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May 20, 2022May 20, 2022

Symbolizing Statements in Propositional Logic

In these notes, I will be discussing the topic “symbolizing statements in propositional (or symbolic) logic.” This is very important because, as I have already said in my earlier post before we can determine the validity of an argument in symbolic logic by applying a specific rule, we need to symbolize the argument first. So, how do we symbolize propositions in symbolic logic?

First, we need to identify the major connective. This is because once we have identified the major connective, we will be able to punctuate the proposition properly.

Second, we have to keep in mind that the variables or constants, such p and q or Y and Z, stand for the entire proposition, and not for the words within the proposition itself.

Third and last, we need to put proper punctuation and negation if necessary.

Let us consider the examples below.

If the squatters settle here, then the cattlemen will be angry and there will be a fight for water rights. (p, q, r)

As we can see, this example is a combination of a conditional proposition and a conjunctive proposition. However, if we analyze the proposition, it becomes clear to us that it is a conditional proposition whose consequent is a conjunctive proposition. Thus, the major connective in this proposition is “then.” Hence, when we symbolize the proposition, we need to punctuate the consequent. So, if we let p stand for “The squatters settle here,” q for “The cattlemen will be angry,” and r for “There will be a fight for water rights,” then the proposition is symbolized as follows:

p ⊃ (q • r)

If either the butler or the maid is telling the truth, then the job was an inside one; however, if the lie detector is accurate, then both the butler and the maid are telling the truth. (p, q, r, s)

This example is indeed a complicated one. But it can be easily symbolized.

If we analyze the proposition, it becomes clear that it is a conjunctive proposition whose conjuncts are both conditional propositions with a component inclusive disjunction and conjunction respectively.

Now, if we let

p stands for “The butler is telling the truth”
q for “The maid is telling the truth”
r for “The job was an inside one” and
s for “The lie detector is accurate”

then we initially come up with the following symbol: p v q ⊃ r • s ⊃ p • q

The symbol above, however, is not yet complete. In fact, it remains very complicated. So, we have to punctuate it.

Since the major connective of the proposition is “however,” then we have to punctuate the component conjuncts. Thus, we initially come up with the following symbol:

[p v q ⊃ r] • [s ⊃ p • q]

However, the symbolized form of the proposition remains complicated because the component conjuncts have not been properly punctuated. As already said, there should only be one major connective in a proposition. So, let us punctuate the first conjunct.

Since it is stated in the first conjunct that the proposition is a conditional proposition whose antecedent is an inclusive disjunction, then we have to punctuate p v q. Thus, we initially come up with the following symbol:

[(p v q) ⊃ r] • [s ⊃ p • q]

And then let us punctuate the second conjunct. Since it is stated in the second conjunct that the proposition is a conditional proposition whose consequent is a conjunctive proposition, then we have to punctuate p • q. Thus, we come up with the following symbol:

[(p v q) ⊃ r] • [s ⊃ (p • q)]

Now, the symbol appears to be complete. Thus, the final symbol of the proposition “If either the butler or the maid is telling the truth, then the job was an inside one; however, if the lie detector is accurate, then both the butler and the maid are telling the truth” is as follows:

[(p v q) ⊃ r] • [s ⊃ (p • q)]

Neither Lucas is hard-working nor is he intelligent. (p, q)

This example is obviously an inclusive disjunction; hence, we may initially symbolize the proposition as p v q. However, the words “Neither…nor” is a signifier of a negation, and these words suggest that the entire proposition is negated. Thus, we finally symbolize the proposition “Neither Lucas is hard-working nor is he intelligent” as follows:

~ (p v q)

Please note that ~ (p v q) is not the same with ~ p v ~ q. And ~ p v ~ q is not the proper symbol of example #3 because the words “Neither…nor” suggest that the proposition has to be completely negated. As we learned in my previous post titled “Punctuating Propositions in Symbolic Logic” (see http://philonotes.com/index.php/2018/02/11/punctuating-propositions-in-symbolic-logic/), when the proposition is completely negated, then the entire proposition has to be punctuated.

But let me explain why ~ (p v q) is not the same with ~ p v ~ q. If we recall, the rules in inclusive disjunction say “The inclusive disjunction is true if at least one of the disjuncts is true.” With this, let us determine the truth value of ~ (p v q) and ~ p v ~ q in order to prove that they are not the same.

Let us assign the truth value “true” for p and “false” for q.

symbolizing propositions in symbolic logic

It is not the case that the manager will resign if she does not receive a salary increase. (p, q)

Please note that since the negation sign “It is not the case” precedes the entire proposition, then the entire proposition has to be negated. Thus, we need to punctuate the entire proposition and put the negation sign outside of it.

As I discussed in one of my previous posts, we learned that 1) the variables provided after the proposition represent the propositions in the entire proposition respectively, and 2) since in the example above the antecedent is written after the consequent, then q must be our antecedent and p our consequent. Hence, we initially come up with the following symbol: ~q ⊃ p. Please note that q is negated because it is clearly specified in the proposition. In other words, the proposition contains a negation sign “not.”

Now, since the negation sign “It is not the case” precedes the entire proposition, then, again, the entire proposition must be negated. Thus, we finally symbolize the proposition “It is not the case that the manager will resign if she does not receive a salary increase” as follows:

~ (~q ⊃ p)

If it is not the case that the professor will take a leave of absence if and only if the administration allows him to, then there must be another good reason why the professor will take a leave of absence. (p, q, r)

In this example, since the negation sign “It is not the case” does not precede the entire proposition, then we do not negate the entire proposition. We only negate the proposition where the negation sign immediately precedes. Thus, the negation sign in the example above only negates the proposition “The professor will take a leave of absence if and only if the administration allows him to.” It does not clearly negate the proposition “There must be another good reason why the professor will take a leave of absence.”

Now, if we analyze the proposition, we notice that:

p stand for “The professor will take a leave of absence”
q for “The administration allows him to” and
r for “There must be another good reason why the professor will take a leave of absence.”

Please note that we do not repeat the variable “p” for the proposition “There must be another good reason why the professor will take a leave of absence” because the thought of the proposition is completely changed. This is because of the addition of the idea “There must be another good reason.” Thus, instead of repeating the variable “p,” we use the variable “r” to represent the proposition “There must be another good reason why the professor will take a leave of absence.”

So, we symbolize the proposition “If it is not the case that the professor will take a leave of absence if and only if the administration allows him to, then there must be another good reason why the professor will take a leave of absence” as follows:

~ (p ≡ q) ⊃ r

May 20, 2022May 20, 2022

How to Symbolize Arguments in Propositional Logic?

In these notes, I will discuss how to symbolize arguments in propositional or symbolic logic, which uses all the basic symbols, especially the use of parentheses. As I have mentioned in my other notes, symbolizing arguments in logic is important because before we can determine the validity of an argument in symbolic logic, we need to symbolize the argument first.

In symbolizing arguments in symbolic logic, we need to do the following:

First, we need to symbolize the argument sentence by sentence.

Second, we have to identify the major connectives in each sentence of the argument. This is important because once we have identified the major connective we will be able to punctuate the sentence or proposition properly.

Third, we need to remember that the variables or constants, such p and q or Y and Z, stand for the entire sentence or proposition, and not for the words within the sentence or proposition itself.

Lastly, we need to put proper punctuations and negation signs if necessary.

Let us consider the example below.

If the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth, then the captain either had good motives for controlling the world or his motives were evil. The airship Albatros had a powerful weapon if and only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground. It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax. It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax. Furthermore, the captain’s motives for controlling the world were not evil. Therefore, his motives were good. (A, D, W, G, E, S, B, P)

As we can see, the argument above is quite long and indeed complicated. But again, we can easily symbolize this argument because, as I already mentioned, we will symbolize this argument sentence by sentence (or proposition by proposition). So, let’s start with the first sentence.

Sentence 1

If we analyze this sentence, it is clear that the major connective is “if…then” or just “then”. Hence, it is a conditional proposition. Now, in symbolizing this sentence, we need to punctuate the antecedent and the consequent.

If we look at the antecedent, we notice that it is a compound proposition whose conjuncts are both conditional propositions. Because there are several connectives in the sentence, then we also need to punctuate the antecedent. Hence, the antecedent (which reads: the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth) is symbolized as follows:

(A ⊃ D) • (D ⊃ W)

As we can see, the consequent of the proposition above is an exclusive disjunction. Thus, we need to underscore the wedge to differentiate it from an inclusive disjunction. The consequent (which reads: the captain either had good motives for controlling the world or his motives were evil) is symbolized as follows: G v E.

Please note that the constants provided at the end of the argument above represent the propositions in the entire argument respectively. Thus, in the first proposition, the constant A stands for “the airship Albatros had powerful weapon”, D stands for “it could destroy objects on the ground”, W stands for “the captain could enforce his will all over the earth”, G stands for “the captain either had good motives for controlling the world”, and E stands for “his motives were evil”.

Now, when we symbolize the entire proposition, we need to punctuate both the antecedent and the consequent because, as the rule says, there should only be one major connective in each proposition. Thus, the proposition above is symbolized as follows:

[(A ⊃ D) • (D ⊃ W)] ⊃ (G v E)

Note: Please apply the principles discussed above in symbolizing the rest of the sentences below. If you have questions or clarifications, please leave a comment below. The PHILO-notes team is happy to respond to them.

Sentence 2

The airship Albatros had powerful weapon if and only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground.

(A ≡ S) • (S ⊃ D)

Sentence 3

It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax.

(D ⊃ W) v (B ⊃ P)

Sentence 4

It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax.

~ (B ⊃ P)

Sentence 5

Furthermore, the captain’s motives for controlling the world were not evil.

~ E

Sentence 6 (which is the conclusion)

Therefore, his motives were good.

In the end, the argument above is symbolized as follows:

How to Symbolize Arguments in Symbolic Logic

May 20, 2022May 20, 2022

Punctuating Statements in Propositional Logic

In these notes, I will briefly discuss the topic “punctuating statements in propositional (or symbolic) logic.”

But why do we need to punctuate propositions in symbolic logic? This is because, in many instances, propositions in symbolic contain more than one connective; but in symbolic logic, all propositions should only have one major connective.

Thus, if there are two or more connectives, then we have to punctuate the proposition accordingly so that the major connective will become clear.

Symbolic logic uses parentheses ( ), brackets [ ], and braces { } as punctuation symbols.

Let us consider the example below.

If the road is wet, then either it rains today or the fire truck spills water on the road. (p, q, r)

As we can see, the example contains three propositions, namely:

1) The road is wet,

2) It rains today, and

3) The fire truck spills water on the road.

And as I already discussed in my previous posts, we learned that the variables provided after the proposition represent the propositions in the entire proposition respectively. Thus, in the example above, p stands for “The road is wet,” q for “It rains today,” and r for “The fire truck spills water on the road.” Hence, initially, the proposition is symbolized as follows:

p ⊃ q v r

However, the symbol above is not yet complete because, at this point, it is not yet clear what type of proposition it is. This is the reason why we need to punctuate the proposition. Please see my previous discussion on “Propositions and Symbols Used in Symbolic Logic” (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/) for some idea on how to symbolize a proposition in symbolic logic.

Now, if we analyze the proposition, it would become clear that it is a conditional proposition whose consequent is an inclusive disjunction. For this reason, we need to punctuate the consequent.

Thus, the proposition “If the road is wet, then either it rains today or the fire truck spills water on the road” is symbolized as follows:

p ⊃ (q v r)

I will discuss more about this when I go to the discussion on “symbolizing propositions” in symbolic logic.

Meantime, let me give examples of a punctuated proposition just to show that statements in propositional or symbolic logic that contain two or more connectives have to be punctuated accordingly. Please see examples below then.

punctuating propositions in symbolic logic

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.