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Semantically, p and q are equivalent if they have the same truth value in every model... Would it not be more appropriate to write if they have the same truth value in every interpretation? I thought that a model is just a set and relations for each n-ary symbol, we can not say whether a formula (containing free variables) is "true" in a model (?). 18.104.22.168 (talk) 13:30, 9 May 2012 (UTC)
It is common to use "model" and "interpretation" essentially as synonyms. In first-order logic, each model has a canonical truth function which tells whether a given formula is true or false in the model. Thus models are a particular way of getting interpretations. Conversely, the completeness theorem shows that every interpretation of a set of sentences is in fact the interpretation obtained from some model.
Many treatments of first-order logic do include a way to tell whether an arbitrary formula is true or false in a given model. One common way they do this is by including a variable assignment in the model, which is used to give values to free variables (this is the method used by Enderton's book, for example). — Carl (CBM · talk) 12:00, 9 April 2013 (UTC)