Axiom System and facts/rules

Question

Im studying First Order Logic and im trying to make order of the various concepts.

I think i have understood what is an axiom system (some people call it "Formal System"). But i have two questions:

1) I haven't understood the difference beetween a theory and a formal system. (however i understood the definition of theory) 2) i havent understood how facts and rules are connected to the concepts of formal system and theory.

Can you help me ?

In the context of logic a "fact" can be some "ground" proposition having a definite truth value, like e.g. $a=a, a=f(b), P(a,f(b))$, and so on. — Mauro ALLEGRANZA, Mar 01 '18 at 10:12
Regarding rules, see Logical programming: "In all of these languages, rules are written in the form of clauses: $H \leftarrow B_1,\ldots, B_n$ and are read declaratively as logical implications: "if $B_1$ and … and $B_n$, then $H$." — Mauro ALLEGRANZA, Mar 01 '18 at 10:15
If i have a set of facts/rules in FOL, they represent a theory ? — Qwerto, Mar 01 '18 at 10:48
In prolog we add a definite program composed of rules and facts. I dont understand if (for the resolution) a formal system is used or not. — Qwerto, Mar 01 '18 at 12:38
@Qwerto formal systems have inference rules, that tell you 'if you have this, then you can get that'. They are sort of like rules of chess: they tell you what you can do ... but not what, at any point, you should be doing. Computer programs are quite different: they follow an algorithm that do specify what to do next. This is true even for Prolog, as the backtracking mechanism in prolog determines what the next 'move' is going to be. So, I wouldn't call that a formal proof system, but rather an algorithm that works with logic statements. — Bram28, Mar 01 '18 at 12:42
My prof. introduced to us an axiom scheme (composed of 5 axioms ) and 2 inference rules — Qwerto, Mar 01 '18 at 12:43
Regarding formal systems, you can see the post formal-language-vs-first-order-language. — Mauro ALLEGRANZA, Mar 01 '18 at 12:59

Mauro ALLEGRANZA · Accepted Answer · 2018-03-01T12:48:48.943

Theory in mathematical logic is used with different (but related) meanings.

In a broad sense, we understood as a theory in e.g. FOL a set of non-logical axioms on top of the deductive calculus, like Peano's axioms or Tarski's axioms.

In the first case, we have the so-called first-order theory of arithmetic.

It is made of: a (first-order) language, logical axioms (one of many equivalent versions of axioms for predicate calculus), a set of rules (usually, at least modus ponens) and the set of specific arithmetical axioms (the first-order version of Peano original axioms).

In model theory we call (complete) theory associated with a structure $A$, denoted: $\text{Th}(A)$:

the set of all first-order sentences over the signature [i.e. language] of $A$ which are satisfied by $A.$

I.e.

$\text{Th}(A) = \{ \sigma \mid A \vDash \sigma \}$.

A third related meaning is:

a theory is a set of sentences closed under logical implication.

That is, a set of sentences $T$ is a theory iff for any sentence $σ$ of the language: $T \vDash σ \text { iff } σ ∈ T$.

If we define $\text{Mod}(T)$ to be the class of all models of the set $T$ of sentences, we have that $\text {Th} (\text { Mod}(T))$ is the set of all sentences which are true in all models of $T$. But this is just the set of all sentences logically implied by $T$, or set of consequences of $T$.

For example, first-order theory of arithmetic is the set of consequences of the axioms for arithmetic.

Good. So, in FOL i can get a theory by the instantiation of a formal system? — Qwerto, Mar 01 '18 at 13:10
@Qwerto - now you are using "formal system" in a loose way :-) See the post axiom-systems-and-formal-systems for a brief characterization of what is a formal sytem. We can have formal systems for many applications: one of them is to logic. — Mauro ALLEGRANZA, Mar 01 '18 at 13:21

Axiom System and facts/rules

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