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Let's take a formula with 'free' variables $x+1=x+1-1$ in some texts I see that if we can write that the formula is true, for all $x$ in a domain of discourse $D$, we give the formula a constant truth value, so for the formula above over $R$, we can universally quantify it over the reals and in some examples the formula is given a value of true. This seems a little bit odd, because a formula like '$x+1=2$ being true is just a statement and doesn't really have much of a meaning until I define what $x$ actually is. Is this just a shorthand for not needing universal quantification. I.E. for any $x$ we know that we will have, $[P(x)]= 1$

user37577
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  • A statement with free variables means: "Whatever x may actually be, the statement is true". This amounts to saying "For all possible values of x". – Natalie Clarius Aug 30 '22 at 23:26
  • Already asked here – Mauro ALLEGRANZA Aug 31 '22 at 05:38
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    In general math practice, when we assert a formula with free variable x we read it as universally quantified. – Mauro ALLEGRANZA Aug 31 '22 at 06:39
  • In logic we define the formal relation of satisfaction for formula with free variables using variable assignment function. – Mauro ALLEGRANZA Aug 31 '22 at 06:40
  • @lemontree In that case why do we bother with universal quantification? Is the 'assignment' of true to P(x) then just equivalent to sort of saying 'whatever x may be, this statement is true' (which is really universal quantification', I feel that the statement is surely just a statement at the end of the day, and any other meaning is extra meaning that we give it in the context? – user37577 Aug 31 '22 at 08:20
  • Let's take 'x is a man' doesn't have much meaning in terms of truth but 'john is a man' (as long as we know who john is) does, we can say that for all men named john that $[P(x(]=1$ but that is universal quantification again, we can define $f(x)=3$ but $f(x)$ just tells us a number goes here, it's value is defined when $x$ is defined (even though it's the same for all $x$) but we know for all $x$ that $f(x)=3$ so we essentially define the symbol to be three, and as it turns out it acts as a constant in our system in all expressions as if the symbol $f(x)$ was just another way of writing '$3$. – user37577 Aug 31 '22 at 08:35
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    "why do we bother with universal quantification?" Because not every formula holds universally. If we omit the universal quantifier, what does it mean $\lnot Px$? Obviously $\forall x \lnot Px$; but in this way how can we express $\lnot \forall x Px$? – Mauro ALLEGRANZA Aug 31 '22 at 09:41
  • @MauroALLEGRANZA obviously, but that's what I mean, by itself a formula is just a formula, we might mean to quantify it universally, but giving it a truth value when we aren't formally giving in an assignment feels more like a shorthand approach. – user37577 Aug 31 '22 at 16:57
  • Yes it is a useful convention... the correct approach is: to use quantifiers. Compare with natural language: can you give a truth value to the syntactically correct statement "it is red"? – Mauro ALLEGRANZA Aug 31 '22 at 17:08
  • yes of course thanks. – user37577 Aug 31 '22 at 17:09

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It is a frequent practice to drop wide-scope quantifiers for a less crowded and more natural reading when the context is clear enough and allows the inferences to be carried out without taking trouble with explicit quantifier operations. So,

$\forall x\forall y\forall z\big((Rxy\wedge Ryz) \rightarrow Rxz\big)$ can be expressed as $(Rxy\wedge Ryz) \rightarrow Rxz$

and

$\exists x(x^{2}-4 = 0)$ as $x^{2}-4 = 0$.

However, if the reader is left grappling with ambiguity, then it is an expository fault.

These are open formulas by syntactic form, while semantically meant to be quantified. They should not be confused with "genuine" open formulas in which free variables occur on purpose. For example, consider the following sentence

$\text{The students }\underbrace{\text{who volunteered to participate in the experiment}}_{relative\: clause}\text{ are going to discuss the results}$

where an open formula represents the relative clause in some linguistic and philosophical formalisms. As this example suggests, some notions of logic whose applications may not be mathematically interesting can find significance in other fields.

A possible structural representation of the formulas in the questions are $f(x) = g(f(x))$ and $f(x) = c$, where $f$ and $g$ are term-forming functions, $c$ is an individual constant (hence, a referring term) and $=$ is the identity predicate. As such, they are well-formed formulas (in standard predicate logic) and their closures should be either clear from the context or supplied explicitly as mentioned in the examples above.

Tankut Beygu
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