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**Tautologies and Contradictions**

In this post, I will briefly discuss tautologies and contradictions in symbolic logic. But please note that this is just an introductory discussion on tautologies and contradictions as my main intention here is just to make students in logic become familiar with the topic under investigation.

On the one hand, a **tautology is defined** as a propositional formula that is **true** under any circumstance. In other words, a propositional expression is a **tautology** if and only if for all possible assignments of truth values to its variables its truth value is always **true. **Thus, a tautology is a proposition that is always true. Consider the following example:

Either the accused is guilty **or** the accused is not guilty. (p)

Obviously, the proposition is a disjunction; yet both disjuncts can be represented by the variable ** p**. Hence, the proposition is symbolized as follows:

*p***v** *~p*

Now, in what sense that this proposition is always true? The truth table below will prove this point.

As we can see in the truth table above, if ** p** is

On the other hand, a **contradiction is defined** as a propositional formula that is always false under any circumstance. In other words, a propositional expression is a **contradiction** if and only if for all possible assignments of truth values to its variables its truth value is always **false**. Thus, again, a contradiction is a proposition that is always **false**. Let us consider the examples below.

Man is both mortal **and** immortal. (p)

Obviously, the proposition is a conjunction; yet both conjuncts can be represented by the variable ** p**. Hence, the proposition is symbolized as follows:

*p ***• ***~p*

Now, in what sense that this proposition is always false? The truth table below will prove this point.

As we can see in the truth table above, if ** p** is

**Note**:

We have provided a video of all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

3. Negations of Statements (see https://www.youtube.com/watch?v=hHrmrh7qYnA)

The following posts may also be of great help in understanding the discussion above:

1. Propositions and Symbols Used in Symbolic Logic (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/)

2. Negation of Propositions in Symbolic Logic (see http://philonotes.com/index.php/2018/02/03/negation-of-propositions/)

3. Conjunctive Statements (see http://philonotes.com/index.php/2018/02/03/conjunctive-statements/)

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**Symbolizing Propositions in Symbolic Logic**

In this post, I will be discussing the topic “symbolizing propositions in 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

**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,”

*p***⊃**** (**** q **•

- 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** stand for “The butler is telling the truth”

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 **

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**)

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**)

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**)

- 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 “

** ~** (

Please note that ** ~** (

But let me explain why ** ~** (

Let us assign the truth value “**true**” for ** p** and “

** **

** **

** **

**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

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:

** ~** (

- 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”

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 “

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:

** ~ **(

**Note to students**:

If you have questions about the topic “symbolizing propositions in symbolic logic,” please feel free to leave a comment below. And if you want us to help you symbolize a complicated proposition, please just write the proposition in the comment box below and we will symbolize it for you.

We have provided a video of all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

3. Negations of Statements (see https://www.youtube.com/watch?v=hHrmrh7qYnA)

The following posts may also be of great help in understanding the discussion above:

1. Propositions and Symbols Used in Symbolic Logic (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/)

2. Negation of Propositions in Symbolic Logic (see http://philonotes.com/index.php/2018/02/03/negation-of-propositions/)

3. Conjunctive Statements (see http://philonotes.com/index.php/2018/02/03/conjunctive-statements/)

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**Punctuating Propositions in Symbolic Logic**

In this post, I will briefly discuss the topic “punctuating propositions in 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,”

*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 propositions in symbolic logic that contain two or more connectives have to be punctuated accordingly. Please see examples below then.

We have provided a video for all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

3. Negation of Statements (see https://www.youtube.com/watch?v=hHrmrh7qYnA)

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**Biconditional Propositions**

Biconditional propositions are compound propositions connected by the words “**if and only if**.” As we learned in the previous discussion titled “Propositions and Symbols Used in Symbolic Logic,” 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

*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

**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

The truth table below illustrates this point.

The truth table above says:

- If
is*p***true**andis*q***true**, then*p**≡*is*q***true**. - If
is*p***true**andis*q***false**, then*p**≡*is*q***false**. - If
is*p***false**andis*q***true**, then*p**≡*is*q***false**. - If
is*p***false**andis*q***false**, then*p**≡*is*q***true**.

Now, suppose we have the example *~p **≡ **q*** .** How do we determine its truth value if

Let me illustrate.

The illustration says that ** p** is

** **

We have provided a video of all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

See also “Propositions and Symbols Used in Symbolic Logic” http://philonotes.com/index.php/2018/02/02/symbolic-logic/

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**Conditional Propositions**

Conditional propositions are compound propositions connected by the words “**If…then**” or just “**then**.” As we learned in the previous discussion titled “Propositions and Symbols Used in Symbolic Logic (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), the symbol for “**if…then**” is a horseshoe. Consider the example below:

**If** it rains today, **then** the road is wet. (p, q)

If we let ** p** stand for “It rains today” and

*p ***⊃*** q*

Please note that the proposition that precedes the connective horseshoe (**⊃**) is called the “antecedent” and the proposition that comes after it is called “consequent.”

Please note as well that there are cases wherein the words “**if…then**” is not mentioned in the proposition, yet the proposition remains a conditional one. Consider the example:

**Passage of the law **means** morality is corrupted.** (p, q)

If we analyze the proposition, it is very clear that it is a conditional proposition because it suggests a “cause and effect” relation. Thus, the proposition can be stated as follows:

**If** the law is passed, **then** morality will be corrupted.

If we let **p** stand for “The law is passed” and **q** for “Morality will be corrupted,” then the proposition is symbolized as follows:

*p ***⊃*** q*

It is also important to note that sometimes the antecedent is stated after the consequent. If this happens, then we have to symbolize the proposition accordingly. Let’s take the example below.

**Morality would be corrupted should the abortion law is passed**. (p, q)

If we analyze the proposition, it is clear that the antecedent is “**Abortion law is passed**” and the consequent is “**Morality would be corrupted**.” Hence, the proposition “Morality would be corrupted should the abortion law is passed” is symbolized as follows:

*q ***⊃*** p*

Again, as I already pointed out in my previous discussion, the variables provided after the proposition represent the propositions in the entire proposition respectively. Thus, in the statement

**Morality would be corrupted should the abortion law is passed. **(p, q)

the variable ** p** stands for “Morality would be corrupted” and

*q* **⊃*** p*

**Rules in Conditional Propositions**

- A conditional proposition is
**false**if the antecedent is**true**and the consequent**false**. - Thus, other than this form, the conditional proposition is true.

The truth table below illustrates this point.

The truth table above says:

- If
is*p***true**andis*q***true**, then*p***⊃**is*q***true**. - If
is*p***true**andis*q***false**, then*p***⊃**is*q***false**. - If
is*p***false**andis*q***true**, then*p***⊃**is*q***true**. - If
is*p***false**andis*q***false**, then*p***⊃**is*q***true**.

As we can see, the rules in conditional propositions say that the only instance wherein the conditional proposition 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

The second row says that ** p** is

The third row says ** p** is

Lastly, the fourth row in the truth table above says ** p** is

We have provided a video for all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

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**Exclusive Disjunction**

In my previous post titled “Inclusive Disjunction” (see http://philonotes.com/index.php/2018/02/06/inclusive-disjunction/), I discussed the nature and characteristics of an inclusive disjunction, including its rules and how to determine its truth-value. In this post, 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

As already mentioned, if we let ** p** stand for “John is singing” and

(

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

(*p***v** ** 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: “

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

(*p***v** ** 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
is*p***true**andis*q***true**, then*p*__v__is*q***false**. - If
is*p***true**andis*q***false**, then*p*__v__is*q***true**. - If
is*p***false**andis*q***true**, then*p*__v__is*q***true**. - If
is*p***false**andis*q***false**, then*p*__v__is*q***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

The illustration says that ** p** is true and

We have provided a video for all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

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**Inclusive Disjunction**

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 post titled “Propositions and Symbols Used in 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
is*p***true**andis*q***true**, then*p***v**is*q***true**. - If
is*p***true**andis*q***false**, then*p***v**is*q***true**. - If
is*p***false**andis*q***true**, then*p***v**is*q***true**. - If
is*p***false**andis*q***false**, then*p***v**is*q***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

The illustration above says that ** p** is true and

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

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

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**Conjunctive Statements**

There are four types of compound statement used in symbolic logic, namely, conjunctive, disjunctive, conditional, and biconditional. In this post, 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 previous post 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 **•** 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

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

*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** •

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:

- If
is*p***true**andis*q***true**, thenis*p • q***true**. - If
is*p***true**andis*q***false**, thenis*p • q***false**. - If
is*p***false**andis*q***true**, thenis*p • q***false**. - If
is*p***false**andis*q***false**, thenis*p • q***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

The illustration above says that ** p** is true and

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

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**Negation of Propositions in Symbolic Logic**

In my previous post titled “Propositions and Symbols Used in 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 this post, I will discuss the topic “negation of propositions in 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:

**No **

**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

*~ 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

*~ 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

*~ 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:

*p*

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

*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

* **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*

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,

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)

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**Propositions and Symbols Used in Symbolic Logic**

Just as in traditional or Aristotelian logic, our main goal in 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:

- Jack is singing, while Jill is dancing.
- 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

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

** p** •

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

*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

*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**

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.

For those who do not have a strong background and orientation in philosophy, please see our post titled “What is Philosophy” (see http://philonotes.com/index.php/2017/12/16/what-is-philosophy/). We also have a video for it (see https://www.youtube.com/watch?v=nRG-rV8hhpU).

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