p(x)⇒∀x.p(x) is contingent?

问题

I've encountered a question asking whether the flowing sentence is valid/contingent/unsatisfiable:

p(x)⇒∀x.p(x) 

I think the answer is the sentence is valid. under section 6.10 of the textbook here http://logic.stanford.edu/intrologic/secondary/notes/chapter_06.htmlsays

a sentence with free variables is equivalent to the sentence in which all of the free variables are universally quantified.

Therefore I think the first relational sentence p(x) is equal to ∀x.p(x) and therefore the sentence is valid, ie. it is always true.

However,the correct answer is that the sentence is contingent viz. under some truth assignment it is true and other some other truth assignment it is false.

So why is the sentence contingent?Is the answer wrong?

回答1:

You have a statement:

p(x)⇒∀x.p(x)

If you universally close the free variable, you get:

∀x.(p(x)⇒∀x.p(x))

in other words:

∀x.(p(x)⇒∀y.p(y))

which is not tautology, but is contingent. In non-technical terms, this reads:

for any x, if p(x) is true, then p(y) is true for all y

or, to transform it into an equivalent form:

(∃x.p(x))⇒(∀y.p(y))

it reads:

if p(x) is true for some x, then p(y) is true for all y

In other words,

p(x) is either always true or always false

回答2:

I think it depends on how you read the sentence.

If you read it as a definition, then it is not contingent.

However, if you read it as pure logic ... then there are actually 2 meanings of x in the statement. The x on the left of the implication is different to the x in the quantification on the right.

p(x) => for all x . p(x)

means the same as

p(x) => for all y . p(y)

and that is clearly contingent. It is not true for all predicates p.

(For example:

1. Let p(x) stand for the predicate "x is left handed"

2. The statement then says:

    X is left-handed implies that everyone is left-handed.
   

   ... which is not a logically valid statement.

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See @sawa's answer for a more "mathematically rigorous" explanation.

来源：https://stackoverflow.com/questions/38417547/px%e2%87%92%e2%88%80x-px-is-contingent