Is a propositional function a proposition in propositional logic?

Question

In mathematical logic, a proposition is defined as a declarative sentence that is either true or false, but not both. Two examples are '1 + 1 = 2' and 'Paris is the capital of France'. I have noticed that in many mathematical logic textbooks the authors use propositional functions as examples of propositions when discussing propositional logic. This seems to be incorrect; see If $x = 1$, then $x + 1 = 5$. Is it a logical proposition?. Here are three such examples:

• Elliot Mendelson's "Introduction to Mathematical Logic"

  "Sentence" is the term used for proposition (page 1): Sentences are combined in various ways to form more complicated sentences. We shall consider only truth-functional combinations, in which the truth or falsity of the new sentence is determined by the truth or falsity of its component sentences.

  Exercise 1.4d Write the sentence, "A sufficient condition for x to be odd is that x is prime," as a statement form, using statement letters to stand for the atomic sentences—that is, those sentences that are not built up out of other sentences. Answer from back of book: (A $\implies$ B), A: x is prime, B: x is odd.

• Chiswell & Hodges "Mathematical Logic"

  "Statement" is the term used for proposition (Page 5): What is a statement? Here is a test. A string S of one or more words or symbols is a statement if it makes sense to put S in place of the '...' in the question Is it true that ...? For example it makes sense to ask any of the questions: Is it true that $\pi$ is rational? Is it true that differentiable functions are continuous? Is it true that $f(x) > g(y)$? So, all of the following are statements: $\pi$ is rational. Differentiable functions are continuous. $f(x) > g(y).$

• Wolf's "A TOUR THROUGH MATHEMATICAL LOGIC"

  On page 8 Wolf states "'x + 3 = 7' is not a proposition because it is not true or false as it stands. Its truth or falsity depends on the value of x, and so it is called a propositional function or predicate".

  Later on page 8 Wolf states "Let us use the word statement to mean any declarative sentence (including mathematical ones such as equations) that is true or false or could become true or false in the presence of additional information".

0. Based on Wolf's definition, 'x+3=7' is a statement and a propositional function? This doesn't make sense at all coming from my understanding of things.
1. Is a propositional function like 'x is prime' considered a proposition in propositional logic?
2. What is truth-functional form?
3. How does truth-functional form relate to propositions? For example in the answer Can tautologies have free variables?, the author says, "'(x is even) ∨ ¬(x is even)' is an open formula, but its truth-functional form B ∨ ¬B is a closed formula (i.e., a sentence) that is tautological." But '(x is even) ∨ ¬(x is even)' is a propositional function rather than a proposition, so how can we use B ∨ ¬B and say that this is a proposition, since the string that B is representing is not a proposition?
4. Is there a mathematical logic textbook that explains all these things clearly. I have not been able to find a text to study mathematical logic because of this confusion.

Answer 1

In predicate logic, a formula with free variables is a propositional function. A formula without free variables is called sentence.

In propositional logic there are no propositional functions because there are no predicates and variables in the syntax. See Mendelson's example: the two mathematical statements "x is prime" and "x is odd" are represented in propositional logic with two statement letters: A and B.

Regarding the "wider" use of "tautology" in predicate logic, the term applies to prop logic, while the corresponding concept for predicate logic is universally valid. But we can extend the concept of tautology to predicate logic considering instances of propositional tautologies, like the example above: $(x=x) ∨ ¬(x=x)$.

Regarding Chiswell & Hodges' example "Is it true that $f(x) > g(y)$?" my point of view is that it must be read in the context of "grammatically correctness":

we can correctly ask if the statement "π is rational" is true, because the predicate "...is true" applies to statements, while we cannot do it with the name "π " because the predicate "...is true" does not apply to names.

Thus, the statement "f(x) > g(y)" is syntactically correct, also if its truth value is undefined, unless we assign values to variables x and y.

See R.Carnap (1929): "In logic we understand a 'statement' to be something which is either true or false. ("True" and "false" are indefinable basic terms.) 'Statement' is not the historical act of speaking, thinking, imagining, but the timeless content."

Answer 2

'x + 3 = 7' is not a proposition. it is called a propositional function

Let us use the word statement to mean any declarative sentence that is true or false or could become true or false in the presence of additional information.

0. Based on Wolf's definition, 'x+3=7' is a statement and a propositional function?

I don't see the author claiming the propositional function 'x + 3 = 7' to be a declarative sentence, or consequently a statement.

"Additional information" refers to axioms.

'(x is even) ∨ ¬(x is even)' is an open formula, but its truth-functional form B ∨ ¬B is a tautological sentence.

3. how can we say thatB ∨ ¬B is a proposition, when B is not representing a proposition?

B is a substitute for, not a renaming of, '(x is even)'. '(x is even) ∨ ¬(x is even)' and B ∨ ¬B are counterparts to, rather than alternative representations of, each other, the latter being the propositional-logic abstraction/skeleton/structure of the former.

2. What is truth-functional form?

To determine the truth-functional form of a predicate-logic formula:

1. working left to right, underline the entire scope of each quantifier, including the quantifier itself
2. underline each remaining atomic formula
3. assign a symbol to every underlined formula, assigning the same symbol to two formulae if and only if they are identical.

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Follow-up reply

following your algorithm, for ∃x(P(x) ∨ Q(x)), I get P.

How would it work for something like this: ∃x(P(x) ∨ Q(y))? What would the scope of x be, given that we have Q(y)?

The scope of ∃ in the predicate '∃x(P(x) ∨ Q(y))' is the entire formula, which contains a free occurrence of $y.$ We can assign the propositional variable ω to this predicate, and the propositional variable ϕ to (instead of recycling the symbol P for) the sentence '∃x(P(x) ∨ Q(x))'.

what is the difference between a propositional variable corresponding to a predicate versus it representing a proposition?

To be more precise: the predicate '∃x(P(x) ∨ Q(y))' is injectively mapped to its truth-functional form ω, which has an injective map to propositions.