.
The second requirement is also met: there is a common term between the equations (universal propositions) which is one negative and the other affirmative, namely:
3. But the third requirement is not met: the other two terms in the equations (that is, universal propositions), namely
are changed in the inequation (that is, particular proposition), namely: F and W.
Hence, the above syllogism is invalid because it does not meet the three requirements for the antilogism of a valid syllogism.
Practice Test
Determine the validity of the arguments or syllogisms below using the antilogism method.
Example 1:
All criminals are guilty of a felony. But some politicians are guilty of a felony. Therefore, some politicians are criminals.
Example 2:
Some drivers are traffic law violators. Some government employees are drivers. Therefore, some government employees are traffic law violators.
Example 3:
Nurses are sweet lovers. But Kit is a nurse. Therefore, Kit is a sweet lover.
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An argument consists of two or more propositions offered as evidence for another proposition. In logic and critical thinking, the propositions that are offered as evidence in the argument are called the premises, while the proposition for which the evidence is offered is called the conclusion. Thus, when one gives an argument, one is providing a set of premises as reasons for accepting his or her conclusion. It is important to note that when one gives an argument, one does not necessarily attack or criticize the other. In this way, an argument can also be viewed as a support of someone’s viewpoint.
Types of Arguments
Arguments can either be inductive or deductive. On the one hand, an inductive argument is one in which it is claimed that if the premises are true, then it is probable that the conclusion is true. Hence, even if all of the premises are true, inductive argument or reasoning allows the conclusion to be false. It is also important to note that inductive arguments go from the specific (or particular) to the general. In other words, inductive arguments make broad generalizations from specific observations. Consider the example below.
Example 1:
Ninety percent of the mongo seeds germinate in day 1. And in day 2, ninety percent of the mongo seeds germinate. Therefore, ninety percent of the mongo seeds germinate.
Based on the example above, we can also say that inductive arguments are based on observations or experiments.
Deductive arguments, on the other hand, is one in which it is claimed that if the premises are true, then the conclusion is necessarily true. And unlike inductive arguments, deductive arguments proceed from the general to the particular. Thus, a deductive argument or reasoning begins with a general statement or hypothesis and then “examines the possibilities to reach a specific, logical conclusion”.
Let us consider the example below.
Example 2:
Anybody who kills a person is guilty of a felony. Jim kills Jack. Therefore, Jim is guilty of a felony.
Syllogisms
Syllogisms are arguments which consist of three propositions which are so related so that when the first two propositions (that is, premises) are posited as true the third proposition (that is, the conclusion) must also be true. In other words, a syllogism is an argument arranged in a specific manner in such a way that it contains a major premise, minor premise, and a conclusion. Consider the classic example of a categorical syllogism below.
Example 1:
All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
How do we determine the major premise, minor premise, and the conclusion?
The major premise is the premise that contains the major term, while the minor premise is the premise that contains the minor term. The conclusion is the third proposition whose meaning and truth are implied in the premises.
How do we determine the major term, minor term, and the middle term?
The major term is the predicate of the conclusion, while the minor term is the subject of the conclusion. The middle term is the remaining term which does not (and cannot) appear in the conclusion.
If we look at the example above, then we know that the major term is “mortal” because it is the predicate of the conclusion and the minor term is “Socrates” because it is the subject of the conclusion. The middle term is “man” or “men” because it is the remaining term and which does not appear in the conclusion. As we can see in the example below, the major term is in red color, the minor term in blue, and the middle term in purple.
Rules of Syllogism
Now that we have presented the key concepts in arguments or syllogisms, let us proceed to the determination of their validity. Logicians have formulated eight (8) rules of syllogism, but of course they can be expanded to 10 or reduced to 6. But let us follow what logicians commonly used, that is, the 8 rules of syllogism. It must be noted that all of the 8 rules of syllogism must be met or satisfied for the argument or syllogism to be valid. If at least one of the 8 rules of syllogism is violated, then the argument or syllogism is invalid.
The 8 rules of syllogism are as follow:
There should only be three terms in the syllogism, namely: the major term, the minor term, and the middle term. And the meaning of the middle term in the firs premise should not be changed in the second premise; otherwise, the syllogism will have 4 terms.
The major and the minor terms should only be universal in the conclusion if they are universal in the premises. In other words, if the major and the minor terms are universal in the conclusion, then they must also be universal in the premises for the argument to be valid. Hence, if the major and minor terms are particular in the conclusion, then rule #2 is not applicable.
The middle term must be universal at least once. Or, at least one of the middle terms must be universal.
If the premises are affirmative, then the conclusion must be affirmative.
If one premise is affirmative and the other negative, then the conclusion must be negative.
The argument is invalid whenever the premises are both negative. This is because we cannot draw a valid conclusion from two negative premises.
One premise at least must be universal.
If one premise is particular, then the conclusion must be particular.
Now, let us apply these 8 rules of syllogism to the arguments below. Let us color the terms to avoid confusion. So, let us assign the color red for the major term, blue for the minor term, and purple for the middle term.
Rule #1 of the 8 rules of syllogism: There should only be three terms in the syllogism, namely: the major term, the minor term, and the middle term.
If we analyze the syllogism above, it would appear that the argument is invalid because it violates rule #1. As we can see, the syllogism above contains 4 terms because the meaning of the middle term “stars” in the first premise is changed in the second premise. The term “stars” in the first premise refers to astronomical bodies or objects, while the term “star” in the second premise refers to celebrities.
Let us consider another example.
As we can see, the syllogism above contains only three terms. Hence, this syllogism is valid in the context of rule #1.
Rule #2 of the 8 rules of syllogism: The major and the minor terms should only be universal in the conclusion if they are universal in the premises.
As we can see, the minor term “terrorist” in the conclusion is universal because of the universal signifier “no”. And since the minor term “terrorist” in the second premise is universal because of the universal signifier “no”, then the syllogism above does not violate rule #2 in the context of the minor term. However, the major term “brilliant” in the conclusion is universal because the proposition is negative; as we already know, the predicate terms of all negative propositions are universal. But if we look at the major term in the first premise, it is particular because, as we already know, the predicate terms of all affirmative propositions are particular. In the end, thesyllogism above is invalid because it violates rule #2. This is what logicians call the “fallacy of illicit major”.
Let us consider another example.
Because the major term “creative” in the conclusion is particular, as it is a predicate term of an affirmative proposition, then it does not violate rule #2 because the rule is not applicable here. As we can see, rule #2 is applicable only to universal minor and major terms. But if we check the minor term “weird people” in the conclusion, we learned that it is universal because of the universal signifier all. Since the minor term “weird people” is universal in the conclusion, then it must also be universal in the second premise for this syllogism to be valid. If we look at the minor term in the second premise, it is particular because it is a predicate term of an affirmative proposition. Therefore, in the end, the syllogism above is invalid because it violates rule #2. This is what logicians call the “fallacy of illicit minor”.
Let us consider a valid argument below in the context of rule #2 of the 8 rules of syllogism.
The syllogism above is valid in the context of rule #2 of the 8 rules of syllogism because rule #2 is not violated. As we can see, the minor term “Greg” in the conclusion is particular; hence, rule #2 is not applicable. Of course, if a rule is not applicable, then it cannot be violated; and if no rule or law is violated, then the argument is automatically valid. Now, if we look at the major term “liar” in the conclusion, it is universal because it is a predicate term of a negative proposition. But because the minor term “liar” is also universal in the first premise because, again, it is a predicate term of a negative proposition, then this argument satisfies rule #2.
Let us consider another valid argument in the context of rule #2 of the 8 rules of syllogism.
Both the minor and major terms in the conclusion of the syllogism above are particular. For this reason, rule #2 of the 8 rules of syllogism is not applicable. Hence, the syllogism is automatically valid in the context of rule #2 of the 8 rules of syllogism.
Rule #3 of the 8 rules of syllogism: The middle term must be universal at least once.
The syllogism above is valid in the context of rule #3 of the 8 rules of syllogism because the middle term “beans” in the first premise is universal. In fact, rule #3 of the 8 rules of syllogism asks that at least one of the middle terms must be universal.
Let us consider another example.
As we can see, both middle terms in the first and second premise are particular. But because rule #3 of the 8 rules of syllogism asks that at least one of the middle terms must be universal, then the syllogism above is invalid.
Rule #4 of the 8 rules of syllogism: If the premises are affirmative, then the conclusion must be affirmative.
The syllogism above is valid because it satisfies rule #4 of the 8 rules of syllogism. As we can see, both premises are affirmative and the conclusion is affirmative.
Let us consider another example.
The syllogism above is invalid because it does not satisfy rule #4 of the 8 rules of syllogism. As we can see, both premises are affirmative, but the conclusion is negative.
Rule #5 of the 8 rules of syllogism: If one premise is affirmative and the other negative, then the conclusion must be negative.
The syllogism above is valid in the context of rule #5 of the 8 rules of syllogism. As we can see, the first premise is affirmative, the second premise is negative, and the conclusion is negative.
The syllogism above is invalid in the context of rule #5 of the 8 rules of syllogism. As we can see, the first premise is affirmative, the second premise is negative, but the conclusion is affirmative. Hence, it violates rule #5 of the 8 rules of syllogism.
Rule #6 of the 8 rules of syllogism: The argument is invalid whenever the premises are both negative.
Obviously, the above syllogism is invalid because both premises are negative.
Rule #7 of the 8 rules of syllogism: One premise at least must be universal.
The above syllogism is valid in the context of rule #7 of the 8 rules of syllogism because it qualifies the rule. As we can see, the first premise is universal.
Rule #8 of the 8 rules of syllogism: If one premise is particular, then the conclusion must be particular.
The first premise of the above syllogism is particular, and the conclusion is particular too. Therefore, this syllogism is valid in the context of rule #8 of the 8 rules of syllogism.
From the list of possible conclusions provided, pick the one the makes the syllogism valid. Write only the letter on the space provided before each number.
_____ 1. All public properties are for common use. Some roads are public properties.
a. Ergo, all roads are for common use. b. Ergo, some roads are for common use. c. Ergo, some roads are not for common use. d. Ergo, no roads are for common use.
_____2. No plunderers are dignified persons. Some politicians are plunderers.
a. Ergo, some dignified persons are not politicians. b. Ergo, some politicians are dignified persons. c. Ergo, some politicians are not dignified persons. d. Ergo, some dignified persons are politicians.
_____3. No bird is a fish. Some fish are sharks.
a. Ergo, some sharks are not birds b. Ergo, some birds are not sharks. c. Ergo, some sharks are birds. d. Ergo, some birds are sharks.
_____4. All government officials are worthy of respect. However, some policemen are government officials.
a. Ergo, some policemen are not worthy of respect. b. Ergo, every policeman is worthy of respect. c. Ergo, some policemen are worthy of respect. d. Ergo, any policeman is not worthy of respect.
_____5. Every Russian born during the Cold War is a communist. Maria Sharapova is a Russian born during the Cold War.
a. Ergo, Maria Sharapova is a devout communist. b. Ergo, Maria Sharapova is non-communist. c. Ergo, Maria Sharapova is not a communist. d. Ergo, Maria Sharapova is a communist.
Practice Test II
Items 6-10 are all invalid syllogisms. Determine which syllogistic rule or rules are violated. Write only the letter on the space provided before each number.
_____6. Some judges are biased. But no Comelec officials are biased. Therefore, no Comelec officials are judges.
a) #2, #7, and #8 b) #3 c) #2 and #8 d) #3 and #8
_____7. All lifeguards are life-savers. But all lifeguards are good swimmers. Therefore, all good swimmers are life-savers.
a) #2 and #3 b) #2 c) #3 d) #2 and #8
_____8. Some war veterans are heroes. But some heroes are traitors. Therefore, no traitors are war veterans.
a) #2, #3, #4, #7, and #8 b) #2, #3, #4, and #8 c) #2, #3, #4, and #7 d) #2, #3, #5, #7, and #8
_____9. Some comedians are amusing. But no serious persons are comedians. Therefore, no serious persons are amusing.
a) #2 and #3 b) #3 and #8 c) #2 and #5 d) #2 and #8
_____10. All sacrifices are rewarding. But some acts of cheating are rewarding. Therefore, all acts of cheating are sacrifices.
a) #2, #3, #4, and #8 b) #2, #3, and #8 c) #2, #3, #4, #7, and #8 d) #2, #3, #4 and #8
