On page 258 of Logic: The Laws of Truth by Nicholas Smith, the notions of necessary truth preservation and validity are defined in the context of predicate logic as follows:
- An argument is necessarily truth-preserving (NTP) iff there is no ww-model in which all its premises are true and its conclusion false.
- An argument is valid iff there is no model in which all its premises are true and its conclusion false.
Earlier in the book (in §1.4), these notions were defined intuitively as follows:
a. An argument is NTP if it is impossible for its premises to all be true and its conclusion false.
b. An argument is valid if it is NTP by virtue of its form.
Note on terminology:
- ww - way of the world (wws - ways of the world)
- ww-model - given an language fragment and an assignment of intensions to the nonlogical symbols of that fragment, a ww-model is a model that is obtained by applying the assigned intensions to a ww. The set of all ww-models is a subset of all possible models.
- By "all possible models", I am referring to the set of models that can be obtained by freely assigning values to the nonlogical symbols of the given fragment.
Questions:
I'm trying to understand how the formal definitions ((1) and (2)) of these notions capture their intuitive definitions ((a) and (b)).
I think I see how the formal definition of validity in (2) captures the intuitive idea in (b): if an argument is NTP by virtue of its form, then it is NTP regardless of its content. Since, in this case, content takes the form of intensions, we can rephrase the previous sentence as saying: for an argument to be valid, it must be NTP regardless of which intensions we assign to its nonlogical symbols, i.e. for any set of intensions, applied to any ww, we never get a model in which all the premises of the argument are true and the conclusion false. This is only possible if there is no model at all (ww-model or other) in which all the premises are true and the conclusion false, which is what definition (2) says.
However, I'm not sure about NTP. Is (1) defined in terms of "ww-models" because when we assign intensions to the nonlogical symbols of some fragment, we cull the space of possible models down from all the models that can be obtained via arbitrary value assigments to only those that can be reached by applying the assigned intensions to a ww (i.e. ww-models)?
Definition (1) also got me thinking about what the formal definition for NTP is in the context of propositional logic. When we say an argument is NTP in propositional logic, are we saying that it is NTP in the actual row of its truth table (the row which represents the world as it really is), i.e. it is not possible for the actual row to have all true premises and false conclusion?
If we cannot give a formal analysis of NTP, then how can we determine whether an argument is NTP?
For example, here are two examples of arguments that are NTP but invalid:
The sun is hot. Therefore, the sun is not cold.(this example is from the book)w is water. Therefore, w is H₂0. (this example is from this article.
But how do we know that these arguments are in fact NTP? For example, for the second argument to be NTP, there can be no ww-model in which the referent of $w$ is in the extension of water and not in the extension of $H_{2}0$. But if we cannot give a complete account of how the world could and could not be (see the last two paragraphs of page 260), then we cannot know which ww-models exist, and therefore, we cannot make any determination as to whether the argument is NTP.
EDIT: I thought about question 1 some more and came up with the following explanation - is it correct?
According to the intuitive notion of NTP, an arg is NTP iff it is impossible for its premises to all be true and its conclusion false. In order for this to be true in predicate logic, there can be no model in which all the premises of the arg are true and its conclusion false. But what model space do we choose - the space of all possible models or the space of ww-models? The NTPness of an argument can be due to its form (see argument (3) on page 15) or its content (see argument (7) on page 15). If we define NTP in terms of all possible models (i.e. an arg is NTP iff there is no model in which all its premises are true and its conclusion false), then our definition would exclude arguments that are NTP due to their content: for such arguments, there is no model in the set of ww-models in which the premises of the arg are all true and the conclusion false. But if we define NTP in terms of all possible models, then this definition would reject such an argument as being NTP if there existed a non-ww model in which the premises of the argument are all true and its conclusion false, even though this model is unreachable given the content (i.e. the assigned intensions) of the arg. But if we define NTP in terms of ww-models (i.e. an arg is NTP iff there is no ww-model in which all its premises are true and its conclusion false), then we correctly identify such an argument as NTP. This definition also captures args that are NTP due to their form, since such arguments are NTP regardless of their content, so it doesn't matter which intensions we assign to their nonlogical symbols, the resulting set of ww-models will never have a model in which all the premises of the argument are true and its conclusion false, so the definition in terms of ww-models will always correctly identify such arguments as NTP.