Logic 3.1 Statements and Logical Connectives

Logic 3.1 Statements and Logical Connectives
3.2 Truth Tables for Negation, Conjunction, and Disjunction 3.3 Truth Tables for the Conditional and Biconditional 3.4 Equivalent Statements 3.5 Symbolic Arguments 3.6 Syllogistic Arguments

Section 3.1 – Statements and Logical Connectives
Studying logic can prepare us to better understand other areas of mathematics and computer programming. It also helps us understand the thought processes involved in learning subjects other than mathematics.

A Little History: Read “History” on page 83 of your text.
Two names to remember are Aristotle, the Father of Logic ( B.C.) and Gottfried Leibniz ( ), the first serious student of symbolic logic. One difference between Aristotelian logic and Symbolic logic is that in symbolic logic, symbols represent written statements.

An English Lesson in Math Class?
Logic involves the English language: Connectives - words such as - and , or , if...then We will be using an “Inclusive Or” when studying logic (remember? soup, salad, or both soup and salad). Statements – can be simple or compound An example of a simple statement – The dog sleeps (notice there is only one thought here). An example of a compound statement – The dog sleeps and the cat rules the house (notice there are two thoughts here).

English Lesson Continued:
Negation – changes the truth value of a statement. An example – Broccoli is green (a true statement), can be negated by adding the word “not” – Broccoli is not green (a false statement). Note: A statement and its negation have opposite truth values. Quantifiers – words such as - all, none, some Note: Negating statements with quantifiers is tricky. Look at the red box and Example 1 on page 85 in the text to review negations with quantifiers.

English Lesson Continued:
Compound Statements – A statement that consists of two or more simple statements and a connective. A negation of a simple statement is also a compound statement. Examples of compound statements: The dog is asleep and the cat rules the house. The dog is not asleep. Connectives include the words - and, or, if…then, if and only if.

Compound Statements: “Not” Statements – are called negations.
The symbol for “not” is  . Given the statement s: The dog is yellow. Then s represents the following two statements – “The dog is not yellow”, or “It is false that the dog is yellow”. “And” Statements – are called conjunctions. The symbol for “and” is  . Other words that are sometimes used for a conjunction are but, however, or nevertheless.

Compound Statements: “Or” Statements – are called disjunctions.
The symbol for “or” is  . “If-Then” Statements – are called conditional statements. The symbol for “if…then” is  . For example p  q would be read “If p then q”. A conditional statement has two parts, the antecedent (the part in front of the arrow) and the consequent (the part that follows the arrow).

Compound Statements: “If and Only If” Statements – are called biconditional statements. The symbol for “if and only If” is  . For example p  q would be read “p if and only if q”. Commas – are used as a grouping symbol in a compound statement. The simple statements on the same side of the comma are grouped together in parenthesis.

Examples of Compound Statements:
Given o: The store is open. d: The line of people extends out the door. Write the following statements symbolically: The store is open and the line of people extends out the door. The line of people does not extend out the door if and only if the store is open. The line of people extends out the door or the store is not open.

Examples of Compound Statements:
Given o: The store is open. d: The line of people extends out the door. Write the following statements symbolically: The store is open and the line of people extends out the door. o  d The line of people does not extend out the door if and only if the store is open. d o The line of people extends out the door or the store is not open. d   o

Examples (with Commas):
Given t: The temperature is 90 degrees. a: The air conditioner is working. h: The apartment is hot. Write the following statements symbolically: If the temperature is 90 degrees or the air conditioner is not working, then the apartment is hot. The temperature is not 90 degrees if and only if the air conditioner is not working, or the apartment is not hot.

Examples (with Commas):
Given t: The temperature is 90 degrees. a: The air conditioner is working. h: The apartment is hot. Write the following statements symbolically: If the temperature is 90 degrees or the air conditioner is not working, then the apartment is hot (t   a)  h The temperature is not 90 degrees if and only if the air conditioner is not working, or the apartment is not hot. ( t   a)   h

Example – Write in Words:
Given w: The water is 70 degrees. s: The sun is shining. f: We will go swimming. Write each symbolic statement in words. w  (s  f ) (s  f )  w

Example – Write in Words:
Given w: The water is 70 degrees. s: The sun is shining. f: We will go swimming. Write each symbolic statement in words. w  (s  f ) The water is not 70 degrees, and the sun is shining or we will go swimming. (s  f )  w If the sun is shining and we will go swimming, then the water is 70 degrees.

Things to Notice: When representing simple statements, try to choose letters that relate to the statements because it makes the statements easier to understand. For example, using w to stand for “The water is 70 degrees”. Note: The text does not always do this. When writing symbolic statements in words, remember the statements may not make a lot of sense (as you probably noticed in the last slide).

Dominance of Connectives:
What are connectives? They are the negation, conjunction, disjunction, conditional, and biconditional. Remember the saying, “Please Excuse My Dear Aunt Sally”? It relates to the order of operations. It lists the order in which the calculations should be completed. Similarly, in symbolic logic, there is a dominance of connectives that is followed when evaluating logic statements. Unfortunately, there is no cute saying to remember the order.

Dominance of Connectives:
Least dominant 1. Negation  2. Conjunction  ; Disjunction  3. Conditional  Most dominant 4. Biconditional  Always evaluate the least dominant connectives first. Start at the top of the list and evaluate negations first, then conjunctions and disjunctions, then conditionals, and finally, biconditionals. This is also found in Table 3.1 on page 91 in the text.

Practice Dominance of Connectives:
Add parenthesis by using the dominance of connectives.  p  r   q  p  q q  p   r q  p  p  q

Practice Dominance of Connectives:
Add parenthesis by using the dominance of connectives.  p  r   q ( p  r ) ( q)  p  q ( p)  q q  p   r q  (p   r) q  p  p  q (q  p) (p  q)

Section 3.2 – Truth Tables for Negation, Conjunction, and Disjunction
A truth table is a device used to determine when a compound statement is true or false. This section covers truth tables involving negations, conjunctions, and disjunctions.

Introduction to Truth Tables:
A truth table is a table that lists all the possible truth values for a given statement. Given only one simple statement, there are two possible truth values. The simple statement can be true (T), or it can be false (F). If p is our statement then it can be represented in the following truth table: p Case 1 T Case 2 F

Introduction to Truth Tables:
Given a compound statement (two simple statements and a connective), there are four possible cases. If p and q are our given simple statements, then they can be represented in the following truth table: Notice there are four cases. p q Case 1 T Case 2 F Case 3 Case 4

Truth Table for Negation:
If p is a true statement, then the negation of p, “not p” would be a false statement. If p is a false statement, then the negation of p is a true statement. p  p Case 1 T F Case 2

Truth Table for Conjunction:
Consider the following statement: We will go out to dinner and we will go shopping. You are being told that both dinner and shopping will occur. Let p be “We will go shopping.” Let q be “We will go to dinner.” The conjunction, p  q, is true only when both p and q are true.

Truth Table for Conjunction:
The truth table for the conjunction would have four cases since there are two simple statements involved. Case 1 is the only case where both statements are true. Thus, p  q is true. p q p  q Case 1 T Case 2 F Case 3 Case 4

Truth Table for Disjunction:
Consider the following job description – The applicant must have a 2-year college degree in civil technology or the applicant must have five years experience in the field. Who can apply? What qualifications must they have?

Consider the following job description – The applicant must have a 2-year college degree in civil technology or the applicant must have five years experience in the field. Who can apply? What qualifications must they have? If I have the college degree can I apply? Yes. If I have five years experience can I apply? Yes. If I have the degree and the experience, can I apply? Yes.

Let p be “The applicant must have a 2-year college degree in civil technology.” Let q be “the applicant must have five years experience in the field.” The disjunction, p  q, is true when one, the other, or both of the simple statements is true. The truth table for the disjunction would have four cases since there are two simple statements involved.

Case 4 is the only case where both statements are false. Thus, p  q is false. p q p  q Case 1 T Case 2 F Case 3 Case 4

Truth Tables: Study Example 1 on page 96. Notice there are four cases in the truth table since it is a conjunction. To determine the column being discussed, look at numbers at the bottom of the table. The original truth values for p and q are placed in the table To construct the truth table, we complete column 1, by copying the truth values for p. Next we complete column 2, by writing the truth values for q. Then we complete column 3, by negating column 2. Last, we complete the conjunction (column 4) by looking at the truth values in columns 1 and We conclude that the compound statement, p  q is only true in Case 2, where originally, p is true and q is false.

Example 2 on page 97: Study Example 2 on page 97 in the text.
Notice that the book no longer writes Case 1, Case 2, etc. in the truth table. We will continue to refer to them as Case 1, Case 2, etc. To construct the truth table, we complete column 1 by negating the truth values for p. Next we complete column 2 by negating the truth values for q. Then we complete column 3, by finding the conjunction of columns 1 and 2. We conclude that the compound statement, p  q, is only true in Case 4, where p was originally false, and q was originally false.

Example 3 on page 99: Study example 3 on page 99.
To construct the truth table, we complete column 1 by copying the truth values for q. Next we complete column 2 by negating the truth values for p. Then we find the disjunction of columns 1 and 2. (Remember, the disjunction is true if one, the other, or both statements is true.) Last, we complete column 4 by negating the truth values in column 3. We conclude that the compound statement (q   p) is only true in Case 2, where p was originally true, and q was originally false.

General Procedure for Constructing Truth Tables:
Read through the “General Procedure for Constructing Truth Tables” on page 99 in the text. These will help you with your homework exercises. Here are a couple examples similar to the homework exercises.

Truth Table Examples: 1. Construct a truth table for the statement
q   p p q   p T F First construct the table. Since there are two simple statements, there should be four cases. Fill in the initial truth values for p and q.

Truth Table Example #1 continued:
1. Truth table for the statement q   p p q   p T F 1 Fill in the column numbered 1 by filling in the truth values for q.

1. Truth table for the statement q   p p q   p T F 1 2 Next, fill in the column numbered 2, ~p, by negating the truth values of p.

1. Truth table for the statement q   p p q   p T F 1 3 2 Now fill in column 3 by finding the conjunction of columns 1 and 2.

1. Truth table for the statement q   p p q   p T F 1 3 2 Now the table is complete!

Truth Table Examples #2:
2. Construct a truth table for the statement  ( p   q ) p q ~(  ~q ) T F First construct the truth table. Since there are two simple statements, there should be four cases. Fill in the initial truth values for p and q.

2. Truth table for the statement  ( p   q ) Next fill in columns 1 and 2 with the truth values for p and ~q. p q ~(  ~q ) T F 1 2

2. Truth table for the statement  ( p   q ) Find the conjunction of columns 1 and 2. Place the results in column 3. p q ~(  ~q ) T F 1 3 2

2. Truth table for the statement  ( p   q ) Find the negation of column 3. Place the results in column 4. p q ~(  ~q ) T F 4 1 3 2

2. Truth table for the statement  ( p   q ) The truth table is complete! p q ~(  ~q ) T F 4 1 3 2

Truth Tables with Eight Cases:
So far, the truth tables we have looked at have contained, at most, two simple statements. What happens when you have a compound statement that contains three simple statements? For example: (p   q)  r . Since each simple statement can have two possible truth values, there are eight cases in the truth table ( 2x2x2 = 8 possible cases !). Look at table 3.10 on page 100 and the second paragraph on page 100 to observe what pattern we use to list all eight possible cases.

Note: You will want to always use this pattern for your original truth values when constructing truth tables. This pattern ensures that you will not repeat any of the cases, and that you will not miss any of the cases.

Determining the Truth Value of a Compound Statement:
Sometimes you are told (or can find) the original truth values of the simple statements. In these cases, you do not need to use a truth table to determine the truth value of the statement. The truth table is used to show all the possible cases, but since you already know the original truth values in this instance, you can substitute the truth values into the statement. Let’s look at some examples.

To determine the truth value of a compound statement first write the statement in symbolic form and determine the truth values of the simple statements. Then find the truth value of the compound statement. = 10 or = 7

To determine the truth value of a compound statement first write the statement in symbolic form and determine the truth values of the simple statements. Then find the truth value of the compound statement. = 10 or = 7 F  T

To determine the truth value of a compound statement first write the statement in symbolic form and determine the truth values of the simple statements. Then find the truth value of the compound statement. = 10 or = 7 F  T T

If p is true, q is false, and r is true determine the truth value of (  r   p )   q

If p is true, q is false, and r is true determine the truth value of (  r   p )   q ( F  F )  T

If p is true, q is false, and r is true determine the truth value of (  r   p )   q ( F  F )  T F  T

If p is true, q is false, and r is true determine the truth value of (  r   p )   q ( F  F )  T F  T T

3.3 Truth tables for the Conditional and Biconditional.
Earlier it was mentioned that conditional and biconditional statements were types of compound statements that connected two simple statements. This section looks at the truth tables for these two statements.

Truth Table for the Conditional:
Remember the conditional  is said, “if p then q”. So, if p happens, then q will follow. Think of the conditional as a promise. The conditional statement is true, unless I have broken the promise. Let p be “I get a bonus.” Let q be “ We will go on vacation.” The conditional statement would be: “If I get a bonus, then we will go on vacation”.

“If I get a bonus, then we will go on vacation”. When would the conditional statement be false? The truth value of this conditional statement would be false if I get the bonus, but we do not go on vacation. Note: If I do not get the bonus, whether we go on vacation or not, I have not broken my promise since I did not get a bonus.

p q p  q Case 1 T Case 2 F Case 3 Case 4 Case 2 is the only case where the conditional is false, since I received the bonus, but we did not go on vacation.

Truth Table for the Biconditional:
The biconditional statement p  q means that p  q and q  p. The biconditional is true only when both p and q have the same truth values.

Truth Table for the Biconditional:
p q p  q Case 1 T Case 2 F Case 3 Case 4 In case 2 and case 3, the truth values of p and q are different, so the truth value of the biconditional is false.

Example Truth table for conditional:
Construct a truth table for ~p  ~q. Construct the truth table. Since there are two simple statements, we need to check four cases. p q ~p  ~q T F

Example Truth Table for Conditional:
Construct a truth table for ~p  ~q. First find the negations of p and q and write them in columns 1 and 2. p q ~p  ~q T F 1 2

Construct a truth table for ~p  ~q. Next, using the truth values in columns 1 and 2, determine the solution, column 3. p q ~p  ~q T F 1 3 2

Construct a truth table for ~p  ~q. The truth table is complete! Notice that case 3 is the only case where the compound statement is false, since “ I received the bonus, but we did not go on vacation”. p q ~p  ~q T F 1 3 2

Example Truth Table for Conditional and Biconditional:
Construct a truth table for ( p  q)  p p q ( p  q )  T F Since there are only two simple statements, p and q, there are four cases in our truth table.

Example Truth Table for Conditional and Biconditional: ( p  q)  p
1 2 First fill in columns 1 and 2 with the truth values of p and q.

1 3 2 Next, find the truth values of the biconditional using columns 1 and 2. Fill them in column 3.

1 3 2 4 Fill in column 4 by filling in the truth values for p.

1 3 2 5 4 Then find the solution using columns 3 and 4, and fill in column 5.

1 3 2 5 4 The truth table is complete!

Study Examples 4 and 5 on page 108 in your text:
Example 4 is a truth table example containing the conditional and biconditional which involves three simple statements. Thus, there are eight cases to check. In example 5, the truth value of a compound statement is found given the initial truth values of the simple statements.

Self-Contradictions, Tautologies, and Implications:
Sometimes there is a pattern in the truth table of a compound statement. A self-contradiction is a compound statement that is always false. Thus, when every truth value in the answer column of a truth table is false, the statement is a self-contradiction.

A tautology is a compound statement that is always true. Thus, when every truth value in the answer column of a truth table is true, the statement is a tautology.

An implication is a conditional statement that is a tautology. (take note of the word conditional in the above definition.) Thus, when every truth value in the answer column of a truth table is true, and the final compound statement was a conditional, then the statement is an implication. The antecedent implies the consequent.

Tautology Example: Look at example 8 on page 111 in the text. Notice in column 3, the answer column, that all of the truth values are true, thus the compound statement is a tautology. Further notice that the answer column was found by finding the conditional of columns 1 and 2, thus, the compound statement is also an implication.

3.4 Equivalent Statements
Equivalent statements are important in the study of logic. Equivalent statements are two statements that have exactly the same truth values in the answer columns of the truth tables.

Equivalent Statements:
Equivalent statements, symbolized by , are two statements that have exactly the same truth values in the answer columns of the truth tables. Equivalent statements are also called logically equivalent. To show two statements are equivalent: Construct a truth table for each statement. If the answer columns are the same, then the statements are equivalent. If the answer columns are not the same, then the statements are not equivalent.

Study Example 1 on page 115 in the text. Notice that because the truth tables have the same answer column, the two statements are considered equivalent.

Determine if ? Construct a truth table for the two compound statements and determine if the answer columns are identical. p q ~( p  q ) ( p  T F

Determine if ? First determine the answer column of the first compound statement, column 3. p q (~ p  q ) ( p  T F 1 3 2

Determine if ? Then determine the answer column of the second compound statement, column . p q (~ p  q ) ( p  T F 1 3 2 4 6 5

Determine if ? Compare the answer columns (3 and 6). Since they are the same, then the original compound statements are equivalent. p q (~ p  q ) ( p  T F 1 3 2 4 6 5

Look at Example 3 on page 116 in the text. In this example, four different compound statements (Table 3.25) are being compared to one compound statement (Table 3.24). Comparing the pink answer columns, note that ,since the answer columns in those two truth tables are the same. Also notice that the answer columns in the other truth tables are not the same. Thus, those statements are not equivalent.

De Morgan’s Laws: The equivalence found in Example 3 is actually one of two special laws called De Morgan’s laws. De Morgan’s Laws Notice how the conjunctions (  ) and disjunctions (  ) are used in these laws.

De Morgan’s Laws: De Morgan’s laws can also be negated to give us the following: These seem reasonable when you look at the conjunctions and disjunctions, and De Morgan’s laws.

One More Rule: One more rule that can be useful:
This rule is useful because it takes a conditional statement and turns it into a disjunction. It also takes a disjunction and turns it into a conditional statement.

Writing an Equivalent Statement Examples:
You have now been given five different laws. How are they useful? They allow you to take a given statement and write an equivalent statement. For example: 1. Use De Morgan’s laws to write an equivalent statement for the sentence – It is false that the ink is red and the pen has a ball point.

Writing an Equivalent Statement Examples:
You have now been given five different laws. How are they useful? They allow you to take a given statement and write an equivalent statement. For example: 1. Use De Morgan’s laws to write an equivalent statement for the sentence – It is false that the ink is red and the pen has a ball point. In symbols our original statement is This is similar to De Morgan’s first law, thus, The ink is not red or the pen does not have a ball point.

Equivalent Statement Examples:
2. Use De Morgan’s laws to write an equivalent statement for the sentence – Felicia ate the ice cream or Felicia did not eat the yogurt.

2. Use De Morgan’s laws to write an equivalent statement for the sentence – Felicia ate the ice cream or Felicia did not eat the yogurt. In symbols the original statement is . This is similar to De Morgan’s first law, thus . Actually this is the first law in reverse order, and c is used instead of ~c. Thus, the equivalent statement is – It is false that Felicia did not eat the ice cream and Felicia did eat the yogurt.

Further examples can be found in the text on pages , examples 6 – 10. Study these examples until you can write equivalent statements easily. You do not have to memorize the laws, just be able to use them.

Variations of Conditional Statements:
A conditional statement can be reordered or negated in a few different ways. Variations of the Conditional Statement: Conditional Converse Inverse Contrapositive Note: Remember that the first part of a conditional statement is called the antecedent and the last part of the statement is called the consequent. These are also summarized in a chart on page 120 in the text.

Variations of Conditional Statements:
The converse, inverse, and contrapositive are three ways of rewriting a conditional statement. They involve reversing the conditional statement, negating the conditional statement, or reversing and negating the conditional statement. Studying the truth tables in Table 3.26, on page 121 in the text, we can determine that the conditional and the contrapositive are equivalent statements since they have the same truth values. The converse and the inverse are also equivalent statements.

Writing the Converse, Inverse, and Contrapositive of a Given Statement:
Given the conditional statement “If it is a rose, then it is a flower” write the converse, inverse, and contrapositive. Converse: If it is a flower, then it is a rose. Inverse: If it is not a rose, then it is not a flower. Contrapositive: If it is not a flower, then it is not a rose. Note: All of these are still conditional statements, they have “If… Then”. It is easy to forget these important words!

Writing the Converse, Inverse, and Contrapositive:
Given the conditional statement “If we do not order pizza, then I will have fortune cookies” write the converse, inverse, and contrapositive symbolically. ( pizza (p) and fortune cookies (f)) Conditional Converse Inverse Contrapositive Notice the inverse starts with p since this is the opposite of ~p, which was in our original statement.

One More Interesting Example:
Study Example 14 on pages 122 and 123 in the text. This example shows how to translate compound statements into symbols, and then make a truth table to determine which of the compound statements are equivalent. There are more of these to practice in the homework exercises.

Section 3.5 – Symbolic Arguments
In this section we will determine the validity of symbolic arguments using truth tables and by comparing the arguments to standard form.

Symbolic Argument: Symbolic argument consists of premises and a conclusion. Premises are statements that we accept as true. When the conclusion follows from a given set of premises, then the argument is considered valid. When the conclusion does not follow from a given set of premises, then the argument is invalid (also called a fallacy).

Symbolic Argument: An example:
If the tree is green, then it is getting enough water. The tree is green. The above two statements are called the premises. If we accept them as true, then we can conclude: It (the tree) is getting enough water.

Symbolic Argument: So the three statements constitute a symbolic argument. If the tree is green, then it is getting enough water. The tree is green. Therefore, it is getting enough water. This argument form is called Law of Detachment. gw g w Note: The symbol  stands for therefore.

Determining a Valid Argument:
Look at Example 1 on page 130 in the text. This example illustrates how to convert statements into symbolic form, how to construct a truth table for the statements, and how to determine whether the argument is valid or invalid. If the statement is a tautology ( the answer column of the truth table has all Ts), then the argument is valid. If the statement is not a tautology, then the statement is invalid. A similar example can be found on page 130, Example 2.

Determining a Valid Argument:
At this point you are probably wondering, “Isn’t there an easier way to do this? Truth tables, truth tables, truth tables.” Luckily, in some instances, there is an easier way. There are four Standard Forms of Arguments listed on page 131 in the text. If your argument follows one of these forms, you can automatically determine that the argument is valid. But if the argument does not follow one of the forms, then you will have to construct a truth table to determine if the argument is valid.

Standard Forms of Arguments:
Law of Detachment (also called Modus Ponens) pq p q If the tree is green, then it is getting enough water. The tree is green. Therefore, it is getting enough water.

Law of Contraposition (also called Modus Tollens) pq ~q ~p If today is Monday, then tomorrow is Tuesday. Tomorrow is not Tuesday. Therefore, today is not Monday.

Law of Syllogism pq q  r p r If it is sunny, then I wash my car. If I wash my car, then I need to buy soap. Therefore, if it is sunny, then I need to buy soap. Notice in the example below how, “I wash my car”, appears at the end of one statement and at the beginning of another.

Disjunctive Syllogism p  q ~p q The shirt is too wide or the shirt is too long. The shirt is not too wide. Therefore, the shirt is too long. Notice the word “or” in this example. p or q. not p. Therefore q.

Valid or Invalid? To use a standard form to determine whether an argument is valid or invalid, the argument must match the standard form exactly. The following examples should help you understand the importance of the above statement. Is the following argument valid or invalid? 1. rs r s

Valid or Invalid? To use a standard form to determine whether an argument is valid or invalid, the argument must match the standard form exactly. The following examples should help you understand the importance of the above statement. Is the following argument valid or invalid? 1. rs r s Valid. Law of Detachment.

Valid or Invalid? 2. xy ~x ~y

Valid or Invalid? 2. xy ~x ~y
Invalid. At first thought, it looks like the argument follows law of contraposition since it has ~x, but for law of contraposition, the second premise (2nd line) is the opposite of the consequent (the “then” or second part of line 1). To be law of detachment, the second premise would be the same as the antecedent (1st part of line 1).

Comparisons to Standard Form: x  y xy x y x ~x ~y  y ~y ~x
Detachment Example #2 Contraposition Notice how the previous question (Example #2) is not exactly like detachment or contraposition.

Valid or Invalid? 3. ~ab ~b a

Valid or Invalid? 3. ~ab ~b a
Valid, law of contraposition. The end of line 1 (the consequent) is being negated in line 2, therefore we conclude the opposite of ~a, which is a. Is this making any sense? Let’s look at a few more examples.

Valid or Invalid? 4. r  s s  ~t ~t  u r  u

Valid or Invalid? 4. r  s s  ~t ~t  u r  u
Valid, law of syllogism. We are actually using law of syllogism two times in this example. Notice how the end of one line, is the beginning of the next line. That should bring to mind law of syllogism.

Valid or Invalid? 5. t  ~u ~t u

Valid or Invalid? 5. t  ~u ~t u
Invalid. At first glance, it looks like disjunctive syllogism since it has an “or” (a disjunction), but the conclusion is incorrect. For this example to be valid, the conclusion would have to be changed to ~u.

Valid or Invalid? Translate the argument into symbols, then determine if the argument is valid or invalid. 6. If you cook the meal, then I will vacuum the rug. I will not vacuum the rug. Therefore, you will not cook the meal. c  v ~v  ~c

Valid or Invalid? Translate the argument into symbols, then determine if the argument is valid or invalid. 6. If you cook the meal, then I will vacuum the rug. I will not vacuum the rug. Therefore, you will not cook the meal. c  v ~v  ~c Valid, law of contraposition.

Valid or Invalid? Translate the argument into symbols, then determine if the argument is valid or invalid. 7. If there is an atmosphere, then there is gravity. If an object has weight, then there is gravity. Therefore, if there is an atmosphere, then an object has weight. a  g w  g a  w

Valid or Invalid? Translate the argument into symbols, then determine if the argument is valid or invalid. 7. If there is an atmosphere, then there is gravity. If an object has weight, then there is gravity. Therefore, if there is an atmosphere, then an object has weight. a  g w  g a  w Invalid, this is not law of syllogism since both “g”s are at the end of the statements.

Valid or Invalid? 8. If the football team wins the game, then Tom played quarterback. If Tom played quarterback, then the team is in second place. Therefore, if the football team wins the game, then the team is in second place. w  t t  s w  s

Valid or Invalid? 8. If the football team wins the game, then Tom played quarterback. If Tom played quarterback, then the team is in second place. Therefore, if the football team wins the game, then the team is in second place. w  t t  s w  s Valid, law of syllogism.

Arguments with Three Premises:
Example 4 on page 132 in the text involves a three premise argument. The validity of the argument is determined using truth tables. It is an interesting example to read, though you will not be asked to actually solve a problem at this level of complexity.

Section 3.6 – Euler Diagrams and Syllogistic Arguments
This section presents another form of argument called syllogistic argument (or syllogism). To determine if a syllogistic argument is valid or invalid, we will construct Euler diagrams.

A Little History: Syllogistic argument is a deductive process of arriving at a conclusion developed by Aristotle around 350 B.C. Aristotle studied the relationships between four types of statements: 1. All _________ are _________. 2. No _________ are _________. 3. Some ______ are _________. 4. Some ______ are not ______.

A Little History: Examples of these four statements are:
1. All musicians are thin. 2. No musicians are thin. 3. Some musicians are thin. 4. Some musicians are not thin.

A Little History: Since Aristotle’s time, other statements have been added, two of which follow: ______ is a _________. ______ is not a ______. Examples of these two statements are: Mitch is a musician. Mitch is not a musician.

Symbolic vs. Syllogistic Argument:
Look at the top of page 136 in the text for an interesting table comparing symbolic arguments to syllogistic arguments. The words or phrases in the arguments differ, as does the method of determining validity. This is a useful table to refer to when you are working on a mixture of the two types of problems.

Venn Diagrams and Aristotle’s Four Statements:
Aristotle’s four statements can be represented by Venn diagrams. Look at the chart near the bottom of page 136 in the text. Note how the Venn diagrams represent the statements: All As are Bs, No As are Bs, Some As are Bs, and Some As are not Bs. Notice the placement of the tiny black box, which represents the element being discussed in the statement.

Euler Diagrams: Euler diagrams are used to determine whether an argument is valid or invalid (a fallacy). Euler diagrams use circles to represent sets in syllogistic arguments. Study Example 1 on page 137 in the text. The solution illustrates how to change written statements into an Euler diagram. It also shows how to use the diagram to determine the validity of the syllogism.

Euler Diagram Examples:
Determine whether the following syllogism is valid or is a fallacy. 1. All As are Bs. All Bs are Cs.  All As are Cs.

Determine whether the following syllogism is valid or is a fallacy. 1. All As are Bs. All Bs are Cs.  All As are Cs. A B C U

Determine whether the following syllogism is valid or is a fallacy. 1. All As are Bs. All Bs are Cs.  All As are Cs. A B C U This argument is valid, since all As are inside circle C.

2. All doctors have college degrees. Tong is a doctor.  Tong has a college degree.

2. All doctors have college degrees. Tong is a doctor.  Tong has a college degree. Let D stand for Doctors. Let C stand for a College degree.

2. All doctors have college degrees. Tong is a doctor.  Tong has a college degree. Let D stand for Doctors. Let C stand for a College degree. D C U Tong This argument is valid.

3. All doctors have college degrees. Tong has a college degree.  Tong is a doctor.

3. All doctors have college degrees. Tong has a college degree.  Tong is a doctor. Let D stand for Doctors. Let C stand for College. Let T stand for Tong.

3. All doctors have college degrees. Tong has a college degree.  Tong is a doctor. Let D stand for Doctors. Let C stand for College. Let T stand for Tong. D C U T

3. All doctors have college degrees. Tong has a college degree.  Tong is a doctor. Let D stand for Doctors. Let C stand for College. Let T stand for Tong. D C U T There are two places that Tong can be placed. Since one of them involves Tong not being a doctor, the argument is Invalid.

4. No dogs are cats. Max is not a cat.  Max is a dog.

4. No dogs are cats. Max is not a cat.  Max is a dog. D C U Max Max

4. No dogs are cats. Max is not a cat.  Max is a dog. D C U Max Max The argument is invalid since there are two possible places for Max. One of which is not a dog.

5. Some politicians are stuffy. Todd Hall is a politician.  Todd Hall is not stuffy.

5. Some politicians are stuffy. Todd Hall is a politician.  Todd Hall is not stuffy. Let T stand for Todd Hall.

5. Some politicians are stuffy. Todd Hall is a politician.  Todd Hall is not stuffy. Let T stand for Todd Hall. U P S T

5. Some politicians are stuffy. Todd Hall is a politician.  Todd Hall is not stuffy. Let T stand for Todd Hall. U P S T The argument is invalid since there are two places that Todd Hall can be, and one of them is “stuffy”.

Look at Example 5 on page 139 in the text. Notice all of the different possible diagrams representing the arguments. Since at least one of the diagrams shows that the conclusion does not follow from the given premises, the argument is invalid.

Look at Example 6 on page 139 in the text. Notice how the diagram represents the statements, and that the conclusion is valid. Also note that the conclusion is false. Obviously neurosurgeons are college graduates. What happened?

Look at Example 6 on page 139 in the text. Notice how the diagram represents the statements, and that the conclusion is valid. Also note that the conclusion is false. Obviously neurosurgeons are college graduates. What happened? When working with Euler diagrams, an argument is valid if the conclusion follows from the given premises. We are concerned whether the conclusion is valid or invalid, not true or false.

Congratulations! You have now completed the PowerPoint slides for Chapter 3.

Logic 3.1 Statements and Logical Connectives

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