26
Formal Logic
An informal answer is that Q is a logical conclusion from P 1 , … , P n whenever
the truth of P 1 , … , P n implies the truth of Q. In other words, when the implication
P 1 ` P 2 ` P 3 ` … ` P n S Q
is true. (Of course, this implication is true if any of the hypotheses is false, but
in an argument we usually care about what happens when all the hypotheses are
true.) Furthermore, this implication should be true based on the relationship of the
conclusion to the hypotheses, not on any incidental knowledge we may happen to
have about Q.
eXAMPLe 9
Consider the following argument:
George Washington was the first president of the United States. Thomas
Jefferson wrote the Declaration of Independence. Therefore, every day has
24 hours.
This argument has the two hypotheses
1. George Washington was the first president of the United States.
2. Thomas Jefferson wrote the Declaration of Independence.
and the conclusion
Every day has 24 hours.
Even though each of the individual hypotheses, as well as the conclusion, is a true
statement, we would not consider this argument valid. The conclusion is merely an
isolated true fact, not at all related to or “following from” the hypotheses.
A valid argument should therefore be true based entirely on its internal structure; it should be “intrinsically true.” Therefore we make the following formal
definition.
defInItIon VaLiD arguMenT
The propositional wff
P 1 ` P 2 ` P 3 ` … ` P n S Q
is a valid argument when it is a tautology.
The argument in Example 9 would be symbolized as
A ` B S C
which is clearly not a tautology.
eXAMPLe 10
Consider the following argument. If George Washington was the first president of
the United States, then John Adams was the first vice president. George Washington was the first president of the United States. Therefore John Adams was the first
vice president.
Formal Logic
An informal answer is that Q is a logical conclusion from P 1 , … , P n whenever
the truth of P 1 , … , P n implies the truth of Q. In other words, when the implication
P 1 ` P 2 ` P 3 ` … ` P n S Q
is true. (Of course, this implication is true if any of the hypotheses is false, but
in an argument we usually care about what happens when all the hypotheses are
true.) Furthermore, this implication should be true based on the relationship of the
conclusion to the hypotheses, not on any incidental knowledge we may happen to
have about Q.
eXAMPLe 9
Consider the following argument:
George Washington was the first president of the United States. Thomas
Jefferson wrote the Declaration of Independence. Therefore, every day has
24 hours.
This argument has the two hypotheses
1. George Washington was the first president of the United States.
2. Thomas Jefferson wrote the Declaration of Independence.
and the conclusion
Every day has 24 hours.
Even though each of the individual hypotheses, as well as the conclusion, is a true
statement, we would not consider this argument valid. The conclusion is merely an
isolated true fact, not at all related to or “following from” the hypotheses.
A valid argument should therefore be true based entirely on its internal structure; it should be “intrinsically true.” Therefore we make the following formal
definition.
defInItIon VaLiD arguMenT
The propositional wff
P 1 ` P 2 ` P 3 ` … ` P n S Q
is a valid argument when it is a tautology.
The argument in Example 9 would be symbolized as
A ` B S C
which is clearly not a tautology.
eXAMPLe 10
Consider the following argument. If George Washington was the first president of
the United States, then John Adams was the first vice president. George Washington was the first president of the United States. Therefore John Adams was the first
vice president.