PHIL 117 EXAM #2

Directions: Download this exam and answer the questions using your word processing software. You may print it, fill it in by hand, and then scan it if you’d like. If you do so, be sure that all answers are legible. When you are finished, please upload the exam to Canvas as either an MSWord document (.docx) or Adobe document (.pdf). Be sure you have answered all of the questions as thoroughly as needed, as per the directions in each section. You will have the full class session to finish. If you use out-of-class sources to answer questions, you must cite the sources you have used. The test is worth 200 points.

True/False Questions (20 points total): Indicate whether the following statements are true or false.

1. Some statements are valid. ________________________________________________________

2. Valid arguments must have true conclusions. ________________________________________________________

3. If an argument has a false conclusion, then it must be invalid. ________________________________________________________

4. The complete truth table for three distinct variables has 8 rows. ________________________________________________________

5. If the antecedent of a conditional statement is false, then the conditional must be true. ________________________________________________________

6. There may be more than one minor logical operator in a statement. ________________________________________________________

7. On a completed truth table, the column under the main operator of a symbolic expression indicates the truth value for that expression.

________________________________________________________

8. A well-formed formula (WFF) is always grammatically correct and never vague.

________________________________________________________

9. The following statement is a well-formed formula: ∼[P ⦁ ∼(∼∼Q → R ⌵ S)].

________________________________________________________

10. If the form of an argument is valid, then all arguments with that form will be valid.

________________________________________________________

Short Answer Questions (30 points total): Write your answer to the following questions. As a rule of thumb, answers may be as short as a single word, but shouldn’t need to be longer than four sentences.

1. What is the definition of validity?

2. What are the two conditions necessary for an argument to be sound?

3. Under what conditions is the biconditional operator true?

4. Under what conditions is the conditional operator false?

5. Under what conditions is the disjunction operator false?

6. Under what conditions is the conjunction operator true?

7. Provide the truth table for the negation operator, using the variable “P”.

8. Explain what it means for a statement to be a tautology and provide an example of a tautological statement.

9. Explain what it means for a statement to be a contradiction and provide an example of a contradictory statement.

10. Explain what it means for a statement to be contingent and provide an example of a contingent statement.

Exercises Section 1 – Symbolizing English Sentences (30 points total): Provide the symbolic form of the arguments below using variables of your choice. (DO NOT CONSTRUCT A TRUTH TABLE. YOU DO NOT NEED TO INDICATE VALIDITY.)

You may copy and paste the following symbols for your truth tables: Negation: ∼ Disjunction: ⌵ Conjunction: ⦁ Conditional: → Biconditional: ↔

1. If Grover Cleveland was the 22nd president of the United States and also the 24th president of the United States, then he served non-consecutive terms in office. Grover Cleveland was the 22nd president of the United States and also the 24th president of the United States. So, Grover Cleveland served non-consecutive terms in office.

2. If Susan is either extraordinarily talented or extraordinarily motivated, then she will receive a 4.0 at Yale. It’s false that Susan is extraordinarily motivated, but she is extraordinarily talented. So, Susan will receive a 4.0 at Yale.

3. Either Smith or Jones has four coins in their pocket. Smith has four coins in his pocket if and only if Cathy paid him on time. Jones has four coins in his pocket if and only if Barbara paid him on time. So, either Cathy paid Smith on time, or Barbara paid Jones on time.

4. We are either alive, or we are not alive. If we are alive, then we do not experience death. If we are not alive, then it’s not the case that we have experiences. If it’s not the case that we have experiences, then we do not experience death. So, either we do not experience death or we do not experience death.

5. Modern science is capable of genetically modifying brain cells to activate when stimulated with light. If this is true, and we had some way to disperse the modified genes through the gene pool, then this technology could be used to effectively mind control people through their television sets. Thus, if we had some way to disperse the modified genes through the gene pool, we could take over the world or we could go eat some burritos at Casa.

Exercises Section 2 – Truth Tables for Statement Types (40 points total): Complete the following two steps. First, construct a complete truth table for the following symbolic statements. Second, indicate whether the statement is a tautology, a contradiction, or contingent.

1. P → ∼P

2. ∼(T ⦁ ∼T) ⌵ S

3. (∼A ⌵ B) ↔ (A → B)

4. [(P ⦁ Q) ⌵ ∼P] → Q

5. ∼[(P ⦁ R) → (R ⌵ ∼R)]

Exercises Section 3 – Truth Tables, Arguments, and Validity (40 points total): Complete the following steps. First, construct a complete truth table for the following arguments. Then indicate whether the argument is valid or invalid. *If it is invalid*, write the truth values for the atomic statements that show the argument to be invalid. You may use additional paper. If you do so, please be sure to clearly mark which question you are answering.

1.

1. P ⦁ Q

2. ∴ P → Q

2.

1. P ↔ (P ⌵ ∼Q)

2. ∼Q

3. ∴ P

3.

1. ∼(A ⌵ ~B)

2. C → B

3. ∴ ∼(C ⌵ A)

4.

1. A → (B ⦁ C)

2. B ↔ (∼C → ∼B)

3. ∴ ∼B ⦁ C

Terminology

Logic: “the study of methods for evaluating whether the premises of an argument adequately support its conclusion.”

· Very broadly, the study of the principles of good reasoning.

Reasoning: A psychological state where one or more beliefs are derived from one or more (typically distinct) beliefs.

Argument: In philosophy, an argument is the fundamental unit of reasoning. It is a series of statements, at least one of which is a premise and at least one of which is a conclusion. The conclusion is always related to the premises by way of an inference.

Inference: The relation between premises and a conclusion. Conclusions are inferred from premises (but never vice versa).

Statement: A statement is a sentence that is either true or false, or “has truth value”. (e.g. declarative sentence, assertion, claim, proposition) It is distinct from questions, commands, and exclamations (which do not have truth value).

Premise: A statement that supports a conclusion; the evidence provided for a claim.

Conclusion: A statement that is supported by premises, the claim that is being made.

Validity: “A valid argument is one in which it is necessary that, if the premises are true, then the conclusion is true.”

· Informally, in a valid argument, the conclusion follows from the premises.

· A property of arguments but not statements.

· Validity is almost always determined by the form of the argument, and not the content.

Formally Valid: An argument that is valid in virtue of its form.

Deduction: A deductive argument is one in which the premises are intended to guarantee the truth of the conclusion

· Deductive arguments are valid or invalid, not strong or weak.

Induction: “An inductive argument is one in which the premises are intended to make the conclusion probable, without guaranteeing it.”

· Inductive arguments are strong or weak, not valid or invalid.

Inductive Strength: An inductive argument is inductively strong if and only if, should all the premises be true, then the conclusion is likely to be true.

Truth (informally):A property of statements but not arguments. A statement is true when it corresponds to reality.

· e.g. “Ronald Reagan is the 44th and current president of the U.S.A.”

· e.g. “Barrack Obama is the 44th and current president of the U.S.A.”

Soundness (informally): A sound argument is one in which all of the premises are true and it is a valid argument.

· If an argument is sound, then the conclusion must be true.

· An unsound argument is one that is either invalid, has at least one false premise, or both.

· A property of arguments but not statements.

Tautology: A statement is a tautology when it cannot possibly be false.

Contradiction: A statement is a contradiction when it cannot possibly be true.

Contingency: A statement is contingent when it can be either true or false.

Logical Consistency: Two statements are consistent when they can both be true.

Logical Equivalence: Two statements are logically equivalent if the truth of either one entails the truth of the other, and the falsity of either one entails the falsity of the other.

e.g. P ↔ ∼∼P

Fallacy: An error in reasoning. (Only applies to arguments, not statements)

Equivocation: “A fallacy that occurs when a word (or phrase) is used with more than one meaning in an argument, but the validity of the argument depends on the word’s being used with the same meaning throughout.”

Circularity (Begging the Question): A fallacy that occurs in an argument when the truth of one or more premises depend upon the truth of the conclusion.

Counterexample: A counterexample to an argument form is a substitution instance in which the premises are true and the conclusion is false.

Premise Indicator Words Conclusion Indicator Words

because

since

for

as

evidenced by

seeing that

in that

the reason being

given that

indicated by

owing to

due to

is inferred from

therefore

so

hence

consequently

thus

accordingly

implies that

entails that

follows that

in conclusion

proves that

we may infer that

deduce that

SYMBOLIZATION AND THE FIVE FAMOUS FORMS

· Argument form: A pattern of reasoning.

· Symbolizing an argument: Symbolize each statement by *uniformly *replacing statements with variables while preserving the most logically sensitive form.

· Substitution instance: an argument that results from uniformly replacing the variables in a symbolic argument with statements or terms.

Examples for symbolized statements:

1) *No operator (atomic statement)*

Statement: “Geese like to beat up kids.”

Symbolized: “A”

2) *Negation (NOT)*

Statement: “It is not the case that geese like to beat up kids.”

Symbolized: “NOT A”

3) *Disjunction (OR)*

Statement: “Either geese like to beat up kids, or seagulls like to poop on people.”

Symbolized: “A OR B”

4) *Conjunction (AND)*

Statement: “Geese like to beat up kids and seagulls like to poop on people.”

Symbolized: “A AND B”

5) *Conditional (IF…THEN)*

Statement: “If geese like to beat up kids, then seagulls like to poop on people.”

Symbolized: “IF A THEN B”

6) *Biconditional (IF AND ONLY IF)*

Statement: “Geese like to beat up kids if and only if seagulls like to poop on people.”

Symbolized: “A IF AND ONLY IF B”

FIVE FAMOUS FORMS

Modus Ponens:

An Argument

1. If voter suppression is an effective means to win swing states, then you should expect voter suppression to be more prevalent in swing states.

2. Voter suppression is an effective means to win swing states.

3. So, you should expect voter suppression to be more prevalent in swing states.

*Symbolized form*:

1. If P, then Q.

2. P.

3. So, Q.

1. If voter suppression is an effective means to win swing states, then you should expect voter suppression to be more prevalent in swing states.

2. It is not the case that you should expect voter suppression to be more prevalent in swing states.

3. So, it is not the case that voter suppression is an effective means to win swing states.

*Symbolized form*:

1. If P, then Q.

2. NOT Q.

3. So, NOT P.

1. If voter suppression is an effective means to win swing states, then you should expect voter suppression to be more prevalent in swing states.

2. If you should expect voter suppression to be more prevalent in swing states, then you should expect lower voter turnout in swing states.

3. So, if voter suppression is an effective means to win swing states, then you should expect lower voter turnout in swing states.

*Symbolized form:*

1. If P, then Q.

2. If Q, then R.

3. So, if P, then R.

1. Either cats are cute, or dogs are dumb.

2. It is not the case that dogs are dumb.

3. So, cats are cute.

OR

1. Either cats are cute, or dogs are dumb.

2. It is not the case that cats are cute.

3. So, dogs are dumb.

*Symbolized form*:

1. Either P or Q.

2. Not P.

3. So, Q.

OR

1. Either P or Q.

2. Not Q.

3. So, P.

Constructive Dilemma

1. Either republicans will win the next election, or democrats will win the next election.

2. If republicans win the next election, then we should expect fewer fiscal regulations.

3. If democrats win the next election, then we should expect more environmental regulations.

4. So, either we should expect fewer fiscal regulations, or we should expect more environmental regulations.

*Symbolized form*:

1. Either P or Q.

2. If P, then R.

3. If Q, then S.

4. So, either R or S.

__Chapter 1 Section 3 Reference Document__

Terms:

· “Categorical Statement” A statement that relates two classes or categories, where a class is a set or collection of things.

· There are four kinds of categorical statements. They have these forms:

#1) [Universal Affirmative]: “All S are P”

#2) [Universal Negative]: “No S are P”

#3) [Particular Affirmative]: “Some S are P”

#4) [Particular Negative]: “Some S are not P”

· EXAMPLE: “All pigs are mammals.” is a universal affirmative.

· EXAMPLE: “No circles are shapes that possess angles.” is a universal negative.

· EXAMPLE: “Some vampires are creatures with souls.” is a particular affirmative.

· EXAMPLE: “Some vampires are not creatures with souls.” is a particular negative.

· “Argument Form”: A pattern of reasoning

· Valid Argument Form: A pattern of reasoning that is *always *valid for any substitution instance.

· Invalid Argument Form: A pattern of reasoning that is *always *invalid for any substitution instance.

· “Substitution Instance”: A substitution instance of an argument form is an argument that results from uniformly replacing the variables in that form with statements or terms.

· “Symbolic Argument”: A symbolic argument is an argument form in which statements or terms have been uniformly replaced with variables, while retaining the most logically sensitive version of the argument.

· “Counterexample”: A counterexample to an argument form is an argument that has obviously true premises and an obviously false conclusion.

Counterexample Method: to construct a counterexample:

1. Symbolize the argument, so that statements and terms have been replaced with variables while the most logically sensitive form of the argument has been preserved.

2. Construct a substitution instance to that symbolic argument such that the premises are all obviously true and the conclusion is obviously false.\

REFERENCE NOTES FOR CHAPTER 7 SECTION 1 [PAGES 279-303]

Terms

· Atomic Statement: A statement that does not have any other statement as a component.

· EXAMPLE: “I like hats.”

· NOTE: Does not have *any* logical operators in the statement.

· NOTE: Will always be symbolized by a single variable, such as “P”.

· Compound Statement: A statement that has at least one atomic statement as a component.

· EXAMPLE: “I do not like hats.”

· EXAMPLE: “Either I like hats, or I like cats.”

· EXAMPLE: “If I like hats, then I like cats.”

· NOTE: Will always have at least one logical operator, and can have more.

· NOTE: May have only one variable, provided a negation is attached to it, such as “~P”.

· Main Logical Operator: The logical operator that governs the largest component or components of a compound statement.

· Minor Logical Operator: A logical operator that governs a smaller component or components.

· Well-formed formula (WFF): A grammatically correct symbolic expression.

· Statement variable: A lowercase letter (e.g. a variable) that serves as a placeholder for any statement.

The Logical Operators Take 2 [BOOK PAGES 281-290]

NEGATION: “NOT” = ∼

EXAMPLE: ∼P

DISJUNCTION: “OR” = ∨

EXAMPLE: P ∨ Q

CONJUNCTION: “AND” = ●

EXAMPLE: P ● Q

CONDITIONAL: “IF…THEN…” = →

EXAMPLE: P → Q

BICONDITIONAL: “…IF AND ONLY IF…” = ↔

EXAMPLE: P ↔ Q

The conclusion symbol (not a logical operator)

Conclusion = “∴” ( or “/ ” )

Disambiguating Statements and Operator Scope

Sometimes, you will have compound statements with multiple operators. When this happens, the English and symbolic expressions might be *ambiguous*, meaning that they could allow two or more interpretations. That’s bad for regular everyday language, and it’s really bad for math, for a variety of reasons. Consider, for example, what would happen if we didn’t have PEMDAS (the mathematical rules for order of operations). It would be impossible to find the ‘correct’ solution to basic arithmetic problems like “3+4 x 5”

So, we need a way to disambiguate potentially ambiguous statements in logic. Consider, for example, the following statement: “It’s not true that if you like cats, then you hate dogs”.

If we were to symbolize this right now, we might write it like this:

IF NOT P, THEN Q

or

∼P ➝ Q But this is technically mistaken. When P is “you like cats” and Q is “you hate dogs”, that expression says that if you don’t like cats, then you hate dogs. But that means something different!

*The statement is saying that a person can like both*. Here is the correct way to symbolize it:

∼(P ➝ Q)

The parenthesis make the negation operator (∼) attach to *the entire conditional statement*, rather than just the “P”. What this means is that the negation needs to be treated as the main operator. Our first mistake was to treat the conditional (➝) as the main operator. That changed the meaning.

Parentheses are used to dictate the scope of operators, and they can determine which operator is the main operator, and which are minors.

Now, here is an example of a scary and confusing expression:

(P ➝ M) v (T ● Q)

What is the *main operator *in this expression? It is whatever operator governs the largest component. In this case, it is the disjunction OR operator, the “v”. This disjunction governs two *compound statements*, each of which is a disjunct. (~P ➝ M) is one disjunct, and (T ● Q) is the other.

NOTE: If there is only one operator outside of all parentheses and brackets, that operator will be the main operator.

NOTE: The negation is only the main operator when: (a) it is the only operator outside of all parentheses and brackets, or (b) it is the only operator in the statement.

Here is a slightly less scary symbolic expression: (~P ➝ Q)

What is the *main operator *in this expression? It is the conditional, the “➝”. The conditional governs the relation between the antecedent (~P) and the consequent (Q). The negation only governs the truth of the atomic statement P.

1. It’s not true that if we have freewill then we are always responsible for our actions.

~(F -> R)

2. Either Cthulhu exists and the world will end, or the world won’t end.

(C * E) v ~E

3. It’s not the case that reality is real and things are as they seem.

~(R * S)

4. I’m Pinocchio and I’m not Pinocchio if and only if I am not a puppet.

P * (~P <-> ~ U)

5. The largest celestial body is not a star, nor is it a planet.

~S v ~P

~(S * P)

6. If nothing exists except the universe (and all else is mere convention) then it is not the case that you and I are different.

(N * C) -> ~D

7. If Pigs can fly but pigs can’t fly then pigs can fly or FSM is real.

(P * ~P) -> (P v R)

__Truth Functions and Truth Table Basics__

NOTE: The Truth Tables below may appear slightly different from when they are on the board or in the book. This is because I am using MSWord Tables to make writing out truth values easier. I do not mind if you write them like they are below, with each variable, statement, and premise given a unique column. Alternatively, you could construct them like they are in the book, which is a bit quicker. See pages 303 – 329

Truth tables are *models*, similar to a graph or Venn diagram, which let us visually represent the truth values of statements.

They express all the logically possible truth values of the logical operators for all the logically possible truth values of the (relevant) statements.

Truth table for a simple statement, ‘x’.

x |

T |

F |

Truth table for two simple statements, x, y (no operator)

x | y |

T | T |

T | F |

F | T |

F | F |

Truth table for 3 distinct statements, x, y, z

x | y | z |

T | T | T |

T | T | F |

T | F | T |

T | F | F |

F | T | T |

F | T | F |

F | F | T |

F | F | F |

NOTE: A truth table for 4 distinct statements would have 16 possible combinations of truth values. A table with 5 statements would have 32. 6 statements would have 64. Truth tables get unwieldy when using more than 3 statements.

Below are the truth tables for our five logical operators.

NOTE: Truth values for logical operators are written under the operators, not the variables.

__NEGATION: True when x is false, false when x is true__

x | ∼x |

T | F |

F | T |

__CONJUNCTION: True only when x and y are both true, false otherwise__

x | y | x ⦁ y |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

__DISJUNCTION: Only false when x and y are both false, true otherwise__

x | y | x ⌵ y |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

__CONDITIONAL: Only false when x is true and y is false, true otherwise__

x | y | x → y |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

__BICONDITIONAL: True when both x and y have same truth value, false when truth values are different.__

x | y | x ↔ y |

T | T | T |

T | F | F |

F | T | F |

F | F | T |

__Complex Statements and Identifying the Main Operator__

Consider the truth table for the following statement: A → B ⦁ ∼A

Since this statement isn’t well formed, we can’t tell which operator is the main operator. When this happens, there will be more than one way to make the truth table. So, we need to know the __order of operations__ when filling in the truth tables. Let’s say the above means (A → B) ⦁ ∼A. The main operator is now the conjunction, “⦁”. Here is a truth table below.

A | B | (A → B) ⦁ ∼A |

T | T | T F F |

T | F | F F F |

F | T | T T T |

F | F | T T T |

I’ve bolded the truth values for the main operator, because they tell you the truth values for the entire statement. The negation, “∼” and conditional, “→” were minor operators. I determined their truth values first. In the above case, it wouldn’t matter which of the minor operators you wrote the truth values for first. But in general, follow these rules:

1. You will always determine the truth value of the main operator __last__ .

2. First, solve for operators within parenthesis or brackets.

3. If there are two operators within a pair of parenthesis, for example (P ⌵ ∼Q), solve for the minor operator first and the main operator last. In the example, the negation is minor so we will solve for it first. After that, we determine the truth value of the main operator, the disjunction.

4. If there are both parenthesis and brackets, for example [(P ⌵ ∼Q) ⦁ Q], start with the innermost parenthesis. So, we would solve for (P ⌵ ∼Q), and then solve for the conjunction ⦁ Q.

5. When two negations are side by side, the main operator is to the far left. So you would solve for the inside negation first, and the outside negation second.

a. For example: ∼∼P

6. If there are two or more operators outside parenthesis (or brackets), and one is a negation, the other operator is the main operator. So, determine truth values for the negation first, and the main operator second.

a. For example: ∼(P ⌵ ∼Q) ⦁ Q {Conjunction is main}

b. For example: P ⌵ ∼Q {Disjunction is main}

c. For example: ∼∼(P ⌵ ∼Q) → ∼∼(Q ⌵ ∼P) {Conditional is main}

7. There are many possible shortcuts one can take when doing truth tables. There is nothing wrong with this once you are familiar with them. On a test, I’ll expect you to be able to show that you know how to construct them fully.

__Arguments and Truth Tables__

Writing the truth table for entire arguments isn’t difficult when you can do it for statements. See the book, Chapter 7.3 for more information. Below, I’ve written truth tables for arguments by merely adding additional columns, one for each premise and one for the conclusion. You can also separate statements by using a comma, which is how they are done in the book.

MODUS PONENS

1. P → Q

2. P

3. ∴ Q

P | Q | 1. P → Q | 2. P | 3. ∴ Q |

T | T | T | T | T |

T | F | F | T | F |

F | T | T | F | T |

F | F | T | F | F |

This is the truth table for Modus Ponens. I’ve included the premises numbers above for clarity, although you do not need to do this. The bolded P Q columns at the left represent all the logically possible truth values for those two variables. The line titled “1. P → Q” represents the first premise of modus ponens. Below it we see the regular truth values for the conditional. Since its form is the same as the x → y truth table above, it has the same truth values. The next column to the right expresses the second premise of Modus Ponens, “2. P”. It has the same truth values as the P column to the far left, since it merely restates a variable without adding any logical operator. The final column is the conclusion, “3. ∴ Q”. It also has the same truth value as the “Q” column to the left. Below are truth tables for other famous forms.

NOTE: A truth table can be made for any argument, valid or invalid.

NOTE: If a statement in an argument isn’t well formed, then you may not be able to determine the truth value for that statement.

MODUS TOLLENS

1. P → Q

2. ∼Q

3. ∴∼P

P | Q | P → Q | ∼Q | ∴∼P |

T | T | T | F | F |

T | F | F | T | F |

F | T | T | F | T |

F | F | T | T | T |

DISJUNCTIVE SYLLOGISM

1. P ⌵ Q

2. ∼Q

3. ∴ P

P | Q | P ⌵ Q | ∼Q | ∴P |

T | T | T | F | T |

T | F | T | T | T |

F | T | T | F | F |

F | F | F | T | F |

HYPOTHETICAL SYLLOGISM

1. P → Q

2. Q → R

3. ∴ P → R

P | Q | R | P → Q | Q → R | ∴ P → R |

T | T | T | T | T | T |

T | T | F | T | F | F |

T | F | T | F | T | T |

T | F | F | F | T | F |

F | T | T | T | T | T |

F | T | F | T | F | T |

F | F | T | T | T | T |

F | F | F | T | T | T |

CONSTRUCTIVE DILEMMA

1. P ⌵ Q

2. P → R

3. Q→ S

4. ∴ R ⌵ S

P | Q | R | S | P ⌵ Q | P → R | Q→ S | ∴ R ⌵ S |

T | T | T | T | T | T | T | T |

T | T | T | F | T | T | F | T |

T | T | F | T | T | F | T | T |

T | T | F | F | T | F | F | F |

T | F | T | T | T | T | T | T |

T | F | T | F | T | T | T | T |

T | F | F | T | T | F | T | T |

T | F | F | F | T | F | T | F |

F | T | T | T | T | T | T | T |

F | T | T | F | T | T | F | T |

F | T | F | T | T | T | T | T |

F | T | F | F | T | T | F | F |

F | F | T | T | F | T | T | T |

F | F | T | F | F | T | T | T |

F | F | F | T | F | T | T | T |

F | F | F | F | F | T | T | F |

__Determining Validity:__

The five famous forms are all valid. And we can see that on the truth table, because there is no line in which the conclusion is false and the premises are all true. (Remember what is needed for a counterexample?)

Validity: An argument is valid means that if the premises are true, then the conclusion must be true.

For truth tables, we can see every single logically possible way the premises can be true. In the above arguments, every time all the premises are true, then the conclusion is also true. This is because they are valid.

Invalid arguments will have __at least one line__ on the truth table in which the premises are all true, and the conclusion is false. This tells us that the argument form is invalid.

For example:

1. A → B

2. B

3. ∴ A

A | B | A → B | B | ∴ A |

T | T | T | T | T |

T | F | F | F | T |

F | T | T | T | F |

F | F | T | F | F |

I’ve bolded the line in which the premises are both true, but the conclusion is false. This occurs when the antecedent “A” is false, and the consequent “B” is true. Think:

1. If it’s raining, then the sidewalks are wet.

2. The sidewalks are wet.

3. So, it is raining.

The sidewalks could be wet for some other reason (and the antecedent false), and hence the conclusion could be false even when the premises are true. It is invalid.

Abbreviated Truth Tables

Relatively quick way to show that an argument is valid or invalid using truth tables.

Useful for arguments containing 4 or more variables.

1. Write out the truth table but do not fill in the possible truth values for the variables.

2. Assume the conclusion is false.

3. Assume the premises (main operator of each) are true.

4. Try to consistently fill in truth values in such a way that the assumptions in 2 and 3 are confirmed. Start with minor operators and then work backwards to the variables.

5. If there is such a way, the argument is invalid. Fill in the truth values for the variables to the left.

6. If there is no consistent way to assign truth values such that the premises are all true, then the argument is valid. Do not fill in values for the variables to the left.

Exercises page 329

Tautology

· If an argument has a tautology for a conclusion, then it is valid *even if the conclusion is unrelated to the premises*.

Contradiction

· If an argument has a contradiction in the premises, then it is valid (but unsound).

Contingency

Logical Equivalency: Two statements are logically equivalent if and only if, for every row of their truth table, the truth value of those two statements is identical.

EX: P ↔ Q, (P → Q) ⦁ (Q → P)

Logical Contradictory: Two statements are logically contradictory if and only if, for every row of their truth table, the truth value of those two statements is different.

EX: P, ∼P

EX: P → Q, ∼(∼P ⌵ Q)

P | Q | P → Q | ∼(∼P ⌵ Q) |

T | T | T | F F T |

T | F | F | T F F |

F | T | T | F T T |

F | F | T | F T T |

· If an argument has contradictory premises, then it is valid (but unsound).

Logical Consistency: Two statements are logically consistent if and only if there exists at least one row of their truth table such that both statements are true.

EX: P, Q

EX: P → Q, P ⌵ Q

Logical Inconsistency: Two statements are logically inconsistent if and only if there does not exist a row of their truth table such that both statements are true.

EX: (P → Q) ⦁ P, ∼Q

· If an argument has inconsistent premises, then it is valid (but unsound).

CH 8 Proofs

· Proof: “A series of steps that leads from the premises of a symbolic argument to its conclusion.”

Eight Rules of Inference

Modus Ponens {MP}

- P→Q
- P
- ∴ Q

Modus Tollens {MT}

- P→Q
- ∽Q
- ∴ ∽P

Disjunctive Syllogism {DS}

- P ⌵ Q
- ∽P
- ∴ Q

Hypothetical Syllogism {HS}

- P→Q
- Q→R
- ∴ P→R

Constructive Dilemma {CD}

- P ⌵ R
- P→Q
- R→S
- ∴Q ⌵ S

Addition {ADD}

- P
- ∴ P ⌵ Q

Conjunction {CONJ}

- P
- Q
- ∴ P ⦁ Q

Simplification {SIMP}

- P ⦁ Q
- ∴P / ∴Q

These rules are used to construct proofs which show that an argument is valid.

Addition {ADD}

· Because the “or” operator is inclusive, if you know the truth of some statement A then you know the truth of any disjunctive statement that contains A.

· EX: “It’s raining. So, Either it’s raining or it’s sunny”

· EX: “Eric is learning logic. So, either Eric is learning logic or Jesus is a fire dragon.”

What seems weird about these conclusions isn’t the logic. The logic is fine, and we could construct a truth table showing that. Rather, no one talks like this. It would serve no purpose in everyday conversation and potentially be misleading or confusing. In other words, it violates a norm of communication, even though the inference is valid.

Conjunction {CONJ} and Simplification {SIMP}

· These rules are rather obvious – so obvious that it might be tempting to skip them when solving proofs. Avoid doing this.

· DO NOT make the mistake of thinking you can conjoin or simplify from anything other than the conjunction operator. It does not work for disjunction, conditionals, or biconditionals.

Proofs and Line Justification

We can show that certain arguments are valid using these rules.

For example:

- (P v Q) → R
- P
- So, R v S
- P v Q
- R
- R v S

QED (quod erat demonstrandum) = (which was to be proven)

But we need to show how we proved it, like showing our work. We do this by adding parenthesis or squirrely brackets to the right of the lines we added, like so:

- P v Q { }

Next, we fill them in with two things:

- The line(s) we used to attain the new statement
- The inference rule we used to attain the new statement

So,

- P v Q {2 ADD}
- R {1,4 MP}
- R v S {5 ADD}

Ned’s Wish List Strategy

· Solve proofs by working backwards

- Where do you want to get to when doing a proof? Answer: the conclusion.
- Using what you have and one of your inference (or equivalence) rules, can you get the conclusion? If so, great! If not, you’ll need at least one extra step. Let’s keep track of what we want by placing it under a “wishlist”.

EXAMPLE:

- A ⦁ (B ⦁ C) WISHLIST: -C
- ∴ C -B ⦁ C
- B ⦁ C {1, SIMP}
- C {3, SIMP}
- [(P ⌵ Q) ⌵ R] ⦁ ∼Q WISHLIST: P, P ⌵ Q, (P ⌵ Q) ⌵ R, ∼R, ∼Q
- ∼Q → ∼R
- ∴P
- (P ⌵ Q) ⌵ R {1 SIMP}
- ∼Q {1, SIMP}
- ∼R {2, 5 MP}
- (P ⌵ Q) {4,6 DS}
- P {5,7 DS}

Proofs and Invalidity

NOTE: You cannot construct a proof for an invalid argument. Were you to try, you’d get stuck.

HANDOUT

BOOK EXERCISES pg 360-363