STATEMENTS: JUSTIFICATION AND TRUTH

PROPOSITIONS (STATEMENTS: What a person may assert, deny, reject, believe, etc. They (but not an argument) can be justified (rational) or unjustified. (irrational). They can also be true (e.g., earth is round) or false (e.g., Bush is taller than Kerry).

RATIONALITY (JUSTIFICATION) AND EVIDENCE: Rationality (justification) is relative to evidence: Different people (or one person at different times) may have different evidence. Thus, what is rational for one person to believe at a time need not be rational for another person to believe at that time.

FALLIBILISM: JUSTIFICATION DOES NOT GUARANTEE THE TRUTH: Statements and beliefs can be rational and well-justified even though they are false.

OBJECTIVE THEORY OF TRUTH (OTT)

  • The Correspondence Principle (CP): A true statement is a statement that describes things as they actually are. It corresponds to the facts.
  • The One Truth Value Principle (OTV): Every statement has exactly one truth value. It is either true or false, but not both.
  • The Objective Truth Principle (OT): A statement's truth value is an objective property of that statement. It is determined by the actual characteristics of the things (in the real world). It is not determined by what people think about it or how they feel about the statement.

RELATIONSHIPS BETWEEN TRUTH AND JUSTIFICATION: SOME EXAMPLES
TRUE
FALSE

JUSTIFIED FOR RICHARD

This morning, Stef had a banana for breakfast.

On Sunday, Stef had a banana for breakfast.

UNJUSTIFIED FOR RICH

Stef drives a Toyota.

The Earth is flat.

JUSTIFIED FOR ANDY

Bush is the current American President.

"This" check is not going to bounce.

UNJUSTIFIED FOR ANDY

Stef has $3 in his pocket.

2 + 7 = 12.

JUSTIFIED FOR STEF

Stef has (at least) $3 in his left pocket.

My bike is waiting for me downstairs

UNJUSTIFIED FOR STEF

Bush would not won.

2 + 7 = 12.

JUSTIFIED FOR SOCRATES

The Acropolis is in Athens.

The Earth is flat.

UNJUSTIFIED FOR SOCRATES

The Earth is round.

Plato is a lousy philosopher.

Notice: the individual statements may change the rows. Justification varies from person to person.
It’s the function of reasons (evidence) people have or lack.
Notice furthermore that the individual statements never change the columns.
Truth is objective. It depends on objective reality ( it depends on objective facts). 

ARGUMENTS: VALIDITY AND SOUNDNESS

 AN ARGUMENT is a set of statements (premises) designed to prove that some statement (the conclusion of an argument) is true. The conclusion is supposed to follow from the premises. It is usually indicated by a connective ‘therefore’, ‘thus’, ‘hence’, etc.
An argument (but not a statement) can be valid or invalid, sound or unsound. The premises and conclusions of arguments can be true or false, justified or not, but never valid or invalid, sound or unsound.

 VALIDITY: VALID ARGUMENTS ARE WELL-FORMED

  • An argument is valid if and only if (hereafter, iff) iits clonusion follows from the premises.
  • That is, it is impossible for the premises of the argument all to be true and the conclusion of the argument to be false.
  • In other words, necessarily, if the premises of a valid argument are all true, then so is its conclusion; the truth of the premises guarantees the truth of the conclusion.
  • An argument is invalid iff it is not valid.

 SOME EXAMPLES OF VALID ARGUMENTS

All premises false and a false conclusion

Some premises false and a false conclusion
(1) Gandhi is a Texan. (F)
(2) All Ts wear sombreros. (F)
(3) Thus, G wears a sombrero. (F)
(1) All hens quack. (F)
(2) Lulu is a hen. (T)
(3) Hence, Lulu quacks. (F)

Some premises false and a true conclusion

All premises false and a true conclusion
(1) All Texans are at least one inch tall.(T)
(2) Stefan is a Texan. (F)
(3) Therefore, Stefan is at least one inch tall. (T)
(1) All women are Romans .(F)
(2) Caesar was a woman. (F)
(3) Thus, Caesar was a Roman (T)

SOUNDNESS AND PROOFS : A valid argument with all true premises must have a true conclusion; the conclusions of such argument cannot be false. Such an argument is a proof; it proves its conclusion.

(1) All men are mortal. (T)
(2) Socrates is a man. (T)
(3) Therefore, Socrates is mortal. (T)
(1) No vegetarians eat meat. (T)
(2) Gandhi was a vegetarian. (T)
(3) Thus, Gandhi did not eat meat. (T)
TYPICAL PATTERNS OF VALID ARGUMENTS

 PATTERN

 TRUE PREMISES

 FALSE PREMISES

(Modus Ponens)
1. If P, then Q.
2. P.
_______
3. Q.

1. If Rene is pregnant, then she is a female.
2. Rene is pregnant.
_______
3. Rene is a female. (1&2)

1. If it's cold, then it's snowing.
2. It's cold.
_______
3. It's snowing. (1&2)

1. All As are Bs.
2. x is an A.
_________
3. x is a B.

1. All humans are mammals.
2. Bill Clinton is a human.
______
3. Bill Clinton is a mammal. (1&2)

1. All animals are mammals.
2. Goldy is an animal.
______
3. Goldy is a mammal. (1&2)

(Modus Tolens)
1. If P, then Q.
2. ~Q.
3. Hence, ~P.

1. If Jay is pregnant, Jay is a female.
2. Jay is not a female.
______
3. He is not pregnant. (1&2)

1. If it's raining, the streets are wet.
2. The streets are not wet.
________
3. It's not raining. (1&2)

1. All As are Bs.
2. x is not a B.
_________
3. x is not an A.

1. All bachelors are men.
2. Rene is not a man.
_____
3. Rene is not a bachelor. (1&2)

1. All who get good grades are students.
2. Jim is not a student.
_____
3. Jim does not get good grades. (1&2)

(Elimination)
1. P or Q.
2. ~P.
3. Hence, Q.

1. Rene is a man or a woman.
2. Rene is not a man.
________
3. Rene is a woman. (1&2)

1. It's raining or it's snowing.
2. It's not raining.
________
3. It's snowing. (1&2)

(Simplification)
1. P and Q..
_______
3. P.

1. Sara knows logic and Stef knows ethics.
_____
2. Sara knows logic. (1&2)

1. Stef owns a Toyota and a book.
_____
2. Stef owns a Toyota. (1&2)

SOME PATTERNS OF INVALID ARGUMENTS

 PATTERN

 TRUE PREMISES AND CONCLUSION

TRUE PREMISES FALSE CONCLUSION

 (Denying Antecedent)
1. If P, then Q.
2. ~P.
_______
3. ~Q.

1. If Jay is pregnant, Jay is a female.
2. Jay is not pregnant.
______
3. Jay is not a female.

1. If it's snowing, then it's cold.
2. It's not snowing.
_______
3. It's not cold.

 (Asserting Consequent)
1. If P, then Q.
2. Q.
_______
3. P.

1. If Rene is pregnant, she is a female.
2. Rene is a female.
______
3. She is pregnant.

1. If it's snowing, the streets are wet.
2. The streets are wet.
______
3. It's snowing.

TWO WAYS TO SHOW THAT AN ARGUMENT IS INVALID

  • Describe a situation in which all the premises of an argument are true but the conclusion is false.
  • Identify the pattern (form) of the argument. Give an example of an argument which (A) has the same form as the original argument; and yet (B) has clearly all true premises but a false conclusion.

If we cannot prove that an argument is invalid we shall assume that it is valid.

 DEFINITIONS

I. LEXICAL DEFINITIONS: rough approximation of the meaning of a word; they elucidate the preexistent meaning of a word.

II. STIPULATIVE DEFINITIONS: do not report any preexistent meaning of a word; rather, they stipulate (create) the convenient new abbreviations.

Example: Let ‘it is drizzmog' mean ‘it is drizzling, humid and foggy'.

III. ANALYTIC DEFINITIONS: Precise accounts of what concepts (words) mean; they display in clear and precise terms the exact meaning of some expression or word. The correct analytical definition must be :

  • (A) immune from counterexamples (it must represent a necessary equivalence between two expressions) and
  • (B) enlightening (i.e., it must use only clear terms, and cannot be circular).

A. Immunity from counterexamples

(DB1) X is a bachelor =def. X is an unmarried man.
(By this definition, widowers and divorces as bachelors. But, in fact, widowers and divorces are not bachelors. So, this definition is false, it not immune to counterexamples)

(DB2) X is a bachelor =def. X is an unmarried man who has never been married.
(This is a better definition, but it still has a problem, namely it counts small children and the Pope as bachelors.)

(DB3) X is a bachelor =def. X is an unmarried man who has never been married and is eligible to be married.

B. The definition also must be enlightening (informative).

(DT) X is a table =def. X is a very tableish piece of furniture.
(This definition does not help us to understand what tables are.)
CONCEPTS

I. SHARP (PRECISE) CONCEPTS
Concepts that can be defined in an extremely precise way:

(DC) X is a circle =def. X is a closed plane curve, equidistant at all points from a given point.

II. VAGUE CONCEPTS
Concepts that cannot be defined with the same precision as the sharp concept; still, in the paradigm cases the objects share the same characteristics. Examples: a river, a table. It's very hard to distinguish precisely a river from a stream, or table (e.g., a laboratory table) from a desk. However, all paradigm (typical, exemplary) rivers will have some common elements -- banks, a bed, some water constrained in its flow by banks and a bed, etc.

III. OPEN CONCEPTS
Concepts which are very vague; it is impossible to give both sufficient and necessary conditions for any open concept. Even paradigm cases share only some but not all characteristics.

  • A game
  • A profession

Nevertheless, there is a family resemblance between games -- not all games are the same, but each game is like some other games.