Is a propositional function a proposition in propositional logic?

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. Here are some examples of this:

• Elliot Mendelson's "Introduction to Mathematical Logic"

Definition of proposition (sentence is 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 1.4 Write the following sentences as statement forms, using statement letters to stand for the atomic sentences-that is, those sentences that are not built up out of other sentences.

d. A sufficient condition for x to be odd is that x is prime.

(A $$\implies$$ B), A: x is prime, B: x is odd.

• Chiswell & Hodges "Mathematical Logic"

Chapter 2: Informal Natural Deduction (Page 5)

Definition of statement (statement is term used for proposition): 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 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 Wolfe 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".

[My question]: So 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.

I have provided three sources where the authors seem to be treating propositional functions as propositions. This seems to be incorrect, see this answer: If $x = 1$, then $x + 1 = 5$. Is it a logical proposition?. Here are my questions regarding this:

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.
• See also this post. In predicate logic, a formula with free variables is a propositional function. A formula without free variables is called sentence. Commented Jul 7 at 21:09
• 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. Commented Jul 8 at 5:52
• 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). Commented Jul 8 at 6:12
• You have already discussed C&H's example reagrding "Is it true that $f(x)>g(y)$"? My personal opinion is that it must be read in the context of "grammatically correctness": we can correctly ask if the statement "$\pi$ is rational" is true, because the predicate "...is true" applies to statements, while we cannot do it with the name "$\pi$" 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. Commented Jul 8 at 7:46
• Re statement, see 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. Commented Jul 11 at 10:24

Too long to post as a comment.

'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.

In this third example though, since the author does not even consider 'x + 3 = 7' a declarative sentence, they aren't contradicting themself, at least not within the excerpt. "Additional information" refers to axioms.

1. 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.

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

how can we use B ∨ ¬B and say that this is a proposition, since the string that B is representing is not a proposition?

Don't think of B as rigidly standing for 'x is even'. If it helps, we can alternatively say truth-functional counterpart/version/structure instead of truth-functional form. Think of the predicate 'x is even' as being transformed into the proposition B.

Follow-up replies

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

Yes, but recycling symbols isn't helping your understanding, so let's assign ϕ instead to the proposition '∃x(P(x) ∨ Q(x))'.

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)?

We can assign ψ to the predicate '∃x(P(x) ∨ Q(y))'.

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

In the previous example, ψ being the truth-functional form of '∃x(P(x) ∨ Q(y))' means that the latter has been mapped to the former, just '7' corresponds to '5' in the one-one mapping $$f(x)=x+2.$$ As a truth-functional form, ψ is free to represent any proposition.

• Thank you! I think I’m getting the idea of truth-functional formulas now or at least how to translate. But one thing that is confusing me is in propositional logic they usually say that variables like P, Q, R, S are propositional variables that represent propositions. But here with truth-functional form we are using them to represent predicates which seems to violate that. Also I have a question about steps 1 and 2. If I had something like this: $\exists x$(P(x) $\lor$ Q(x)). Would this be P $\lor$ Q? Or would it just be P? Commented Jul 8 at 14:29
• If it’s not too much trouble, could you explain what we use the propositional variables for? For example you stated they could correspond to a predicate. They also could represent propositions. I guess what is the difference between a propositional variable corresponding to predicate vs it representing a proposition. Also following your algorithm for ∃x(P(x) ∨ Q(x)) I get P I believe because the scope is all the way to the last parentheses. How would it work for something like this: ∃x(P(x) ∨ Q(y))? For this example in this problem what would the scope of x be because we have Q(y)? Commented Jul 8 at 18:33
• @Dr.J I've just replied in an addendum. Commented Jul 9 at 5:14