Theory vs. Deductive Theory

Question

Looking through some notes on model and computability theory, I noticed that the definition of the term 'theory' changed between them; in particular, the model theory (based on Hodges' text) defined a theory as "a set of sentences of a formal language", while the computability theory defined it as "a set of sentences of a formal language closed under the consequence relation in question".

This seems like a non-trivial difference, and wikipedia gives them separate names: 'theory' for the first, 'deductive theory' for the second. Is this standard? If not (and it's common to use both definitions), is there a reason the ambiguity isn't a problem?

Answer 1

Logicians are divided with regards to the definition of a "theory." Some take it to be just any set of sentences. Others take it to be a set of sentences closed under consequence. So you'll have to take caution when an author speaks of a "theory." Ultimately, it doesn't really make a huge difference: everything you can do with one definition you can do with the other. For instance, if you define theories as the former, and you want to talk about the latter, you can just talk about Cn($T$), i.e. the set of consequences of $T$. We're usually concerned with Cn($T$), so which definition you pick is a matter of preference.

Answer 2

It does not make a huge difference : when the first formalism writes $T \vdash \varphi$, the second writes $\varphi \in T$ ; just substitute one to the other to get from one formalism to the other (and conversely).

But it might be important to distinguish the notions of theory and closed under consequences theory when talking about axiomatizability. Typically, this is a problem of the form : given a theory $T$, can we find a theory (in the former sense) $T_0$ such that $T_0 \vdash T$ (i.e. for every sentence $\varphi$, $T \vdash \varphi$ implies $T_0 \vdash \varphi$) and with some good properties about $T_0$. Here you do not want $T_0$ to be closed under consequences (actually you often search for a minimal set of sentences axiomatizing $T$).

As an example, you can ask if a theory $T$ is axiomatizable by a theory with sentence of the form $\forall x_1\ldots x_n \exists y_1\ldots y_m \varphi_0$ with $\varphi_0$ is a quantifier-free formula (with free variables among the $x_i, y_j$) [1]. But $T$ can contain (or derive) some formula not of this form. E.g., the theory $T$ of groups in the language $\{ \times, 1 \}$ admits such an axiomatisation (verify it) although $T \vdash \exists x \forall y \ ( xy = y \wedge yx = y)$ (existence of the neutral).

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[1] Such theory $T$ has models with good behaviour when making increasing union of them.