Statements vs Formulas

In A Tour Through Mathematical Logic, Wolf states that

Every formula of a first-order language is a statement in the sense of Section 1.2, but not conversely.

In Section 1.2, Propositional Logic, he mentions the following for statements:

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.

Question: When can a statement not be a formula?

• "George Washington was the first President of the United States," is a statement that is not a formula. Jun 7, 2019 at 2:33
• Can't we view it as a nullary formula?
– Atom
Jun 7, 2019 at 2:36
• I don't see how it is part of a formal language in any way, but I'm no logician. Jun 7, 2019 at 2:39
• As far as I can tell, not having read the text, the author just means a formula is a formal thing (that either was or will be defined), while a statement is informal (which includes formal a special case). Jun 7, 2019 at 3:32
• @DerekElkins The text is here and no, I don't think that is what he meant. Jun 7, 2019 at 14:20

$$P(x)$$ is a formula, but not a statement.

Formulas can have free variables, and indeed $$P(x)$$ has a free (unquantified) variable $$x$$. Because it is not quantified, we cannot assign a truth-value to it, and so it is not a statement.

Indeed, note that I can quantifiy the $$x$$ either existentially or universally (or, if I had different quantifiers, differently yet). But, $$\exists x \ P(x)$$ is clearly a different statement than $$\forall x \ P(x)$$: I can easily come up with a domain and interpretation of $$P(x)$$ where $$\exists x \ P(x)$$ is true but $$\forall x \ P(x)$$ is false. So, I can indeed not assign a truth-value just to $$P(x)$$, so it is not a statement.

$$P(x)$$ is a formula though: it does follow the syntactical rules (the 'grammar' if you want) for expressions in FOL. I don;t have the text, but presumably the author lays out exactly those rules in defining formulas, and you can check for yourself that $$P(x)$$ does follow that definition. We sometimes therefore call formulas 'well-formed formulas' or wff's for short.

Something like $$)xP)((,)$$ is clearly not a formula ... that's just a string of symbols, but otherwise gibberish

• Are you sure you're not confusing "statement" with "sentence"? (The two words have the same translation in Danish and probably other languages too). The definition the OP quotes allows for "... could become true or false in the presence of additional information" which seems to allow for free variables. Jun 7, 2019 at 14:17
• The author of the OP's text is saying the opposite. He says every formula is a statement and not conversely. I believe the OP is looking for a statement that is not a formula. Jun 7, 2019 at 14:18
• @HenningMakholm Exactly! That's what it means!
– Atom
Jun 7, 2019 at 14:35
• @Atom Aye, my bad. Yes, I clearly did not read the quotes correctly! Well, .... then I don't really know what the author intended ...Will delete Answer soon ... Jun 7, 2019 at 16:31