Section 2

LOGIC

Arguments

• An argument is an attempt to establish or prove a conclusion on the basis of one or more premises. Example:

Arguments

• An argument is an attempt to establish or prove a conclusion on the basis of one or more premises. Example: all historians are golfers, and my buddy Ric is an historian, so he must be a golfer.

Arguments

• To make it easier to distinguish premises and conclusion, an argument can be put in standard form. 1. All historians are golfers.

2. Ric Dias is an historian.

Therefore: Ric Dias is an historian.

Arguments

• In standard form, the premises are each assigned a number, and the conclusion is separated by a horizontal line. 1. All historians are golfers.

2. Ric Dias is an historian.

Therefore: Ric Dias is an historian.

Logical Consistency

• A set of claims is logically consistent only if it is conceivable that all the claims are true at the same time.

Logical Consistency

• Logically consistent: • Logically inconsistent

Logical Consistency

• Logically consistent: – – My grandfather lived in Jonesboro.

My grandfather is dead. • Logically inconsistent

Logical Consistency

• Logically consistent: – – My grandfather lived in Jonesboro.

My grandfather is dead. • Logically inconsistent – – My grandfather lives in Jonesboro.

My grandfather is dead.

Logical Consistency requires careful thinking • Logically Inconsistent 1. Abortion is wrong because it is wrong to take a human life.

2. Capital punishment is right because it is a just punishment for murder.

Logical Consistency requires careful thinking • Logically Consistent 1. Abortion is wrong because it is wrong to take an innocent human life.

2. Capital punishment is right because it is a just punishment for murder.

Supplying Missing Premises

1. All human beings are mammals.

2. All mammals are warm blooded.

Therefore: Socrates is warm blooded.

Supplying Missing Premises

1. Socrates is a human being.

2. All human beings are mammals.

3. All mammals are warm blooded.

Therefore: Socrates is warm blooded.

Logical Possibility vs. Causal Possibility • A state of affairs is causally possible if it does not violate any of the laws of nature.

Logical Possibility vs. Causal Possibility • A state of affairs is causally possible if it does not violate any of the laws of nature. • It is causally possible that the Twins will win the World Series this year.

Causal Possibility

• A state of affairs is causally impossible if it violates any of the laws of nature. • • It is causally possible that the Twins will win the World Series this year. It is not causally possible for Justin Morneau to hit a baseball into outer space (at least, not quite).

Logical Possibilty

• A statement is logically impossible if it involves a contradiction.

Logical Possibilty

• A statement is logically impossible if it involves a contradiction. – It is logically possible that George W. Bush lost the popular vote but was elected President.

Logical Possibility

• A statement is logically impossible if it involves a contradiction. – It is logically possible that George W. Bush lost the popular vote but was elected President. – It is logically impossible that Bush lost the electoral college vote but was elected President.

Logical Possibility

• Philosophy is primarily concerned so much with conceptual analysis.

Logical Possibility

• • Philosophy is primarily concerned so much with conceptual analysis. Proving or ruling out logical possibility is almost always more important than causal possibility.

Lexical vs. Philosophical definitions.

• A lexical definition tells us how a word is frequently used

Lexical vs. Philosophical definitions.

• A lexical definition tells us how a word is frequently used • Coke = a soft drink; cocaine

Lexical vs. Philosophical definitions.

• A philosophically rigorous definition attempts to precisely say what something is and isn’t.

Lexical vs. Philosophical definitions.

• A philosophically rigorous definition attempts to precisely say what something is and isn’t. • A triangle is a closed figure consisting of three line segments linked end-to-end.

Necessary & Sufficient Conditions • A condition in this context means anything that is true of something.

Necessary & Sufficient Conditions • A condition in this context means anything that is true of something.

– So: being alive is a condition of yourself, if indeed you are reading or hearing this.

Necessary & Sufficient Conditions • A condition in this context means anything that is true of something.

– So: being alive is a condition of yourself, if indeed you are reading or hearing this. – Being a closed figure is a condition of being a triangle.

Necessary Conditions

•

A condition q is necessary for p if it is impossible for something to be p

without being q.

Necessary & Sufficient Conditions • •

A condition q is necessary for p if it is impossible for something to be p

without being q. Here p stands for some concept, and q for some condition that has to be true of that concept.

Necessary & Sufficient Conditions • •

A condition q is necessary for p if it is impossible for something to be p

without being q. Example: – Being an animal is a necessary condition for being a mammal.

Necessary Conditions

• • Being an animal is a necessary condition for being a mammal. It must be an animal (q) if it is a mammal (p).

Necessary Conditions

• •

A condition q is necessary for p if it is impossible for something to be p

without being q. Example: – Being an animal is a necessary condition for being a mammal.

Necessary Conditions

Animals Mammals

Necessary Conditions

• •

A condition q is necessary for p if it is impossible for something to be p

without being q. Notice that the converse is not true – Being a mammal is not a necessary condition for being a animal.

Necessary Conditions

fish animal mammal fox

Sufficient Conditions

•

A condition q is sufficient for p if it is impossible for something to be q

without being p.

Sufficient Conditions

• •

A condition q is sufficient for p if it is impossible for something to be q

without being p. Here “sufficient” means enough.

Sufficient Conditions

• •

A condition q is sufficient for p if it is impossible for something to be q

without being p. Example: – Being a triangle is sufficient for having three angles.

Sufficient Conditions

• •

A condition q is sufficient for p if it is impossible for something to be q

without being p. Example: – Being a triangle is sufficient for having three angles. – Being a mammal is a sufficient condition for being an animal.

Sufficient Conditions

• If you know that q is a triangle, that’s enough to know that q has condition p (it has three angles.

Sufficient Conditions

• • If you know that q is a triangle, that’s enough to know that q has condition p (it has three angles. So q is a sufficient condition for p.

Sufficient Conditions

• • If you know that q is a triangle, that’s enough to know that q has condition p (it has three angles. – So q is a sufficient condition for p. If you know that q is a mammal, that’s enough to know that q is an animal. – So again q is a sufficient condition for p.

Necessary Conditions

fish fox fern

Assignment

• Work the Rauhut exercises on page 23.

Counterexamples

• In philosophy, as in science, all useful concepts must be tested.

Counterexamples

• • In philosophy, as in science, all useful concepts must be tested. One way of testing a definition is by challenging the necessary and sufficient conditions implied in the definition.

A Famous Counterexample

• Man is a featherless biped.

A Famous Counterexample

• • Man is a featherless biped. Counter example: then a plucked chicken would be a man.

A Common Counterexample

1. It is wrong to kill a human being when one can avoid it.

2. One does not have to execute criminals.

Therefore: capital punishment is wrong.

A Common Counterexample

It is wrong to kill a human being when one can avoid it.

Counter Examples: 1. Self defense 2. Just War

The Structure of Arguments

• Every argument has two components:

The Structure of Arguments

• Every argument has two components: 1. A conclusion: some assertion that the argument tries to establish (prove); and

The Structure of Arguments

• Every argument has two components: 1. A conclusion: some assertion that the argument tries to establish (prove); and 2. One or more premises: reasons that are offered to support that claim.

Deductive vs. Inductive Arguments • Every deductive argument tries to show that the premises, taken together, are sufficient to establish the conclusion.

Deductive vs. Inductive Arguments • Every deductive argument tries to show that the premises, taken together, are sufficient to establish the conclusion.

Deductive vs. Inductive Arguments 1. If something happened before I was born, I cannot be responsible for it.

Deductive vs. Inductive Arguments 1. If something happened before I was born, I cannot be responsible for it.

2. The assassination of Abraham Lincoln happened before I was born.

Deductive vs. Inductive Arguments 1. If something happened before I was born, I cannot be responsible for it.

2. The assassination of Abraham Lincoln happened before I was born.

Therefore: I cannot be responsible for the assassination of Lincoln.

Deductive vs. Inductive Arguments An inductive argument attempts to show that a set of premises makes a conclusion more or less likely.

Deductive vs. Inductive Arguments 1. My great grandfather died of a heart attack.

Deductive vs. Inductive Arguments 1. My great grandfather died of a heart attack.

2. My grandfather died of a heart attack.

Deductive vs. Inductive Arguments 1. My great grandfather died of a heart attack.

2. My grandfather died of a heart attack.

3. My father died of a heart attack.

Deductive vs. Inductive Arguments 1. My great grandfather died of a heart attack.

2. My grandfather died of a heart attack.

3. My father died of a heart attack.

Therefore I will die of a heart attack.

Deductive vs. Inductive Arguments • In a successful deductive argument, the premises are logically sufficient to establish the conclusion.

Deductive vs. Inductive Arguments • • In a successful deductive argument, the premises are logically sufficient to establish the conclusion.

No matter how good an inductive argument is, the premises are never sufficient to logically establish the conclusion.

Assignments and Exercises

• • Rauhut, pp. 28-32. Work the problems out on paper and have it with you in class.

Judging Deductive Arguments

• Every deductive argument tries to show that the premises, taken together, are sufficient to establish the conclusion.

Valid Deductive Arguments

• A valid argument is a good deductive argument: the premises are in fact sufficient to logically guarantee the conclusion.

Valid Deductive Arguments

• A valid argument is a good deductive argument: the premises are in fact sufficient to logically guarantee the conclusion.

Valid Deductive Arguments

• A valid argument is a good deductive argument: the premises are in fact sufficient to logically guarantee the conclusion.

1. All communists are enemies of freedom.

Valid Deductive Arguments

• A valid argument is a good deductive argument: the premises are in fact sufficient to logically guarantee the conclusion.

1. All communists are enemies of freedom.

2. Stalin was a communist leader.

Valid Deductive Arguments

• A valid argument is a good deductive argument: the premises are in fact sufficient to logically guarantee the conclusion.

1. All communists are enemies of freedom.

2. Stalin was a communist leader.

Therefore: Stalin was an enemy of freedom.

A Diagram

Enemies of freedom Communists Stalin

Invalid Deductive Arguments

• An invalid argument is deductive argument that fails.

Invalid Deductive Arguments

• • An invalid argument is deductive argument that fails.

The Premises do not logically guarantee the conclusion.

Invalid Deductive Arguments

• • An invalid argument is deductive argument that fails.

The Premises do not logically guarantee the conclusion.

1. All communists are enemies of freedom.

Invalid Deductive Arguments

• • An invalid argument is deductive argument that fails.

The Premises do not logically guarantee the conclusion.

1. All communists are enemies of freedom.

2. Adolph Hitler was not a communist leader.

Invalid Deductive Arguments

• • An invalid argument is deductive argument that fails.

The Premises do not logically guarantee the conclusion.

1. All communists are enemies of freedom.

2. Adolph Hitler was not a communist leader.

Therefore: Hitler was not an enemy of freedom.

A Diagram

Enemies of freedom Communists Hitler

Sound Deductive Arguments

A deductive argument is sound if and only if: 1) the argument is valid; and

Sound Deductive Arguments

A deductive argument is sound if and only if: 1) the argument is valid; and 2) the premises are in fact true.

Sound Deductive Arguments

• A deductive argument is sound if: 1) the argument is valid; and 2) the premises are in fact true. 1. All human beings have two biological parents.

Sound Deductive Arguments

• A deductive argument is sound if: 1) the argument is valid; and 2) the premises are in fact true. 1. All human beings have two biological parents.

2. Bill is a human being.

Sound Deductive Arguments

• A deductive argument is sound if: 1) the argument is valid; and 2) the premises are in fact true. 1. All human beings have two biological parents.

2. Bill is a human being.

Therefore: Bill has two biological parents.

Unsound Arguments

• This is very important!

Unsound Arguments

• • This is very important!

An argument can be valid but unsound.

Unsound Arguments

• • • This is very important!

An argument can be valid but unsound.

That would mean that it’s premise(s) logically guarantee the conclusion;

Unsound Arguments

• • • • This is very important!

An argument can be valid but unsound.

That would mean that it’s premise logically guarantee the conclusion; But: at least one premise is false.

Unsound Arguments

• • • • • This is very important!

An argument can be valid but unsound.

That would mean that it’s premise logically guarantee the conclusion; But: at least one premise is false.

In such a case, an argument may be valid, but nonetheless have a false conclusion.

Unsound Arguments

1. All historians are space aliens.

2. Ric Dias is an historian.

Therefore: Ric Dias is a space alien.

Unsound Arguments

1. All historians are space aliens.

2. Ric Dias is an historian.

Therefore: Ric Dias is a space alien. • Why is the argument valid?

Unsound Arguments

1. All historians are space aliens.

2. Ric Dias is an historian.

Therefore: Ric Dias is a space alien. • • Why is the argument valid?

Because the logical meaning of the premises guarantees the logical meaning of the conclusion.

Unsound Arguments

1. All historians are space aliens.

2. Ric Dias is an historian.

Therefore: Ric Dias is a space alien. • • Why is the argument valid?

If the premises were true, the conclusion would have to be true.

Unsound Arguments

1. All historians are space aliens.

2. Ric Dias is an historian.

Therefore: Ric Dias is a space alien. • Why is the argument valid?

• Why is the argument unsound? Because premise one is demonstrably false.

Assignment:

• Rauhut, p 33.

Forms of Deductive Arguments

• You need to know four basic forms of valid arguments.

Forms of Deductive Arguments

• • You need to know four basic argument forms.

To recognize the forms we will use symbols for the basic terms.

Forms of Deductive Arguments

1. If Ric Dias is an historian, then he is a space alien.

2. Ric Dias is an historian.

Therefore: Ric Dias is a space alien. 1. If p, then q. 2. p. Therefore: q.

Modus Ponens

1. Modus ponens has the form described to the right. 1. If p, then q. 2. p. Therefore: q.

Modus Ponens

1. Modus ponens has the form described to the right. 2. If an argument really has that form, then the argument is valid. 1. If p, then q. 2. p. Therefore: q.

Modus Ponens

1. Modus ponens has the form described to the right. 2. If an argument really has that form, then the argument is valid. 3. Modus ponens means “affirming mode.” 1. If p, then q. 2. p. Therefore: q.

Modus Tollens

1. Modus tollens has the form described to the right. 1. If p, then q. 2. Not q.

Therefore: not p.

Modus Tollens

1. If it’s a mammal, then its an animal. 2. It’s not an animal.

Therefore: it’s not a mammal. 1. If p, then q. 2. Not q.

Therefore: not p.

Modus Tollens

1. Modus tollens is always valid. 1. If p, then q. 2. Not q.

Therefore: not p.

Disjunctive syllogism

1. Either p or q. 2. Not q.

Therefore: p.

Disjunctive syllogism

1. Either p or q. 2. Not q.

Therefore: p. 1. Either KCB was my father, or I am a bastard.

2. I am not a bastard. Therefore: KCB was my father.

Hypothetical syllogism

1. If p then q. 2. If q then r.

Therefore: if p then r. 1. If I pray every day, then I will go to Heaven.

2. If I go to Heaven, then I will get to talk to Socrates.

Therefore: If I pray every day, I will get to talk to Socrates.

The Four Forms

Modus Ponens 1. If p, then q. 2. p. Therefore: q. Disjunctive Syllogism 1. Either p or q. 2. Not q.

Therefore: p. Modus Tollens 1. If p, then q. 2. Not q.

Therefore: not p. Hypothetical Syllogism 1. If p then q. 2. If q then r.

Therefore: if p then r.

Assignment

• Rauhut, p. 37-38.

Evaluating Inductive Arguments

• Inductive arguments can only make a conclusion more likely;

Evaluating Inductive Arguments

• • Inductive arguments can only make a conclusion more likely; they cannot guarantee that the outcome follows from the premises.

Enumerative Inductive Arguments • In this type of argument, the larger the number of examples or cases that confirm the conclusion,

Enumerative Inductive Arguments • • In this type of argument, the larger the number of examples or cases that confirm the conclusion, The more likely the conclusion is said to be.

Enumerative Inductive Arguments 1. All the ravens every observed in all countries and throughout history have been black.

Therefore: all ravens are black.

Analogical Inductive Arguments

These arguments have this form: 1. X is like Y 2. X has property p; Therefore: Y will have property p.

Analogical Inductive Arguments

These arguments have this form: 1. X is like Y 2. X has property p; Therefore: Y will have property p. • Example: 1. Terrorism is like Cancer.

2. To survive cancer, you must act aggressively and early.

Therefore: we should strike at terrorists will our forces as soon as we detect them.

Analogical Inductive Arguments

1. Analogical Arguments are generally very weak.