Rule of Universal Instantiation
Universal Modus Ponens and Universal Modus Tollens
Validity of Arguments with Quantified Statements and Checking Validity with Diagrams

Rule of Universal Instantiation top
The rule of universal instantiation says that

If some property is true of everything in a domain, then
it is true of any particular thing in the domain.

This is the fundamental tool of deductive reasoning.

Ex.
Use universal instantiation to simplify,

r^k+1 × r,

where r is a particular real number and k is a particular integer. From algebra we know the following universal statements are true:

1. For all real numbers x and all integers m and n, x^m × xⁿ = x^m + n.
2. For all real numbers x, x¹ = x.

The solution is then

r^k+1 × r = r^k+1 × r¹ = r^{(k+1) + 1} = r^{k + 2}.

The reasoning behind step 1 and step 2 is outlined below.

Universal Modus Ponens and Universal Modus Tollens top

Universal Modus Ponens
" x, if P(x) then Q(x)
P(a) for a particular a
\ Q(a).

Universal modus ponens has two premises, one major and one minor with at least one of the premises quantified. An argument form of this sort, using two premises to draw a conclusion, is called a syllogism. Note that universal modus ponens could have been stated as " P(x), Q(x) from which the conclusion could be drawn by universal instantiation alone. But as previously mentioned, it is more natural to follow " with Þ .

When we combine universal instantiation with modus tollens we have universal modus tollens, the heart of proof by contradiction.

Universal Modus Tollens
" x, if P(x) then Q(x)
Ø Q(a) for a particular a
\ Ø P(a).

See example 2.3.3 on p. 102.

Proofs

If p ® q is true, can you determine the value of Øp Ú (p ® q). Explain your answer.

If a conditional, p ® q, is a tautology, where p and q may be compound statements involving any number of propositional variables, we say that q logically follows from p. Suppose a conditional of the form

(p₁ Ù p₂ Ù … Ù p_n) ® q

is a tautology. Then the conditional is true regardless of the truth values of any of its components. In this case, we say that q logically follows from p₁ Ù p₂ Ù … Ù p_n. This could be written in argument form as

p₁
p₂
M
p_n
^____
\ q

The proof does not show that q is true, but simply shows that q has to be true if the premises, p_i, are all true.

Now consider the argument:

If you invest in the stock market, then you will get rich.
If you get rich, then you will be happy._____________
\ If you invest in the stock market, then you will be happy.

Is this argument form valid?

This is a valid argument in the form of a Hypothetical syllogism:

p ® q
q ® r
^________
\ p ® r

Validity of Arguments with Quantified Statements and Checking Validity with Diagrams top
Definition
To say that an argument form is valid means the following: No matter what particular predicates are substituted for the predicate symbols in its premises, if the resulting premise statements are all true, then the conclusion is also true.

An argument is called valid if, and only if, its form is valid.

One way to shed light on the validity of an argument is to look at diagrams. Diagrams are not rigorous proofs. Diagrams are merely tools that may aid in clearing up relationships.
See example 2.3.6 on p. 105.

One reason so many people make converse and inverse errors is that the forms of the resulting arguments would be valid if the major premise were a biconditional rather than a simple conditional. In standard English, the biconditional naturally replaces the simple conditional with little or no consequence. Consider the argument:

All the town criminals frequent the Den of Iniquity bar.
John frequents the Den of Iniquity bar.
\ John is one of the town criminals.

The conclusion of this argument is invalid -the result of a converse error. The conclusion may be false even when the premises are all true. We have guilt by association in this argument. While suspicion may be warranted based on the premises, it would be wrong to convict John.

Consider the argument:

If taxes are lowered, then income rises.
My income rises._____________
\ My taxes are lowered.

Is this argument form valid? This argument is of the form:

" x P(x) ® Q(x)
Q(a)
^________
\ P(a)

again exhibiting the converse error.