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Syllabus
Lecture Notes
Exercises

Lecture Notes Unit 1

Contents

(In lazy mode, lowercase letters, except ‘v’, are automatically changed to uppercase.)
WTF Cat

Basic Definitions

Logic is the science of the correctness or incorrectness of reasoning.

Or, more to the point

Logic is the study of the evaluation of arguments.

A statement is a declarative sentence, or part of a sentence, that can be either true or false.

How many squares?

How many statements are there in the example below?

Boston is the largest city in Massachusetts, and Springfield is the second largest.

Answer: Three: the two halves plus the whole

A proposition is what it is meant by a statement, the idea or notion it expresses.

(This might be the same for different sentences—e.g. translations from one language into another.)

An argument is a collection of statements or propositions, some of which are intended to provide support or evidence in favor of one of the others.

The premises of an argument are those statements or propositions in it that are intended to provide the support or evidence.

The conclusion of an argument is that statement or proposition for which the premises are intended to provide support.

(The intention need not be fulfilled.)

Example Arguments

First, identify the conclusion. Note that it need not be the last sentence.

God is defined as the most perfect being. A perfect being must have every trait or property that it’s better to have than not to have. It is better to exist than not to exist. Therefore, God exists.

Deductive or Inductive? Deductive

Hillary Clinton must be a communist spy. She supports socialized health care, and everyone who supports socialized health care is a communist spy.

Deductive or Inductive? Deductive

It has rained more than 15 inches per year in Amherst every year for the past 40 years. So you can safely bet it will rain more than 15 inches in Amherst this year.

Deductive or Inductive? Inductive 

Professor Chappell said that the ratio of female to male students in the class was exactly 3:1. This means that there are 112 female students in the class, because there are 148 students in the class total.

Deductive or Inductive? Inductive. (Because of the “said that”.)

Google has a feature to search only pages changed in the past month. It stands to reason that Bing allows you to limit searches to pages changed in the past month as well, since the two search engines tend to provide the same features.

Deductive or Inductive? Inductive 

1 is prime. 3 is prime. 5 is prime. 7 is prime. Therefore, all odd integers between 0 and 8 are prime.

Deductive or Inductive? Deductive

Jason isn’t an NRA member. Almost 90% of NRA members are Republicans, and Jason isn’t a Republican.

Deductive or Inductive? Inductive 

Induction and Deduction

Distinction is often taught in a different, and mostly outdated, way.

Using current terminology, it has to do with strength of the intended evidence.

A deductive argument is one in which the author intends the evidence to be so strong that it is impossible for the premises to be true and the conclusion false, or that the conclusion follows necessarily from the premises.

An inductive argument is one in which the author intends the evidence only to be so strong that it is improbable that the premises could be true and the conclusion false, or that the conclusion is likely true if the premises are true.

This course is almost entirely focused on deductive logic.

(Let us consider the example arguments above; notice that the prime numbers example is deductive despite reasoning from the specific to the general, and that the Jason example is inductive despite reasoning from the general to the specific.)

Strength and Weakness

A strong inductive argument is for which it actually is the case that the conclusion would probably be true if the premises were true.

A weak inductive argument is an inductive argument that is not strong.


Validity and Soundness

A valid deductive argument is one for which it actually is impossible for the premises to be true and the conclusion false, or for which the conclusion follows necessarily from the premises.

An invalid deductive argument is a deductive argument that is not valid.

A rough test for validity:

  1. First imagine that the premises are true—whether or not they actually are.
  2. Ask yourself, without appealing to any other knowledge you have, could you still imagine the conclusion being false?
  3. If you can, the argument is invalid. If you can’t, then the argument is valid.

Validity is not about the actual truth or falsity of the premises.

It’s only about what would follow from the premises if they were true.

A valid argument can have false premises.

All toasters are items made of gold.
All items made of gold are time-travel devices.
Therefore, all toasters are time-travel devices.

It’s hard to imagine these premises as true.

But if they were true, the conclusion would have to be as well.

Validity is about the process of reasoning.

There’s more to an argument’s being a good one than validity.

A good argument must also have true premises.

A factually correct argument is an argument with (all) true premises.

A sound argument is an argument that is both valid and factually correct.

A good argument is a sound argument.

What’s Possible?

ValidInvalid
False premises, False conclusion possible possible
False premises, True conclusion possible possible
True premises, False conclusion impossible possible
True premises, True conclusion possible possible

Sound arguments always have true conclusions.

Homework: Exercise Set 1A

Argument Form

Example 1:

All tigers are mammals.
No mammals are creatures with scales.
Therefore, no tigers are creatures with scales.

Example 2:

All spider monkeys are elephants.
No elephants are animals.
Therefore, no spider monkeys are animals.

These arguments have the same form:

All A are B.
No B are C.
Therefore, no A are C.

All arguments with this form are valid.

Example 3:

All Jedis are one with the Force.
Yoda is one with the Force.
Therefore, Yoda is a Jedi.

Example 4:

All basketballs are round.
The Earth is round.
Therefore, the Earth is a basketball.

These have the form:

All A are F.
X is F.
Therefore, X is an A.

All arguments with this form are invalid.

Homework: Exercise Set 1B

The Counterexample Method

A recap of two points:

Together these mean:

For any argument, if you can find another with

  1. The same form
  2. True premises and a false conclusion

then both arguments are invalid.

This is called finding a counterexample.

The basketball/Earth argument could be used as a counterexample to show the invalidity of the Jedi/Yoda argument.

Is this valid?

All bandersnatches are toves.
Some borogoves are toves.
Therefore, some bandersnatches are borogoves.

Unsure? Try to find a counterexample, such as:

All fish are aquatic animals.
Some mammals are aquatic animals.
Therefore, some fish are mammals.

If an argument is valid, it is impossible to find a counterexample. For example:

All bandersnatches are toves.
Some borogoves are bandersnatches.
Therefore, some toves are borogoves.

Homework: Exercise Set 1C

Sentential Logic

Symbolic logic is the study of the evaluation of arguments through the use of mathematically-inspired logical notation.

Mathematicans use (e.g.) the signs “+” and “=” to stand for the mathematical concepts of addition and equality.

Logicians use signs such as “∨” and “~” to stand for the logical concepts of disjunction and negation.

Sentential logic (also called propositional logic) is the simplest species of symbolic logic; it is the study of truth-functional statement connectives.

Statement connectives are words used to make complex (or molecular) statements out of simpler (atomic) ones.

Example English statement connectives:

and, or, but, if, only if, unless, not, yet (etc.)

Atomic and Molecular Statements

Example Molecular Statements:

I live in Amherst and I hate living there.

If Twilight was a good movie, then I’m crazy.

You shouldn’t take this class unless you are prepared to work hard.

(On screen, the atomic statements are red; the statement connectives blue.)

In sentential logic

So the first example would be written:

L & H

L” and “H” abbreviate the simple statements and “&” means “and”.

Functions

We borrow the notion of a function from math.

Crudely put, a function in math has one or more numbers as input, and a number as output.

Mathematical examples:

The function square root takes 4 as input and gives 2 as output. (√4 = 2)

The function + takes 5 and 7 as inputs and gives 12 as output. (5 + 7 = 12)

The technical names for input and output are argument and value.

Sentential logic deals with functions that operate on truth and falsity rather than on numbers.

We describe truth and falsity as the truth values of statements.

Truth Functions

The statement connectives of sentential logic can be understood as truth functions.

They take the truth values of simpler statements as inputs and yield the truth values of molecular statements as outputs.

Complex statements with “and” are true when both sides are true, and false otherwise.

Amherst is in Massachusetts and Boston is in Massachusetts.

Amherst is in Massachusetts and Chicago is in Massachusetts.

Let A be “Amherst is in MA”, B be “Boston is in MA” and C be “Chicago is in MA”.

  1. A & B is TRUE.
  2. A & C is FALSE.

Negation ~

The simplest truth function is negation (“not”).

It is written “~”. (This sign is called a tilde.) This is placed before the statement to which it applies.

Its output is the opposite of its input.

A ~A
T F
F T

This sign is used to translate “not”, “it is not true that”, “it is false that”, “it is not the case that”, etc.

Some other logic books use the signs “−” or “¬”.

(This is not the same as the mathematical concept of negative.) “I am not 8 feet tall.” ≠ “I am −8 feet tall.”

Conjunction &

Conjunction (“and”), unlike negation, has two inputs.

Conjunction is written “&”. This sign is called an ampersand. It goes between the two statements it connects (the conjuncts).

There are four possible combinations for the two inputs.

A B A & B
T T T
T F F
F T F
F F F

This translates “and”, “but”, “moreover”, “however”, “although”, “yet”, etc.

Some other books use the signs “•” or “∧”.

Disjunction ∨

Disjunction (“or”) is written “∨”. This sign is usually called a wedge.

Its two inputs are called its disjuncts.

A B AB
T T T
T F T
F T T
F F F

Translates “or”, “either … or …” and “unless”.

This leaves open the possibility that both sides are true. This called the inclusive or.

The word “or” is perhaps sometimes used another way in English. Compare:

  1. Either the Yankees will be AL champs or the Mets will be NL champs. (Inclusive or)
  2. Either the Red Sox will be AL champs or the Yankees will be AL champs. (Exclusive or)

Material Implication →

Material implication, also called the material conditional is written “→”. This sign is called the arrow.

A B AB
T T T
T F F
F T T
F F T

The if-part of a conditional is called the antecedent, and the then-part is called the consequent.

Some other books use the signs “⊃” or “⇒”.

This is used to translate “if … then …”, “… only if …” and “… implies that …”. But there are differences.

Since → is a truth-function, it behaves a bit differently from “if … then …”.

If Mitt Romney is the president (M), then a Democrat is running the country (D). (false)

MD (true)

If Kevin grew up in Milwaukee (G), then Kevin lived in Minnesota (L). (Unclear, but seems false.)

GL (true)

Material Equivalence ↔

Material equivalence, also called the material biconditional, is written “↔”. This sign is called a double arrow.

A B AB
T T T
T F F
F T F
F F T

Used to translate “… if and only if …”, its abbreviation “… iff …”, and “… just in case …”. But again, there are differences.

Hillary Clinton is president (H) if and only if Bernie Sanders is president (B). (false)

HB (true)

Some other books use “≡” instead.

Complex Statements

Complex statements can have more than two atomic parts

  1. The election was held on November 7th 2000, and either Bush won the election or Gore won the election.
    E & (BG)
  2. If you think ’N Sync was good or you think the Backstreet Boys were talented, then you’re crazy.
    (NB) → C
  3. I hate Justin Timberlake, but if you like Fergie, then if you don’t like Britney Spears, then we can still be friends.
    H & [L → (~BF)]

To translate these we need to use multiple connectives.

We need the parentheses for the same reason we need them in math.

The placement of parentheses determines the order in which functions are applied.

This order can matter, just as in math.

(12 ÷ 3) ÷ 4 = 1 but
12 ÷ (3 ÷ 4) = 16.

Let A, B and C be true, and X, Y and Z be false. Then

(AB) & Y is FALSE, but
A ∨ (B & Y) is TRUE.

And ~YC is TRUE, but ~(YC) is FALSE.

Evaluating Complex Statements

Work from inside parentheses outwards.

Negations apply only to what comes immediately after, and are calculated prior to anything inside the same number of parentheses.

Examples (Again, A, B, C are true; X, Y, Z are false.)

  1. ~A∨ (B & C)
    True or false? True Main connective?

  2. ~(YZ) & (AY)
    True or false? False Main connective? &

  3. ~[C → (AY)] → X
    True or false? True Main connective? Second →

  4. ~[~(C ∨ ~~A) & B]
    True or false? True Main connective? First ~

  5. A ∨ ~A
    True or false? True Main connective?

  6. Y ∨ ~Y
    True or false? True Main connective?

The main connective (or main operator) of a statement is the one used last in the calculation, having the whole statement as its scope.

Homework: Exercise Set 2A, Exercise Set 2B, Credit Exercises 1.1

Truth Tables

Suppose you don’t know the truth values of P and Q.

What can you know about P → (QP)? A lot!

There are four possibilities for P and Q: both are true, P is true and Q is false, or vice versa, or both are false.

Writing the possibilities under the atomic statements makes a table.

P (Q P)
T   T   T
T   F   T
F   T   F
F   F   F

(Here we repeat the same possibilities under both occurrences of “P”, since they abbreviate the same statement.)

By comparing the highlighted columns we can determine the truth value for the conditional sub-statement on the right.

P (Q P)
T   T T T
T   F T T
F   T F F
F   F T F

We can now use the column we just calculated, along with the column under the first P to calculate the truth value of the whole.

(I’ll change the colors to indicate which columns we’re now looking at.)

P (Q P)
T T T T T
T T F T T
F T T F F
F T F T F

Let us now highlight this final column we just filled in.

P (Q P)
T T T T T
T T F T T
F T T F F
F T F T F

The column under the main operator (here highlighted) is very important.

It tells you the truth value of the whole statement.

In this case, it tells you that this statement cannot be false.

Tautologies and Self-Contradictions

A tautology is a statement that is true for every possible assignment of truth values to its atomic parts.

P → (QP) is a tautology.

A self-contradiction is a statement that is false for every possible assignment of truth values to its atomic parts.

P & ~P is a self-contradiction.

P & ~ P
T F F T
F F T F

A contingent statement is a statement that is true for some (one or more) truth value assignments, and false for others.

P ↔ (PQ) is a contingent statement.

P (P Q)
T T T T T
T F T F F
F F F T T
F F F T F

How to Draw a Truth Table

  1. Count the number of distinct atomic statements.
  2. For n atomics, we 2n rows. (Doubles with each new one.)
    • For 1 atomic, we need 2 rows.
    • For 2 atomics, we need 4 rows.
    • For 3 atomics, we need 8 rows.
    • For 4 atomics, we need 16 rows, etc.
  3. For the first atomic, make the first half true, second half false.
  4. For the next, do half as many trues consecutively as the previous, then the same number of Fs, and repeat.
  5. Last one should alternate T, F, T, F, etc.

Suppose a statement has P, Q and R as distinct letters. You need eight rows. For P make the first four rows T, and the second four F. For Q, do two Ts, then two Fs, repeat. For R, alternate T/F.

The Process in Action

Try it yourself:

Or click here to see Prof. Klement’s answer.

Remember:

  1. Count the letters.
  2. First letter: half Ts, half Fs.
  3. Cut each half in half for next letter; repeat.
  4. Last letter alternates Ts and Fs.
  5. Rest of table, inside parentheses to out.
  6. Check final column.

Ready for 4 atomics (16 rows)?

Try it yourself:

Or click here to see Prof. Klement’s answer.

Homework: Exercise Sets 2C/3A

Logical Equivalence

Logically equivalent statements are those that necessarily have the same truth value (the same for every possible truth value assignments to their atomic parts).

We can test for logical equivalence with a combined truth table for two statements. Their final columns should match exactly.

Do ~(P & Q) and ~P & ~Q mean the same?

Try it yourself:

Or click here to see Prof. Klement’s answer.

No. These differ in truth value when P is true and Q is false, or vice-versa.

Another example.

Important: This is done like one table, not two, so P and Q are treated the same in the two statements.

Try it yourself:

Or click here to see Prof. Klement’s answer.

These are true on precisely the same rows.

In other words, they are logically equivalent.

Homework: Exercise Set 3B

Truth Tables for Arguments

We can also test the validity of an argument with a combined truth table for the premises and conclusion.

We do a combined table for the argument.

If there is any possibility that (even one row where) all the premises are true, and the conclusion false, the argument is invalid. Otherwise the argument is valid.

Try it yourself:

Or click here to see Prof. Klement’s answer.

Here there are two rows—the second and fifth—where the premises are all true and the conclusion false.

However, if not a single row has all true premises and a false conclusion, the argument is valid.

Another example. Try it yourself:

Or click here to see Prof. Klement’s answer.

This argument is valid. (We don’t know whether or not it is sound.)

Homework: Exercise Set 3C

A Table for a Real Argument

If there is a God (G), then God created everything in the universe (C). If God created everything in the Universe, then everything in the universe is good (E). It’s not the case that everything in the universe is good. Therefore, there is not a God.

Try it yourself:

Or click here to see Prof. Klement’s answer.

This argument is valid. I’ll let you think about whether or not it’s sound.


Translations

The first step is to assign a letter to each simple statement.

Usually we’ll use the first letter in the statement.

If we are going dancing, then Jessica should change her shoes and Mark should take a shower.

Use: “W” for “we are going dancing”, “J” for “Jessica should change her shoes”, and “M” for “Mark should take a shower”.

The final translation is W → (J & M).

However, you must use different letters for different statements.

Pick something to help remember the difference.

Either I’ll buy a Chevy, or I’ll buy a Ford.

Becomes: CF

Use the same letter twice only if the same simple statement is repeated.

Either the Belchertown bus is late, or the Belchertown bus is not late and the schedule is outdated.

Becomes: B ∨ (~B & S)

Translating Conjunctions

“&” translates “and”, “but”, “yet”, “although”, “however”, “moreover”, “whereas”, etc.

(These differ in connotation only.)

The “&” always goes in the middle, even if the English word begins the sentence.

Examples:

  1. Peter is intelligent but he voted for Trump.
    Translation: I & V

  2. Philosophers loves truth whereas rhetoricians love eloquence.
    Translation: P & R

  3. Although Quebec is in Canada, the people in Quebec speak French.
    Translation: Q & S

Translating Disjunctions

“∨” translates “or” and the phrase “either … or …”.

Examples:

  1. Jenna bought a copy of the book from the bookstore, or she downloaded it online.
    Translation: BD

  2. Either Peter overslept or he forgot about the meeting.
    Translation: OF

If context suggests the exclusive sense of “or”, the translation must be more complex.

Either you can keep dating Sanjukta or you can keep dating Kalinda [but not both].

Translation: (SK) & ~(K & S)  or S ↔ ~K

Translating Negations

“~” translates anything used to negate a sentence: “not”, “it is not the case that …”, “it is not true that …”, “it is false that …”.

It can be hard to spot if it appears mid-sentence, or in a contracted form as part of a “——n’t” word.

Examples:

  1. It is not true that Boston is boring.
    Translation: ~B

  2. It isn’t raining.
    Translation: ~R

  3. Kanye West is no genius.
    Translation: ~K (where K is “Kanye West is a genius.”)

Translating Conditionals

“→” loosely translates all of “if … then …”, “… if …”, “… only if …”, “… provided that …”, “… on the condition that …”, “… in case …”, “… implies that …”.

The tricky thing with “→” is that the order matters.

Examples:

  1. If Anna goes out, then Ken babysits.
    Anna goes out only if Ken babysits.
    Anna’s going out implies that Ken babysits.
    Provided that Anna goes out, Ken babysits.
    In case Anna goes out, Ken babysits.
    ALL translate to: AK

  2. Anna goes out if Ken babysits.
    Anna goes out provided that Ken babysits.
    Anna goes out on the condition that Ken babysits.
    ALL translate to: KA

Tips for getting the order of conditionals right:

Translating Biconditionals

“↔” translates the whole phrase “if and only if”, its abbreviation “iff”, as well as the phrase “just in case”.

Examples:

  1. Annemarie will make it to the conference if and only if her car is running well.
    Translation: AC

  2. Kevin will have a date this weekend just in case hell freezes over.
    Translation: KH

Hidden Conjunctions and Disjunctions

Grammatically non-compound sentences can express molecular propositions when words like “and” or “or” join names or predicates.

Examples:

  1. Massachusetts and Connecticut are in New England.
    This means: Massachusetts is in New England (M) and Connecticut is in New England (C).
    Translation: M & C

  2. Sean is either Irish or Scottish.
    This means: Either Sean is Irish (I) or Sean is Scottish (S).
    Translation: IS

Statements with “and”/“or” between names or predicates cannot always be broken into separate atomic statements.

Naomi and Kathy are roommates.

Is this a hidden conjunction?

Does this mean Naomi is a roommate and Kathy is a roommate?

It might, but more likely it means that they are roommates to each other.

It is inappropriate to translate this as N & K, since it does not express two separate thoughts. Make it just R.

You need to consider each case individually.

Homework: Exercise Set 4A

Neither … Nor … and Unless

“Neither A nor B” can be translated as either “~(AB)” or “~A & ~B”.

“Unless” roughly means “if not”.

So, “unless A, B” can be put as “~AB”.

(Like “if”, reverse the order if it’s in the middle!)

Examples:

  1. Neither the sun shone nor the stars twinkled.
    Translation: ~(ST)  or ~S & ~T

  2. Minnesota is neither an Eastern nor a Western state.
    Translation: ~(EW)  or ~E & ~W

  3. Unless you stop starting at me, I’ll throw a taco at you.
    Translation: ~ST  or ST

  4. Sarah works at the library, unless she’s been fired.
    Translation: ~FW  or WF

Necessary and Sufficient Conditions

A is a sufficient condition for B” means that A guarantees B, or that if A then B.

A is a necessary condition for B” means that B can be true only if A is, or that if B is true, A must be.

(The difference is the same as between if and only if.)

“p is necessary and sufficient for q” can be translated with “↔”.

Examples:

  1. Averaging above 50% is a sufficent condition for passing.
    Translation: AP

  2. Buying a ticket is a necessary condition for winning the jackpot.
    Translation: WB  or ~B → ~W

  3. Knowing Björk is necessary and sufficient for loving Björk.
    Translation: KL

Homework: Exercise Set 4B, Credit Exercises 1.2

Translating Complex Statements

For these we need to combine approaches.

  1. You’ll ENJOY Scary Movie provided you have a SENSE of humor and you LIKE horror movies.
    Translation: (S & L) → E

  2. Unless ALLIE and ERIN go to the party, I don’t WANT to go.
    Translation: ~(A & E) → ~W

To properly place parentheses, make use of:

More examples:

  1. It’s not true that I am BALD, and I RESENT the insult.
    Translation: ~B & R

  2. It’s not true that I’m BALD and I’m LAZY.
    Translation: ~(B & L)

  3. If they go to the STORE at 7pm then they’ll arrive HOME at 9pm, but I’ll be GROWING hungry by 8pm.
    Translation: (SH) & G

  4. If they go to the STORE at 7pm, then they’ll arrive HOME at 9pm and they will MISS the start of the movie.
    Translation: S → (H & M)

Homework: Exercise Set 4C, Exercise Set 4D

Translating Whole Arguments

Guidelines and tips for translating an entire argument:

  1. Identify the conclusion first. (Remember it needn’t be last.)
  2. Use the same letter for the same simple statement.
  3. Use different letters for different simple statements.

Try it yourself:

Or click here to see Prof. Klement’s answer.

Another example argument. Try it yourself:

Or click here to see Prof. Klement’s answer.

Homework: Credit Exercises 1.3

Practice Translations

Try them yourself, or see Prof. Klement’s answer.


Review

What’s On Exam 1?

The practice exam available on our website provides a near perfect model of the exam.

Review: True/False

  1. All valid arguments are sound.
    Answer: False

  2. All sound arguments are valid.
    Answer: True

  3. All arguments with all true premises and true conclusions are valid.
    Answer: False

  4. All valid arguments with true conclusions are sound.
    Answer: False

  5. All invalid arguments with false premises have true conclusions.
    Answer: False

REVIEW: Syllogisms

Valid? Factually correct? Sound?

  1. No novels are books.
    Some books are refrigerators.
    Therefore, all novels are refrigerators.
    Valid? No  Factually correct? No  Sound? No

  2. All poets are authors.
    All novelists are authors.
    Therefore, some poets are novelists.
    Valid? No  Factually correct? Yes  Sound? No

  3. All diamonds are gems.
    Some gifts are not gems.
    Therefore, some gifts are not diamonds.
    Valid? Yes  Factually correct? Yes  Sound? Yes

  4. All camels are snowmobiles.
    Some staplers are camels.
    Therefore, some staplers are snowmobiles.
    Valid? Yes  Factually correct? No  Sound? No

REVIEW: Truth Tables for Statements

Try it yourself:

Or click here to see Prof. Klement’s answer.

Another problem. Try it yourself:

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REVIEW: Truth Tables for Arguments

Try it yourself:

Or click here to see Prof. Klement’s answer.

Another problem. Try it yourself:

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REVIEW: Combined Problem

Try it yourself:

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Homework: Unit 1 Practice Exam


© 2017 Kevin C. Klement