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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

  1. An If-then statement is false if the antecedent is true and the consequent false.
  2. Thus, other than this form, the If-then statement is true.

The truth table below illustrates this point.

If-then statement

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 true.
  4. 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.

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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

  1. A conditional statement is false if the antecedent is true and the consequent false.
  2. Thus, other than this form, the conditional statement is true.

The truth table below illustrates this point.

conditional statement

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 true.
  4. 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.

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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:

  1. Either John is singing or he is dancing, but not both.
  2. 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.

exclusive disjunction

The truth table above says:

  1. If p is true and q is true, then p v q is false.
  2. If p is true and q is false, then p v q is true.
  3. If p is false and q is true, then p v q is true.
  4. 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:

exclusive disjunction

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.

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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

  1. An inclusive disjunction is true if at least one of the disjuncts is true.
  2. 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.

inclusive disjunction

 The truth table above says:

  1. If p is true and q is true, then p v q is true.
  2. If p is true and q is false, then p v q is true.
  3. If p is false and q is true, then p v q is true.
  4. 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:

inclusive disjunction

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:

inclusive disjunction

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.”

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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:

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

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

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


Symbolizing Conjunctive Statements

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

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

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

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

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

p • ~q

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

Bread and butter is a perfect combination.

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

p

However, if we have the example

“John and Mary are watching TV”

then we have to symbolize this as:

p • q

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


Rules in Conjunction

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

The truth table below illustrates this point.

conjunctive statements

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:

conjunctive statements

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

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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.

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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.

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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.

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Appeal to Authority

An appeal to authority is a common type of fallacy that arises when one who has the difficulty in confronting or understanding complicated questions will seek refuge to the ideas, concepts, principles or judgments of a person who enjoys a reputation as an expert or an authority of the matter at issue. In other words, an appeal to authority is a fallacious argument in which the testimony of someone believed to be an authority is cited in support of a conclusion. It must be noted that the person being cited here is not, in fact, an expert or an authority on the matter or for some reason should not be relied upon. Thus, the fallacy of appeal to authority occurs when the authority cited is not qualified in the relevant matters or, less typically, is not free from adverse influences. Thus, the arguer is relying upon the assertions of someone who is not truly in a position to know.

Let us consider the following examples:

  1. I know your doctor says you need your appendix removed, but according to the famous herbalist Mar Lopez, people with your symptoms just need a change in their diet, plus a daily intake of MX3 capsule. So, forget about having your appendix removed.
  2. Augustine said there is no salvation outside the Catholic Church. That’s reasonable enough for me.

The underlying idea of such arguments is that some statement p is true because some authority q has said it is true. The argument’s basic structure is this:

Authority p asserts that q.
Therefore, q.

Here, we see immediately that such an argument is neither valid nor inductively strong, since the mere fact that someone asserts q neither makes it so nor makes it probable. Typically, however, the arguer believes more than the mere fact that p asserts that q. The arguer very likely is assuming such things as that p is someone who knows what he or she is talking about regarding q, or that p is speaking without bias, or that p is telling the truth. If those or similar assumptions are well founded, then the appeal to authority p may constitute a good argument, that is, non-fallacious reasoning. It must be noted that not all appeals to authority are fallacious. In fact, some appeal to authority may be inductively strong. After all, we should accept the testimony of qualified and unbiased experts, for there are indeed experts in their own right.

Now, to identify an appeal to authority fallacy, we ask two questions:  1) Is the authority, in fact, a qualified authority about matters related to q? and 2) Is there any good reason to believe that the authority may be biased in matters related to q?

Regarding Example #1 above, we should ask whether Mar Lopez is qualified to claim that proper diet and daily intake of MX3 capsule will render the removal of someone’s appendix unnecessary. So, is Mar Lopez an expert in human anatomy? Can he provide a scientific proof that proper diet and daily intake of MX3 capsule will render the removal of someone’s appendix unnecessary?

Regarding Example #2, we should ask whether St. Augustine, although he was a famous Catholic theologian, has proofs that heaven and hell really exist. As a matter of fact, issues about heaven and hell are very complicated ones. In fact, nobody has proven that indeed heaven and hell exist. If this is the case, how can we meaningfully talk about salvation?

A common variation on the appeal to authority is an appeal to a magazine or newspaper article or a radio or TV program. Consider this example:

“They have found a cure for cancer. I read about it in The New York Times.”

In such case, we ought to ask the same question: Is the source cited a reliable one in this matter? Ordinarily, we should be very suspicious of medical breakthroughs reported in The New York Times, though not of such breakthroughs reported in, say, the Journal of the American Medical Association. On the other hand, we would not expect to get reliable advice on the news or current events in a medical journal. Hence, the appeal to authority fallacy occurs when an argument is supported by reference to a publication or program not known for specialization on the subject.

In summary, not all appeals to authority are fallacious. The appeal to authority fallacy only occurs when an arguer appeals to someone who is not an expert in the field for which he or she is cited as support or who is not unbiased.

To recognize the appeal to authority fallacy, we need only look for an argument based primarily on the premises that some person (or some publication) reports that q is true. The fallacy occurs when the person (or publication) is not relevantly qualified or is not speaking without bias.

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Mood and Figure of a Syllogism: Categorical Logic

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A syllogism can be distinguished from other syllogisms by its form, that is, the mood and figure of a syllogism. 

A syllogism’s form is determined by the mood and figure of the argument. 

On the one hand, a Mood refers to the kinds of propositions that syllogistic arguments contain, whether A, E, I or O. 

On the other hand, a Figure refers to the position of the middle term in the premises.

Note that the validity or invalidity of the syllogism depends exclusively upon its form and is completely independent of its specific content or subject matter.

Example:

All Filipinos are humans.
All Cebuanos are Filipinos.
Therefore, all Cebuanos are humans.

The mood and figure of a syllogism can be easily known if the letter S, P, and M are used to represent minor term, major term, and middle term respectively. Thus, the above syllogism will have this:

S – Cebuanos

P – humans

M – Filipinos

Using now the letters corresponding for each term, the syllogistic skeleton of the syllogism can be represented as:

All M are P.
All S are M.
Therefore, all S are P.

The mood of the argument is AAA since the premises and conclusion are all universal affirmative.

As previously stated, figure refers to the middle term’s position in the premises. There are four possible positions of the middle term in the premises, namely, diagonal to the right (figure 1), vertical to the right (figure 2), vertical to the left (figure 3), and diagonal to the left (figure 4).

It is interesting to note that with the four basic statement forms (A, E, I, and O) and four ways of positioning the middle term, it is possible to construct 256 different syllogistic arguments.

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