Universal Instantiation
| Universal Modus Ponens |
Universal Modus Tollens
Using Universal Modus Ponens and Universal
Modus Tollens
If a property or condition is true for all elements in a domain, then it is true for any particular element in the domain. For instance,
Every thing that is alive is growing older.
John is alive.
John is growing older.
The truth of this statement follows naturally from the general, or universal, truth of the properties associated with a given domain. (In this case, the property that every thing in this world is either alive, or it isn't.) Furthermore, if a property is true of everything in a given domain, then it is also true for any particular thing in that domain.
Combining universal instantiation and modus ponens produces the rule of universal modus ponens. To understand this, consider the following famous syllogism.
All men are mortal.
Socrates is a man.
Socrates is mortal.
A syllogism is an argument form containing 2 premises - the major premise (All men are mortal.) and a minor premise (Socrates is a man.) - and at least one of the premises is quantified.
Drawing from Review 1.6 of predicates, let x be an element of the domain of all men D, let a be a particular element, Socrates, in domain D, let P be the predicate symbol for "x is a man", and Q be the predicate symbol for "x is mortal". Universal modus ponens will then assume the following argument form.
Formal Version |
Informal Version |
x
D, if P(x)
then Q(x) P(a) for a particular a Q(a) |
If x makes P(x) true,
then x makes Q(x) true. (If x is a man, then x is mortal.) P(a) is true. (Socrates is a man.) a makes Q(x) true. (Therefore, Socrates is mortal.) |
However, not all arguments will conveniently assume the "If x, then y" format of an implied statement: To be able to effectively use universal modus ponens, you must be able to identify it.
Pigs can't fly.
Wilbur is a pig.
Therefore, Wilbur can't fly.
With a little thought, it becomes apparent that the statement "Pigs can't fly" doesn't mean that some pigs can't fly, or that most pigs can't fly -- it means that ALL pigs can't fly. Rewriting this in a formal, logical format yields:
x
all living things
D, if x is a pig, x can't fly.
Wilbur is a pig.
Wilbur can't fly.
Practice Exercises |
Sometimes one of the easiest methods to prove or disprove an argument is proof by contradiction - showing an argument is invalid by finding an example whereby the argument produces a contradiction. Like Universal Modus Ponens, the form of Universal Modus Tollens is a combination of universal instantiation with modus tollens, and is the mainstay of the proof by contradiction method
Formal Version |
Informal Version |
x
D,
if P(x) then Q(x) ~Q(a) for a particular a ~P(a) |
If x makes P(x) true, then x
makes Q(x) true. Q(a) is false. a makes P(x) false. |
Practice Exercises |
Using Universal Modus Ponens and Universal Modus Tollens
The following is an example of using Universal Modus Ponens in a proof:
To prove: any integer which is a multiple of 4 is even.
n Z , if n is a multiple of 4, then n is even.
Suppose n is particular but arbitrarily chosen integer which is multiple of 4. Then n = 4m for some integer m.
n = 4m
n = 2 x 2 x m = 2(2m) by
factoring out 2.
Substituting p for 2m, you get:
n = 2p
If an integer (n) equals twice some other integer (p),
then that number is even.
n equals twice some integer.
n is
even.
The next example uses Universal Modus Tollens to draw a conclusion.
All good drivers are very alert.
People who are drunk are not very alert.
Therefore, people who are drunk are not good drivers.
While using Universal Modus Ponens or Universal Modus Tollens as a proof or to draw a conclusion, one must be careful not to make an inverse or converse error.
Practice Exercises |