Lecture Notes 1: Logic, Arguments and the Basics of Sentential Logic

Basic Definitions

Logic can be defined in different ways. Here are some common definitions.

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

Logic is the study of the evaluation of arguments.

Here is the definition I prefer.

Logic is the study of relationships involving the truth or falsity of statements or propositions that obtain in virtue of their forms.

The last one is hard to understand until you’ve studied some logic already.

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

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.)

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

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 (click on it). Note that it need not be the last sentence.

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Distinction is often taught in a different, and mostly outdated, way.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

		Valid	Invalid
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.

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.

The Counterexample Method

A recap of two points:

• Deductive arguments with the same form are either both valid or both invalid.
• Valid arguments with true premises always have true conclusions.

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.

Basics of Sentential Logic

Symbolic logic is logic carried out with 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 black; the statement connectives blue.)

In sentential logic

• We use uppercase letters to abbreviate entire atomic statements.
• We use symbols as statement connectives to join atomic statements together to form molecular statements.

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 addition 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 called a wedge.

Its two inputs are called its disjuncts.

𝒜	ℬ	𝒜 ∨ ℬ
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 an arrow.

𝒜	ℬ	𝒜 → ℬ
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 …”.

• There doesn’t need to be any causal or conceptual link between A and B for A → B to be true.
• A →-statement is always true when the part before the arrow is false.

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

M → D (true)

• A →-statement is always true when the part after the arrow is true.

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

G → L (true)

Material Equivalence ↔

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

𝒜	ℬ	𝒜 ↔ ℬ
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)

H ↔ B (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 & (B ∨ G)
2. If you think ’N Sync was good or you think the Backstreet Boys were talented, then you’re crazy.
   (N ∨ B) → 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 → (~B → F)]

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

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

And ~Y ∨ C is TRUE,
but ~(Y ∨ C) 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. ~(Y ∨ Z) & (A ↔ Y)
   True or false? False Main connective? &
   
   
3. ~[C → (A ∨ Y)] → 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.

Some Review from Earlier

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