68
Formal Logic
Note that if we attempt to symbolize this argument in propositional logic, we get
A ` B S C, which is not a valid argument. Propositional logic is simply not expressive enough to capture the interrelationships among the parts of this argument
that serve to make it valid.
A proof sequence is
1. (4x)[L(x) S I(x)]
hyp
2. (E x)[L(x) ` D(x)]
hyp
3. L(a) ` D(a)
2, ei
4. L(a) S I(a)
1, ui
5. L(a)
3, sim
6. I(a)
4, 5, mp
7. L(a) ` D(a) ` I(a)
3, 6, con
8. L(a) ` I(a) ` D(a)
7, comm
9. (E x)[L(x) ` I(x) ` D(x)]
8, eg
Once again, it is the form of the argument that matters, not the content.
pRaCtiCe 27 Show that the following argument is valid: “All rock music is loud music. Some rock music
exists, therefore some loud music exists.” Use predicates R(x) and L(x).
Conclusion
We’ve now finished our study of formal logic. What has been accomplished?
The goal of formal logic, often called symbolic logic, is to make arguments as
meaningless as possible! The symbolic notation of propositional and predicate
logic allows us to symbolize arguments. An argument cast in symbolic notation
removes any possibility that we will be swayed by our opinions or our external
knowledge about the topic of the argument, and we can concentrate solely on its
structure to determine its logical validity. Furthermore, the derivation rules allow
the proof of an argument’s validity to be produced by symbol manipulation. A
proof requires no external knowledge, only a careful adherence to the forms and
restrictions of the rules. In theory, then, producing a proof sequence should be
almost mechanical. Again, one objective of practice with this mechanical process
of applying rules is that it will ultimately transform into a habit of logical thinking
in everyday life.
Nonetheless, you may still feel that it is difficult to produce a proof sequence.
Practice does make the process easier, because after a while, you become familiar
with the various forms an argument might take and you recognize which rules
you should try to apply. At any rate, you should at least find it easy at this point to
check whether a proposed proof sequence is logically correct.
Philosophers through the ages believed logical thinking to be one of the highest achievements of human existence. One additional example, however, will
show how even the most careful application of logic can be frustrated.
Formal Logic
Note that if we attempt to symbolize this argument in propositional logic, we get
A ` B S C, which is not a valid argument. Propositional logic is simply not expressive enough to capture the interrelationships among the parts of this argument
that serve to make it valid.
A proof sequence is
1. (4x)[L(x) S I(x)]
hyp
2. (E x)[L(x) ` D(x)]
hyp
3. L(a) ` D(a)
2, ei
4. L(a) S I(a)
1, ui
5. L(a)
3, sim
6. I(a)
4, 5, mp
7. L(a) ` D(a) ` I(a)
3, 6, con
8. L(a) ` I(a) ` D(a)
7, comm
9. (E x)[L(x) ` I(x) ` D(x)]
8, eg
Once again, it is the form of the argument that matters, not the content.
pRaCtiCe 27 Show that the following argument is valid: “All rock music is loud music. Some rock music
exists, therefore some loud music exists.” Use predicates R(x) and L(x).
Conclusion
We’ve now finished our study of formal logic. What has been accomplished?
The goal of formal logic, often called symbolic logic, is to make arguments as
meaningless as possible! The symbolic notation of propositional and predicate
logic allows us to symbolize arguments. An argument cast in symbolic notation
removes any possibility that we will be swayed by our opinions or our external
knowledge about the topic of the argument, and we can concentrate solely on its
structure to determine its logical validity. Furthermore, the derivation rules allow
the proof of an argument’s validity to be produced by symbol manipulation. A
proof requires no external knowledge, only a careful adherence to the forms and
restrictions of the rules. In theory, then, producing a proof sequence should be
almost mechanical. Again, one objective of practice with this mechanical process
of applying rules is that it will ultimately transform into a habit of logical thinking
in everyday life.
Nonetheless, you may still feel that it is difficult to produce a proof sequence.
Practice does make the process easier, because after a while, you become familiar
with the various forms an argument might take and you recognize which rules
you should try to apply. At any rate, you should at least find it easy at this point to
check whether a proposed proof sequence is logically correct.
Philosophers through the ages believed logical thinking to be one of the highest achievements of human existence. One additional example, however, will
show how even the most careful application of logic can be frustrated.