If x = 1, then x + 1 = 5. Is it a logical proposition?

Solution 1:

• The formula $$x=1$$ is variously called a predicate, a propositional function, and an open formula. You're right about this formula not being a proposition: a predicate typically has a varying truth value. (The predicate $x=x$ is definitely true though, and is thus a (logical) validity.)

  Similarly, $$\text{if }x = 1,\text{ then }x + 1 = 5\tag{*}$$ is also a predicate. It has the opposite truth value as the first predicate above.

• On the other hand, the formula $$\text{for each $x, \, \big($if }x = 1,\text{ then }x + 1 = 5\big)\tag{#}$$ is a (false) proposition, also called a sentence and a closed formula. As you've pointed out, (in each interpretation) every proposition has a definite truth value.

What distinguishes a predicate from a proposition is whether the formula contains any free variable, that is, whether each variable in it is quantified by either $\forall$ or $\exists.$ (As illustrated above, a formula that has a fixed truth value is not necessarily a proposition!)

In practice though, predicates in mathematical writing are typically treated as being implicitly universally quantified. This is the real issue in the given example: the predicate $(*)$ is actually intended to be understood as the proposition $(\#).$

Solution 2:

A proposition is a statement that is "true" or "false" The statement "x= 1" is not a "proposition" because whether it is true or false depending on what x is.

But there is not a condition that the hypothesis and conclusion of a proposition, separately, be propositions. The statement "if x= 1 then x+ 1= 5" is a proposition because it is false.