Logic

LOGICSection 2

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.


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


consistent:– My grandfather

lived in Jonesboro.– My grandfather is

dead.

• Logically inconsistent


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 Premises1. All human beings are mammals.2. All mammals are warm blooded.Therefore: Socrates is warm blooded.

Supplying Missing Premises1. 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


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

• Coke = a soft drink; cocaine


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


• 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.


• 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.


• 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.


• 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.


• 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).



• Example:– Being an animal is a necessary

condition for being a mammal.

Necessary Conditions

Animals

Mammals



• Notice that the converse is not true– Being a mammal is not a necessary

condition for being a animal.


fish

fox

animal

mammal

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.



• Example:– Being a triangle is sufficient for having

three angles.



• 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.


that’s enough to know that q has condition p (it has three angles.

• So q is a sufficient condition for p.


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.


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 Counterexample1. 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 CounterexampleIt is wrong to kill a

human being when one can avoid it.

Counter Examples:1. Self defense2. Just War

The Structure of Arguments• Every argument has two

components:


components:1. A conclusion: some assertion that the

argument tries to establish (prove); and


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.




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.



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

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


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


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.




3. My father died of a heart attack.Therefore I will die of a heart attack.


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


• 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


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.




2. Stalin was a communist leader.




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.



• The Premises do not logically guarantee the conclusion.









2. Adolph Hitler was not a communist leader.





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


A deductive argument is sound if and only if:

1) the argument is valid; and 2) the premises are in fact true.


• A deductive argument is sound if:


2) the premises are in fact true.

1. All human beings have two biological parents.






2. Bill is a human being.






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.• That would mean that it’s premise(s)

logically guarantee the conclusion;


unsound.• That would mean that it’s premise

logically guarantee the conclusion; • But: at least one premise is false.


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 Arguments1. All historians are space aliens.2. Ric Dias is an historian.Therefore: Ric Dias is a space alien.

Unsound Arguments1. All historians are

space aliens.2. Ric Dias is an

historian.Therefore: Ric Dias

is a space alien.

• Why is the argument valid?




is a space alien.


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




is a space alien.


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




is a space alien.


• 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.


• You need to know four basic argument forms.

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


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


Therefore: Ric Dias is a space alien.

1. If p, then q. 2. p. Therefore: q.

Modus Ponens1. Modus ponens

has the form described to the right.


Modus Ponens1. Modus ponens

has the form described to the right.

2. If an argument really has that form, then the argument is valid.


Modus Ponens1. 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.”


Modus Tollens1. Modus tollens has

the form described to the right.

1. If p, then q. 2. Not q.Therefore: not p.

Modus Tollens1. 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 Tollens1. Modus tollens is

always valid. 1. If p, then q. 2. Not q.Therefore: not p.

Disjunctive syllogism1. Either p or q. 2. Not q.Therefore: p.

Disjunctive syllogism1. 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 syllogism1. 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 FormsModus Ponens1. If p, then q. 2. p. Therefore: q.

Modus Tollens3. If p, then q. 4. Not q.Therefore: not p.

Disjunctive Syllogism1. Either p or q. 2. Not q.Therefore: p.

Hypothetical Syllogism3. If p then q. 4. 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,


• 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.


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 Y2. X has property p; Therefore: Y will have property p.


These arguments have this form:

1. X is like Y2. 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.


1. Analogical Arguments are generally very weak.

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