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May 19, 2022May 19, 2022

Conjunctive Statements in Propositional Logic

There are four types of compound statements used in symbolic logic, namely:

1) conjunctive,

2) disjunctive,

3) conditional, and

4) biconditional

In these notes, I will focus only on conjunctive statements.

A conjunctive statement or conjunction is a compound statement connected by the word “and.” The component statements in a conjunction are called conjuncts. Let us consider this example:

Roses are red and jasmines are white.

Obviously, the above statement is a conjunction because it is connected by the word “and.” The first statement “Roses are red” is the first conjunct and the statement “Jasmines are white” is the second conjunct.

In my notes titled “Propositions and Symbols Used in Symbolic Logic” (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), the symbol for “and” is • (dot). Now, if we let p stand for “Roses are red” and q for “Jasmines are white,” then the statement “Roses are red and jasmines are white” is symbolized as follows:

p • q

In some cases, a conjunctive statement does not use the word “and” as connective. Sometimes, the following words are used as connectives of a conjunctive statement:

But

However

Nevertheless

Even though

Whereas

Although

While

Still

Yet

Consider the following examples:

Chocolate is delicious, but it is not a good food for people with diabetes.
Lucas is playing, while Rob is studying.
The teacher was already shouting, yet the students remain very noisy.

In cases where there are no words that signify a conjunction, a comma (,) or a semi-colon (;) may indicate that the statement is a conjunction. Consider the example below:

Although the human person is mortal, she can live long.

Symbolizing Conjunctive Statements

I have been symbolizing statements above and in my previous posts, but it is not until now that I will specifically talk about symbolizing statements.

Firstly, logicians usually put the variables or constants that will represent the statement right after the statement per se. Consider the examples below:

Chocolate is delicious, but it is not a good food for people with diabetes. (p, q)

Please note that the variables provided after the statement represent the component statements respectively. Thus, in the example above, the variable p represents the first component statement “Chocolate is delicious,” while q represents the second component statement “It is not a good food for people with diabetes.”

Secondly, when symbolizing statements, we need to put proper punctuations and negation if necessary. Thus, in the example above, the statement “Chocolate is delicious” is represented by p, while the statement “It is not a good food for people with diabetes” is represented by q. If we are not careful, we may symbolize the statement as follows: p • q. However, if we analyze the statement, we notice that the second component contains a negation sign “It is not the case.” Hence, the statement “Chocolate is delicious, but it is not a good food for people with diabetes” is symbolized as follows:

p • ~q

It is important to note that sometimes the word “and” is not truth-functional, that is, it does not connect two independent propositions. Thus, if this occurs, we should symbolize the proposition simply as a simple proposition. Consider the following example:

Bread and butter is a perfect combination.

Obviously, the “and” in the example above is not truth-functional because it does not connect two truth-functional propositions or sentences. This is because we cannot say that “Bread is a perfect combination” and “Butter is a perfect combination.” Hence, the proposition “Bread and butter is a perfect combination” is symbolized simply as:

However, if we have the example

“John and Mary are watching TV”

then we have to symbolize this as:

p • q

This is because the “and” here is truth-functional, that is, it connects two independent propositions or sentences. For sure, it is possible for us to say “John is watching TV” and “Mary is watching TV.” In other words, both John and Mary are watching TV.

Rules in Conjunction

A conjunction is true if and only if both conjuncts are true.
If at least one of the conjuncts is false, then the conjunction is false.

The truth table below illustrates this point.

The truth table above says:

1) If p is true and q is true, then p • q is true.

2) If p is true and q is false, then p • q is false.

3) If p is false and q is true, then p • q is false.

4) If p is false and q is false, then p • q is false.

Now, given the rule in conjunction, how do we determine the truth-value of the conjunctive statement p • ~q?

Let us suppose that the truth-value of p is true and q is false. So, if p is true and q false, then the statement p • ~q is true. To illustrate:

The illustration above says that p is true and q is false. Now, before we apply the rule in conjunction in the statement p • ~q, we need to simplify ~q first because the truth-value “false” is assigned to q and not to ~q. If we recall our discussion on the rule in negation, we learned that the negation of false is true. So, if q is false, then ~q is true. Thus, at the end of it all, p • ~q is true if p is true and q is false.

May 18, 2022December 9, 2024

Propositions and Symbols Used in Propositional Logic

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Just as in traditional or Aristotelian logic, our main goal in propositional logic (or symbolic logic) is to determine the validity of arguments. But because arguments are composed of propositions, and because we need to symbolize the argument first before we can determine its validity using a specific rule, we need therefore to discuss the types of proposition and symbols used in symbolic logic.

Please note that symbolic logic uses only declarative statements or propositions because any other types of proposition are not truth-functional, that is, they cannot be either true or false. For example, the interrogative proposition “What is your name?” is not truth-functional because we cannot assign any truth-value to it, that is, it cannot be either true or false.

In similar manner, the exclamatory proposition “What an exciting journey!” cannot be used in symbolic logic because, again, we cannot assign a truth-value to it. Hence, again, we can only employ declarative propositions in symbolic logic because they are the only types of proposition that can either be true or false. Think, for example, of the proposition “Donald Trump is a racist president.” Depending on the context, we may say “Yes, it is true that Donald Trump is a racist president,” or we may say “It is false that Donald Trump is a racist president.”

There are two types of declarative proposition used in symbolic logic, namely, simple and compound proposition.

On the one hand, a simple proposition is one that is composed of only one proposition. For example, “Donald Trump is the president of the United States.” As we can see, this proposition has only one component.

On the other hand, a compound proposition is composed of two or more propositions, such as:

1) Jack is singing, while Jill is dancing.

2) If the road is wet, then either it rains today or the fire truck spills water on the road.

As you notice, the first example is made up of two propositions, namely:

Jack is singing.
Jill is dancing.

The second example, on the other hand, is composed of three propositions, namely:

The road is
It rains today.
The fire truck spills water on the road.

Now, logicians usually use the lower case of the English alphabet p through z to symbolize propositions. They are called variables. The upper-case A through Z are called constants. For example, if we let p stand for the proposition “Jack is singing,” then it is symbolized as p. Thus, instead of saying “Jack is singing,” we just say p.

The symbol •(dot), which is read as “and,” is used to symbolize the connective of a conjunctive proposition. As I will discuss in the succeeding posts, a conjunctive proposition is connected by the word “and.” Let’s take, for example, the proposition “Jack is singing and Jill is dancing.” If we let p stand for “Jack is singing,” and q for “Jill is dancing,” then the proposition “Jack is singing and Jill is dancing” is symbolized as follows:

p • q

The symbol v (wedge), which is read as “Either…or” or just “or” is used to symbolize the connective of a disjunctive proposition. As I will discuss in the succeeding posts, disjunctive propositions are connected by the words “Either…or” or simply “or.” If we let p stand for “Jack is singing” and q for “Jill is dancing,” then the proposition “Either Jack is singing or Jill is dancing” is symbolized as follows:

p v q

Please note that the proposition above is an inclusive disjunction. There is another way to symbolize an exclusive disjunction. But I will discuss this other type of disjunctive proposition when I go to the four types of compound propositions.

The symbol ⊃ (horse shoe), which is read as “If…then” or just “then” is used to symbolize the connective of a conditional proposition. As I will discuss in the succeeding posts, conditional propositions are connected by the words “If…then” or just “then.”

Now, if we let p stand for “Jack is singing” and q for “Jill is dancing,” then the proposition “If Jack is singing, then Jill is dancing” is symbolized as follows:

p ⊃ q

The symbol ≡ (triple bar), which is read as “If and only if,” is used to symbolize the connective of a biconditional proposition. As I will discuss in the succeeding posts, biconditional propositions are connected by the words “If and only if.” If we let p stand for “Jack is singing” and q for “Jill is dancing,” then the proposition “Jack is singing if and only if Jill is dancing” is symbolized as follows:

p ≡ q

The symbol /∴ (forward slash and triple dots) is read as “therefore.” This is symbol is used to separate the premises and the conclusion in an argument. For example, if the premises in the argument are 1) p ⊃ q, 2) p and the conclusion is q, then the argument is symbolized as follows:

p ⊃ q
p /∴ q

Lastly, the symbol ~ (tilde), which is read as “not,” is used to negate a proposition. As I will show later, any proposition can be negated. Thus, the proposition “Jack is not singing” is symbolized as follows:

~ p

Below is the summary of some of the basic symbols used in symbolic logic.

May 18, 2022May 18, 2022

Notes in Informal Fallacies

Ignoratio Elenchi: ignorance of what is required to refute or establish a conclusion.

Kinds of Fallacies (Ignoratio Elenchi):

1) Argumentum Ad Hominem (Argument against the man): one which ignores the real claims or issues in the argument so that what is emphasized is the character, personality, or belief of the opponent.

Example: Your honor, it is impossible for us not to believe that the accused of this murder case in not guilty, because the father and grandfather of the accused had been convicted of murder several years ago. And besides, the accused is of bad moral reputation.

2) Argumentum Ad Ignorantiam (Appeal to Ignorance): arises when an argument is taken as true wherein it has not yet proven to be false, or an argument is false because this has not been proven as true.

Example: The existence of heaven must be true since nobody has ever successfully defended that it is just a product of imagination.

3) Argumentum Ad Verecundiam (Appeal to Authority): arise 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.

Example: Anybody who does not go to church will not be saved according to St. Augustine.

4) Argumentum Ad Populum (Appeal to People): arises when one who, instead of concentrating on the relevant facts of the argument, gives more emphasis to the emotions and opinions of the people as basis of his conclusion.

Example: Budweisser is better than any other beer in the world because 90% of the Americans drink it.

5) Argumentum Ad Misericordiam (Appeal to Pity): arises when an appeal to evidence is replaced by an appeal to pity, mercy, or sympathy.

Example: Marco should not be given a failing mark in symbolic logic since he has taken the subject 3 times.

6) Argumentum Ad Baculum (Appeal to Force): arises when one appeals to intimidation, or use force in order to gain acceptance of his propositions or arguments.

Example: Parent to his son/daughter: You should study nursing; otherwise, I will not send you to College.

7) Accident : in this fallacy, general rules are applied to particular cases wherein such rule are applicable.

Example: The mark one gets in symbolic logic measures the kind of intelligence one has. Chiara failed in symbolic logic. Ergo, Chiara has low intelligence.

8) Converse Accident : just he opposite of the fallacy of accident

Example: Takyo is rich. But Takyo is from the America. So, all Americans are rich.

9) Tu Quoque : arises when one answers a charge of wrongdoing by a similar charge to his oppoent.

Example: If (the) father smokes, therefore, there is nothing wrong if (the) son smokes too.

10) False Cause: arises when one assigns as the cause those facts that merely preceded or accompanied the effect

Example: Satanas cuts the acacia tree near the Silliman AS building. The following day, he died. Therefore, the cutting of the acacia tree is the cause of Satanas’ death.

11) Non Sequitor: arises when an argument, the conclusion categorically lacks connection with the proposition.

Example: Jelo is a Sillimanian, therefore, he is a good debater.

May 18, 2022May 18, 2022

Logic: Informal Fallacies

In these notes, I will discuss some of the most common types of informal fallacies. These include:

1) Appeal to Authority (Argumentum ad Verecundiam)

2) Appeal to People (Argumentum ad Populum).

3) Appeal to Force (Argumentum ad Baculum)

4) Appeal to Pity (Argumentum ad Misericordiam)

5) Appeal to Ignorance (Argumentum ad Ignorantiam)

6) Argument against the Man (Argumentum ad Hominem)

7) False Cause

8) Slippery Slope

9) Either/Or Fallacy or False Dichotomy

10) Fallacy of Equivocation

11) Hasty Generalization

12) Fallacy of Composition

13) Fallacy of Division

14) We will update this page for more types of informal fallacies.

But before we discuss these common types of fallacious arguments, let us first briefly define the term fallacy.

A fallacy is generally understood as a kind of error in reasoning. Both deductive and inductive arguments can be fallacious. Some fallacious arguments are detectable by an examination of the form of the argument. Hence, they are called formal fallacies.

The techniques that logicians used in determining the validity of arguments in traditional and symbolic (or advanced) logic, such as the 8 rules of syllogism, and the truth table and partial truth table methods, enable them to recognize inconsistencies or errors in reasoning. These are called formal fallacies. All other types of fallacies are called informal fallacies, and they can be detected by an examination of the content of the argument itself.

May 18, 2022May 18, 2022

Appeal to Authority

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.

May 18, 2022December 9, 2024

Mood and Figure of a Syllogism: Categorical Logic

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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.

May 18, 2022December 9, 2024

Eduction (Conversion of Propositions): Categorical Logic

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)

May 18, 2022May 18, 2022

Conversion of Propositions: Categorical Logic

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.
Some clergymen are not abstainers.
All geniuses are weird.
Some soldiers are not patriotic.
Some non-Filipinos are communists.

May 18, 2022December 9, 2024

Traditional Square of Opposition: Categorical Logic

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.

Note: Please email learnphilosophy@philonotes.com for the answers.

May 18, 2022December 9, 2024

Square of Opposition: Categorical Logic

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.