Section 1.2 Propositional Logic
35
Formal logic is not necessary to prove the validity of propositional arguments. A
valid argument is represented by a tautology, and truth tables provide a mechanical
test for whether a wff is a tautology. So, what was the point of all of this? In the next
section we will see that propositional wffs are not sufficient to represent everything
we would like to say, and we will devise new wffs called predicate wffs. There is no
mechanical test for the predicate wff analogue of tautology, and in the absence of such
a test, we will have to rely on formal logic to justify arguments. We have developed
formal logic for propositional arguments as a sort of dry run for the predicate case.
In addition, the sort of reasoning we have used in propositional logic carries
over into everyday life. It is the foundation for logical thinking in computer science,
mathematics, the courtroom, the marketplace, and the laboratory. Although we have
approached logic as a mechanical system of applying rules, enough practice should
ingrain this way of thinking so that you no longer need to consult tables of rules, but
can draw logical conclusions and recognize invalid arguments on your own.
The argument says that if the hypotheses are true, then the conclusion will be true.
The validity of the argument is a function only of its logical form and has nothing
to do with the actual truth of any of its components. We still have no idea about
whether the diary is really missing. Furthermore, the argument “Skooses are pink,
but if Gingoos does not like perskees, then skooses are not pink; therefore Gingoos
does like perskees,” which has the same logical form, is also valid, even though it
does not make sense.
S e c t I o n 1 . 2 Review
tecHnIQueS
• Apply derivation rules for propositional logic.
• Use propositional logic to prove the validity of a
verbal argument.
MAIn IdeAS
• A valid argument can be represented by a wff of
the form P 1 ` P 2 ` P 3 ` … ` P n S Q that is a
tautology.
• A proof sequence in a formal logic system is a sequence of wffs that are either hypotheses or derived
from earlier wffs in the sequence by the derivation
rules of the system.
• The propositional logic system is complete and
correct; valid arguments and only valid arguments
are provable.
W
W
eXeRcISeS 1.2
For Exercises 1–4, what inference rule is illustrated by the argument given?
1. If Martina is the author, then the book is fiction. But the book is nonfiction. Therefore Martina is not the
author.
2. If the business declares bankruptcy, then all assets must be confiscated. The business declared bankruptcy.
It follows that all assets must be confiscated.
pRaCtiCe 14 Use propositional logic to prove that the following argument is valid. Use statement letters
S, R, and B.
If security is a problem, then regulation will be increased. If security is not a problem, then business on the Web
will grow. Therefore if regulation is not increased, then business on the Web will grow.
35
Formal logic is not necessary to prove the validity of propositional arguments. A
valid argument is represented by a tautology, and truth tables provide a mechanical
test for whether a wff is a tautology. So, what was the point of all of this? In the next
section we will see that propositional wffs are not sufficient to represent everything
we would like to say, and we will devise new wffs called predicate wffs. There is no
mechanical test for the predicate wff analogue of tautology, and in the absence of such
a test, we will have to rely on formal logic to justify arguments. We have developed
formal logic for propositional arguments as a sort of dry run for the predicate case.
In addition, the sort of reasoning we have used in propositional logic carries
over into everyday life. It is the foundation for logical thinking in computer science,
mathematics, the courtroom, the marketplace, and the laboratory. Although we have
approached logic as a mechanical system of applying rules, enough practice should
ingrain this way of thinking so that you no longer need to consult tables of rules, but
can draw logical conclusions and recognize invalid arguments on your own.
The argument says that if the hypotheses are true, then the conclusion will be true.
The validity of the argument is a function only of its logical form and has nothing
to do with the actual truth of any of its components. We still have no idea about
whether the diary is really missing. Furthermore, the argument “Skooses are pink,
but if Gingoos does not like perskees, then skooses are not pink; therefore Gingoos
does like perskees,” which has the same logical form, is also valid, even though it
does not make sense.
S e c t I o n 1 . 2 Review
tecHnIQueS
• Apply derivation rules for propositional logic.
• Use propositional logic to prove the validity of a
verbal argument.
MAIn IdeAS
• A valid argument can be represented by a wff of
the form P 1 ` P 2 ` P 3 ` … ` P n S Q that is a
tautology.
• A proof sequence in a formal logic system is a sequence of wffs that are either hypotheses or derived
from earlier wffs in the sequence by the derivation
rules of the system.
• The propositional logic system is complete and
correct; valid arguments and only valid arguments
are provable.
W
W
eXeRcISeS 1.2
For Exercises 1–4, what inference rule is illustrated by the argument given?
1. If Martina is the author, then the book is fiction. But the book is nonfiction. Therefore Martina is not the
author.
2. If the business declares bankruptcy, then all assets must be confiscated. The business declared bankruptcy.
It follows that all assets must be confiscated.
pRaCtiCe 14 Use propositional logic to prove that the following argument is valid. Use statement letters
S, R, and B.
If security is a problem, then regulation will be increased. If security is not a problem, then business on the Web
will grow. Therefore if regulation is not increased, then business on the Web will grow.