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May 20, 2022May 20, 2022

Negation of Statements in Propositional Logic

In my other notes titled “Propositions and Symbols Used in Propositional (or Symbolic) Logic” (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), I discussed the two basic types of a proposition as well as the symbols used in symbolic logic. I have also briefly discussed how propositions can be symbolized using a variable or a constant.

In these notes, I will discuss the topic “negation of statements in propositional (or symbolic) logic” or the way in which propositions or statements in symbolic logic are negated.

To begin with, we have to note that any statement used in symbolic logic can be negated. And as I have already mentioned in the previous discussion, symbolic logic uses ~ (tilde) to symbolize a negative proposition.

But how do we know that the statement is negative?

A statement is negative if it contains at least one of the following signifiers:

Not

It is false

It is not the case

It is not true

For example, let us consider the following statements:

Either no students are interested in the party or it is not the case that the

administration requires the students to attend the party.

If the company does not increase the salary of the workers, then the union will go on strike to press its various demands.
The professor will not be absent if and only if he is not sick.

As we notice, example #1 is a compound statement, and both component statements contain the negation signs “no” and “it is not the case.” For this reason, when we symbolize the entire statement, then both component statements should be negated. Hence, if we let p stand for “No students are interested in the party” and q for “It is not the case that the administration requires the students to attend the party,” then the statement “Either no students are interested in the party or it is not the case that the administration requires the students to attend the party” can be symbolized as follows:

~ p v ~ q

In example #2, only the first component statement contains the negation sign “not.” Hence, only the first statement should be negated. Thus, if we let p stand for “The company does not increase the salary of the workers” and q for “The union will go on strike to press its various demands,” then the statement “If the company does not increase the salary of the workers, then the union will go on strike to press its various demands,” is symbolized as follows:

~ p ⊃ q

In example #3, both component statements contain a negation sign “not.” Thus, when symbolized, both component statements have to be negated. Hence, if we let p stand for “The professor will not be absent” and q for “He is not sick,” then the statement “The professor will not be absent if and only if he is not sick” is symbolized as follows:

~ p ≡ ~ q

Now, sometimes a statement can be double (or even triple) negated. In other words, the statement contains two or more negation signs. If this happens, then the statement has to be symbolized accordingly. Consider this example: “It is not true that the professor is not sick.” If we let p stand for the entire statement, then it is symbolized as follows:

~~ p

However, since a double negation implies affirmation, then the statement can also be symbolized as follows:

In some cases, contradictory words, such as “kind and unkind” and “mortal and immortal,” may signify negation if and only if it is clearly specified in the statement; otherwise, the statement should not be negated. Consider the following examples:

Lulu is generous, while Lili is unkind.
Either George is kind or Bert is unkind.

In example #1, the word “unkind” does not clearly signify negation. Thus, the statement “Lili is unkind” is not a negative statement. Let us symbolize example #1. If we let p stand for “Lulu is generous” and q for “Lili is unkind,” then the proposition “Lulu is generous, while Lili is unkind” is symbolized as follows:

p • q

However, the word “unkind” in example #2 above clearly signifies negation because of the presence of the contradictory words “kind and unkind” in the statement. Now, if we let p stand for “George is kind” and q for “Bert is unkind,” then the statement “Either George is kind or Bert is unkind” is symbolized as follows:

p v ~q

This is because the statement “Either George is kind or Bert is unkind” can also be stated in this manner: “Either George is kind or Bert is not kind.”

Rule in Negation

The negation of a true statement is false; while the negation of a false statement is true.

Obviously, the rule in negation says that if a particular statement is true, then it becomes false when negated. And if a particular statement is false, then it becomes true when negated. The truth table below illustrates this point.

Let us determine the truth-value of a negative statement by applying the rule in negation.

Consider the example below.

It is not the case that the administration requires the students to attend the party.

Again, if we let p stand for the statement “The administration requires the students to attend the party,” then the statement is symbolized as p. However, since the statement contains a negation sign “It is not the case,” then the statement is negative. Thus, the statement has to be symbolized as follows:

~ p

Now, if we assume that the statement “The administration requires the students to attend the party” is true, that is, the administration did indeed require the students to attend the party, then the statement “It is not the case that the administration requires the students to attend the party” is absolute false. To illustrate:

Please note that when we assign a truth-value to a statement, we assign it to the statement without the negation sign. Thus, if we have the statement ~ p, and if we assign, for example, True value to the statement, we assign it to p and not to ~ p.

May 19, 2022May 19, 2022

Biconditional Statements in Propositional Logic

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 only if 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.

May 19, 2022May 19, 2022

If-then Statements in Propositional Logic

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.

May 19, 2022May 19, 2022

Conditional Statements in Propositional Logic

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.

May 19, 2022May 19, 2022

Exclusive Disjunction in Propositional Logic

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 p v q. 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 p v q. 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 p v q, then we arrived at

(p v q) • ~ (p • q)

Thus, the symbol for the exclusive disjunction “Either John is singing or he is dancing, but not both” is:

(p v q) • ~ (p • q)

However, logicians used a more simplified symbol for the phrase “but not both.” They used the underlined wedge v to symbolize “but not both.” Thus, the exclusive disjunction “Either John is singing or he is dancing, but not both” is symbolized as follows:

p v q

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:

(p v q) • ~ (p • q)

or, simply,

p v q

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 v q?

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 v q 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.

May 19, 2022May 19, 2022

Inclusive Disjunction in Propositional Logic

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.

As I discussed in my other notes titled “Propositions and Symbols Used in Propositional or Symbolic Logic (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), the symbol for the connective “Either…or” is v (wedge).

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:

J v R

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.”

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.