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Previously I asked about assignment functions, and it related the idea that certain formulas do not exist as either true or false until we assign to them as they contain free variables.

In this case, for each assingment function the formula may be 'satisfied'. In the context of atomic formulas I can understand how they can be 'true' as all symbols are well defined.

To what extent does a formula being satisfied under a given assignment make it 'true'? Is the idea of 'true' that if the statement is given under that assignment the information given is correct? How do we limit our truth to within a particular assignment?

Confused
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    In FOL, we have "TRUE in a *strucure*"; see [Interpretation (logic)](https://en.wikipedia.org/wiki/Interpretation_(logic)#Interpretations_of_a_first-order_language). Thus, we say that the formula is *satisfied* (in that structure). We say that a formula is *satisfiable* if there is some structure (suitable for the language of the formula) that satisfies the formula. – Mauro ALLEGRANZA Feb 03 '23 at 12:04
  • See [Connection between interpretation, variable assignment and truth valuation](https://math.stackexchange.com/questions/2339624/connection-between-interpretation-variable-assignment-and-truth-valuation) and [Interpretation of a formula and truth](https://math.stackexchange.com/questions/52056/interpretation-of-a-formula-and-truth). – Mauro ALLEGRANZA Feb 03 '23 at 12:06
  • See also [Understanding 'Assignment'](https://philosophy.stackexchange.com/questions/90908/understanding-assignment) and [How do assignment functions work?](https://philosophy.stackexchange.com/questions/96621/how-do-assignment-functions-work) – Mauro ALLEGRANZA Feb 03 '23 at 12:08
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    The key-point for *satisfaction* and *truth* is the interpretation of a formal language. If we consider the simple formula "everything is P", it is not True or False *per se*: if we set a domain of humans and predicate P is interpreted with "is mortal", then the formula is True. If instead we consider the same domain with the interpretation of P as "is a male" the formula is False. So always (for formal languages) satisfied and true wrt an interpretation. – Mauro ALLEGRANZA Feb 03 '23 at 12:12
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    A formula being satisfied makes it true to no extent for the simple reason that formulas with free variables are analogs of unfinished sentences, like "_is red". The replacement for truth is [satisfiability](https://en.wikipedia.org/wiki/Satisfiability), which answers whether there is a way to finish the sentence (fill in the blanks) to make it true. If an assignment makes the formula true than the *formula* gives correct information about the assigned. In this case, that they are red. Formulas with more blanks describe (correctly or not) relations of the assigned rather than their properties. – Conifold Feb 03 '23 at 12:43

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