Biconditional statements are compound propositions connected by the words “if and only if.”
The symbol for “if and only if” is a ≡ (triple bar). Let’s consider the example below.
I will take a leave of absence if and only the administration allows me to. (p, q)
If we let p stand for “I will take a leave of absence” and q for “The administration allows me to,” then the biconditional proposition “I will take a leave of absence if and only if the administration allows me to” is symbolized as follows:
p≡ q
Please note that the connective “if and only if” should not be confused with “only if.” The connective “only if” is a connective of a conditional proposition. Let’s take the example below:
I will take a leave of absence onlyif the administration allows me to. (p, q)
We have to take note that the proposition that comes after the connective “only if” is a consequent. Thus, if we let p stand for “I will take a leave of absence” and q for “The administration allows me to,” then the proposition is symbolized as follows: p⊃q.
Rules in Biconditional Propositions
A biconditional proposition is true if both components have the same truth value.
Thus, if one is true and the other is false, or if one is false and the other true, then the biconditional proposition is false.
As we can see, the rules in biconditional propositions say that the only instance wherein the biconditional proposition becomes true is when both component propositions have the same truth value. This is because, in biconditional propositions, both component propositions imply each other. Thus, the example above, that is, “I will take a leave of absence if and only if the administration allows me to” can be restated as follows:
If I will take a leave of absence, then the administration allows me to; and if the administration allows me to, then I will take a leave of absence.
Thus, the symbol p≡ q means p is equal to q, and q is equal to p.
The truth table below illustrates this point.
The truth table above says:
If p is true and q is true, then p ≡ q is true.
If p is true and q is false, then p ≡ q is false.
If p is false and q is true, then p ≡ q is false.
If p is false and q is false, then p ≡ q is true.
Now, suppose we have the example ~p ≡ q. How do we determine its truth value if p is true and q is false?
Let me illustrate.
The illustration says that p is true and q is false. Now, before we apply the rules in biconditional in the statement ~p ≡ q, we need to simplify ~p first because the truth value “true” is assigned to p and not to ~p. If we recall our discussion on the rule in negation, we learned that the negation of true is false. So, if p is true, then ~p is false. Thus, at the end of it all, ~p ≡ q is true.
An if-then statement or conditional statement is a type of compound statement that is connected by the words “if…then”. Logicians usually used horseshoe (⊃) as the symbol for “if…then”. In some cases, logicians used the mathematical symbol “greater-than” (>) instead of a horseshoe.
Let us consider the example below:
If the company closes down, then obviously many workers will suffer. (p, q)
If we let p stand for the statement “The company closes down” and q for the statement “Obviously many workers will suffer”, then the conditional statement is symbolized as follows:
p⊃q
If we use the greater-than symbol, then the statement above is symbolized as follows:
p>q
It is important to note that the statement that precedes the connective horseshoe (⊃) is called the “antecedent” and the proposition that comes after it is called “consequent.” Hence, in the example above, the antecedent is “The company closes down”, while the consequent is “Obviously many workers will suffer”.
It is also important to note that there are cases wherein the words “if…then” is not mentioned in the statement, yet it remains a conditional one. Let us consider the following example:
Provided that the catalyst is present, the reaction will occur. (p, q)
If we analyze the statement, it is very clear that it is conditional because it suggests a “cause and effect” relation. Thus, the statement can be stated as follows:
If the catalyst is present, then the reaction will occur. (p, q)
If we let p stand for the statement “Provided that the catalyst is present” and q for “The reaction will occur”, then the statement is symbolized as follows:
p⊃q
It is equally important to note that sometimes the antecedent is stated after the consequent. If this happens, then we have to symbolize the statement accordingly. Let us take the example below.
The painting must be very expensive if it was painted by Michelangelo. (p, q)
If we analyze the statement, it is clear that the antecedent is “It was painted by Michelangelo” and the consequent is “The painting must be very expensive”.
Now, if we let p stand for “The painting must be very expensive” and q for “It was painted by Michelangelo”, then statement “The painting must be very expensive if it was painted by Michelangelo” is symbolized as follows:
q ⊃p
Please note that we symbolized the statement “The painting must be very expensive if it was painted by Michelangelo” as q ⊃p because in symbolizing if-then or conditional statements, we always write the antecedent first and then the consequent. By the way, please note that the variables provided after the statement represent the statements in the entire statement respectively. Thus, in the statement
The painting must be very expensive if it was painted by Michelangelo. (p, q)
the variable p stands for the statement “The painting must be very expensive” and q stands for the statement “It was painted by Michelangelo”. Again, since q is our antecedent and p is our consequent, and since in symbolizing if-then statement we need to write the antecedent first and then the consequent, so the statement “The painting must be very expensive if it was painted by Michelangelo” is symbolized as follows:
q ⊃p
Rules in If-then Statements
An If-then statement is false if the antecedent is true and the consequent false.
Thus, other than this form, the If-then statement is true.
The truth table below illustrates this point.
The truth table above says:
If p is true and q is true, then p ⊃ q is true.
If p is true and q is false, then p ⊃ q is false.
If p is false and q is true, then p ⊃ q is true.
If p is false and q is false, then p ⊃ q is true.
As we can see, the rules in If-then statements or conditional statements say that the only instance wherein the conditional statement becomes false is when the antecedent is true and the consequent false. Let us consider the example below.
If it rains today, then the road is wet.
Now, the first row in the truth table above says that p is true and q is true. So, obviously, p ⊃ q is true. This is because, if it is true that “it rains today,” then it must also be true that “the road is wet.”
The second row says that p is true and q is false. So, p ⊃ q must be false. This is because if it is true that “it rains today” then it must necessarily follow that “the road is wet.” However, it is said that q is false, that is, the road is not wet; hence, the conditional statement is false. Again, it is impossible for the road not to get wet if it rains.
The third row says p is false and q is true. If this is the case, then p ⊃ q is true. This is because if it is false that it rains today (in other words, it does not rain today), it does not necessarily follow that the road is dry. Even if it does not rain, the road may still be wet because, for example, a fire truck passes by and spills water on the road.Lastly, the fourth row in the truth table above says p is false and q is false. If this is the case, then p ⊃ q is true. This is because, based on the example above, it says “it does not rain today” and the “road is not wet.” So, obviously, the conditional statement is true.
A conditional statement or conditional proposition (sometimes referred to as if-then statement) is a compound statement that is connected by the words “If…then” or just “then.” Most logicians used the sign horseshoe (⊃) to mean “if…then”. Let us consider the example below.
If the airship Albatros has a powerful weapon, then it could destroy objects on the ground. (S, T)
If we let S stand for “The airship Albatros has a powerful weapon” and T for “It could destroy objects on the ground,” then the statement above is symbolized as follows:
S ⊃ T
It must be noted that the statement that comes before connective horseshoe (⊃) is called the “antecedent” and the statement that comes after it is called “consequent.”
It must be noted as well that there are instances wherein the words “if…then” are not mentioned in the statement, yet the statement remains a conditional one. Let us analyze the statement below:
Passage of the law means morality is corrupted. (S, T)
If we analyze the statement above, it is obvious that it is a conditional statement because it implies a “cause and effect” relationship. Thus, the statement can be restated in the following manner:
If the law is passed, then morality will be corrupted.
If we let S stand for “The law is passed” and T for “Morality will be corrupted,” then the proposition is symbolized as follows:
S ⊃ T
It is also important to note that sometimes the antecedent is stated after the consequent. If this occurs, then we have to symbolize the statement accordingly. Let us consider the statement below.
The forest will be destroyed should the logging law is passed. (S, T)
If we analyze the statement, it is obvious that the antecedent is “The logging law is passed” and the consequent is “The forest will be destroyed.” Hence, the statement “The forest will be destroyed should the logging law is passed” is symbolized as follows:
T ⊃ S
As we can notice, the variables provided after the statement represent the component statements in the entire statement respectively. Thus, in the statement
The forest will be destroyed should the logging law is passed. (S, T)
The variable S stands for “The forest will be destroyed” and T stands for “The logging law is passed.” Again, since T is our antecedent and S is our consequent, and since in symbolizing a conditional statement we need to write the antecedent first and then the consequent, so the statement “The forest will be destroyed should the logging law is passed” is symbolized as follows:
T⊃ S
Rules in a Conditional Statement
A conditional statement is false if the antecedent is true and the consequent false.
Thus, other than this form, the conditional statement is true.
The truth table below illustrates this point.
The truth table above says:
If p is true and q is true, then p ⊃ q is true.
If p is true and q is false, then p ⊃ q is false.
If p is false and q is true, then p ⊃ q is true.
If p is false and q is false, then p ⊃ q is true.
As we can observe, the rules in a conditional statement say that the only instance wherein the conditional statement becomes false is when the antecedent is true and the consequent false. Let us take this statement:
If the airship Albatros has a powerful weapon, then it could destroy objects on the ground. (S, T)
Now, the first row in the truth table above states that p is true and q is true. So, obviously, p ⊃ q is true. This is because, if it is true that “The airship Albatros has a powerful weapon,” then it must also be true that “It could destroy objects on the ground.”
The second row states that p is true and q is false. So, p ⊃ q must be false. This is because if it is true that “The airship Albatros has a powerful weapon” then it should necessarily follow that “It could destroy objects on the ground.” However, it is stated that q is false, that is, the “It could not destroy objects on the ground”; therefore, the conditional statement is false. For sure, it is not sound to conclude that the airship Albatros does not have the capability to destroy objects on the ground given that it has a powerful weapon. Hence, again, the conditional statement is false.
The third row states that p is false and q is true. If this is the case, then p ⊃ q is true. This is because if it is not true that “The airship Albatros has a powerful weapon”, then it does not necessarily follow that it could not destroy objects on the ground. In fact, even if the airship Albatros does not have a powerful weapon, it is still possible for the airship Albatros to destroy objects on the ground.
Finally, the last row in the truth table above states that p is false and q is false. If this is the case, then p ⊃ q is true. This is because, based on the example above, it states that “The airship Albatros does not have a powerful weapon” and that “it could not destroy objects on the ground.” Hence, obviously, the conditional statement is true.
In my other notes titled “Inclusive Disjunction in Propositional Logic”, I discussed the nature and characteristics of an inclusive disjunction, including its rules and how to determine its truth-value. In these notes, I will focus on exclusive disjunction.
An exclusive disjunction is a type of disjunction that is connected by the words “Either…or, but not both.” As we already know, the symbol for the connective of a disjunctive statement is v (wedge). However, an exclusive disjunction is symbolized differently from an inclusive disjunction. Consider the following examples below:
Either John is singing or he is dancing, but not both.
Either John is sleeping or he is studying.
Example #1 is clearly an exclusive disjunction because of the words “but not both.” Please note that it is possible for John to be singing and dancing at the same time (hence, inclusive), but because of the qualifier “but not both,” which clearly emphasized the point that John is not singing and dancing at the same time, then the statement is clearly an exclusive one.
Now, if we let p stand for “John is singing” and q for “He is dancing,” then the statement “Either John is singing or he is dancing, but not both” maybe symbolized as pvq. However, this is faulty because it does not clearly specify what the statement “Either John is singing or he is dancing, but not both” states. So, how do we symbolize example #1 above?
As already mentioned, if we let p stand for “John is singing” and q for “He is dancing,” then we can come up with pvq. But it’s not yet complete. We need to take into consideration the phrase “but not both.” If we recall the discussion on conjunctive statements, we know that the symbol for “but” is • (dot), and in the discussion on negative statements, we learned that the symbol for a negation is ~ (tilde). Now, the word “both” in the statement refers to “John is singing (p)” and “He is dancing (q).”
Thus, the phrase “but not both” is symbolized as follows: •~ (p • q). If we add this symbol to the previous statement pvq, then we arrived at
(pvq) •~ (p • q)
Thus, the symbol for the exclusive disjunction “Either John is singing or he is dancing, but not both” is:
(pvq) •~ (p • q)
However, logicians used a more simplified symbol for the phrase “but not both.” They used the underlined wedge vto symbolize “but not both.” Thus, the exclusive disjunction “Either John is singing or he is dancing, but not both” is symbolized as follows:
pvq
Please note that the symbol p v q is read as follows: “p or q, but not p and q.”
In some cases, the exclusive disjunction does not contain the phrase “but not both,” but if we analyze the statement, it denotes exclusivity. Let us consider example #2, which reads:
Either John is sleeping or he is studying.
Although the statement does not contain the phrase “but not both,” it is pretty obvious that it is not possible for John to be sleeping and studying at the same time. Hence, example #2 above is an exclusive disjunction.
If we let p stand for “John is sleeping” and q for “He is studying,” then the statement “Either John is sleeping or he is studying” is symbolized as follows:
(pvq) •~ (p • q)
or, simply,
pvq
Rules in Exclusive Disjunction
An exclusive disjunction is false if both disjuncts have the same truth-value.
Thus, for an exclusive disjunction to be true, one disjunct must true and the other false, and vice versa.
The truth table below illustrates this point.
The truth table above says:
If p is true and q is true, then p v q is false.
If p is true and q is false, then p v q is true.
If p is false and q is true, then p v q is true.
If p is false and q is false, then p v q is false.
Now, given the rule in exclusive disjunction, how do we, for example, determine the truth-value of the exclusive disjunction ~ p vq?
Let us suppose that the truth-value of p is true and q is true.
So, if p is true and q true, then the statement ~ p vq is true.
To illustrate:
The illustration says that p is true and q is true. Now, before we apply the rule in exclusive disjunction in the statement ~p v q, we need to simplify ~p first because the truth-value “true” is assigned to p and not to ~p. If we recall our discussion on the rule in negation, we learned that the negation of true is false. So, if p is true, then ~p is false. Thus, at the end of it all, ~p v q is true if p is true and q is true.
A disjunction or disjunctive statement is a compound statement or proposition that is connected by the words “Either…or” or just “or.”
And the component statements in a disjunction are called “disjuncts.” There are two types of disjunctive statements used in symbolic logic, namely: inclusive and exclusive disjunction. In this post, I will only focus on inclusive disjunction.
Inclusive disjunction uses the connective “Either…or, perhaps both.” Consider the example below.
Either Jake is sleeping or Robert is studying, perhaps both. (J, R)
If we let J stand for “Jake is sleeping” and R for “Robert is studying,” then the statement “Either Jake is sleeping or Robert is studying, perhaps both”is symbolized as follows:
JvR
Please note that the constants J and R do not just represent Jake and Robert respectively; rather, they represent the entire statement. Thus, J represents “Jake is sleeping,” while R represents “Robert is studying.”
It must also be noted that in most cases, the phrase “perhaps both” in an inclusive disjunction is not written in the statement. Thus, in determining whether the statement is an inclusive or an exclusive disjunction, we just need to analyze the statement per se. Let us consider this example:
Either Jake is sleeping or Robert is studying.
As we notice, the statement does not contain the phrase “perhaps both.” But if we analyze the statement, it is clear that it is an inclusive disjunction because it is possible for the two component statements, namely, “Jake is sleeping” and “Robert is studying,” to occur at the same time. (Please note that I will discuss the nature and characteristics of an exclusive disjunction in my next post.)
Rules in Inclusive Disjunction
An inclusive disjunction is true if at least one of the disjuncts is true.
If both disjuncts are false, then the inclusive disjunction is false.
In other words, the rules say that the only condition wherein the inclusive disjunction becomes false is when both disjuncts are false. This is because the connective “Either…or” directly implies that either of the disjuncts is possible. Thus, in an inclusive disjunction, we just need one disjunct to be true in order for the entire disjunctive statement to become true. The truth table below illustrates this point.
The truth table above says:
If p is true and q is true, then p v q is true.
If p is true and q is false, then p v q is true.
If p is false and q is true, then p v q is true.
If p is false and q is false, then p v q is false.
Now, given the rules in inclusive disjunction, how do we, for example, determine the truth-value of the inclusive disjunction p v~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 v~q is true. To illustrate:
The illustration above says that p is true and q is false. Now, before we apply the rules in inclusive disjunction in the statement p v~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 v~q is true if p is true and q is false.
Alternatively, we can determine the truth-value of the inclusive disjunction p v~q in the following manner:
The illustration above says that if we assign the truth-value true for p, then we can conclude right away that the inclusive disjunction is true because one of the disjuncts is already true. If we recall, the rule in inclusive disjunction says “An inclusive disjunction is true if at least one of the disjuncts is true.”
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:
p
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.
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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:
pvq
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.
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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.
