Use of propositional logic connectives in the meta-language

I have a doubt that might seem a bit confusing so i will try to explain it the clearer i can.

Suppose we have an expression "A o B" in the meta-language, where 'o' refers to those logical connectives we are used to ( and, or, implication, equivalence ). I want to know exactly what someone means when he ( an author or maybe someone comenting about logic ) uses that expression, regardless of the context it is used.

Lets take as an example, the implication ... When someone uses the expression "A ->B".
I find that i cant find some fixed meaning or interpretations to an implication, when used in the meta-language, i get confused between 2 meanings or interpretations.

Sometiems i get the interpretation that when that expression is used in our meta-language, the person who used it, is affirming that that expression is true ( affirming that theres no interpretion that makes it false ( A cant be 1 while b is 0 ) ) .
An example would be someone using the expression "It's clear that (x is in N ) -> (x is in Z)".

But sometimes i get some different interpretation from that expression.Sometimes it seems like that when that expression is used, the person who used it, is more neutral ... just like introducing an expression, that can be both true and false, depending on which valuation or model we use.
An example would be someone using the expression "The only case in which A -> B is false is when A = 1 and B=0 ".

So, we have the same expression ( an implication ) in our meta-language that seems to have two really distinct interpretations or meanings, depending on the context it is used.

Does these 2 meanings really exist ? Or am i making some huge confusion about it ?

Thanks a lot in advance

It seems to me that it is clear for you the difference between the two "contexts" : the first one is the langauge you are "speaking of", while the second one is the meta-language you are "speaking with".

This distinction cause us no problem if we are studying, let say, latin grammar: we use english (the meta-language) to explain and discuss latin grammatical rules and we do not (usually) use latin words in the english statements we use to explain latin rules, except as citations.

With logic, the situations is, prima facie, exactly the same; there is only a "little difference" : we cannot avoid to "use logic" in explaining logic rules.

So we have exactly the issue you have described; in logic (as the language I'm "speaking of") I want to introduce the logical connectives, like :

$\lor$ ("or") and $\rightarrow$ ("if ..., then _").

In the discussion about them (the language I'm "speaking with") I cannot avoid logical arguments at all; so I must rely on the connectives, like "or" and "if ..., then _", and on their function in natural language.

There is "circularity" ? Of course ...

There is difference in the underlying rules ? This is a thorny issue.

In logic, we regiment the connectives with their truth-functinal definition (in a "classical setting"); this is a sort of "modelling" that introduce a simplification: in a mathematical environment, this simplification is very useful.

Consider, for example, the connective $\lor$: in his truth-functional model, $p \lor q$ is true when both $p$ and $q$ are true; in a "natural language environment" it is not usual to assert an alternative with both disjuncts true. The usual situation where we use "or" is in a sentence like "the ball is white or the ball is black" (the alternative are mutually exclusive) and we perceive as "strange" a sentence like "the ball is black or the ball is black".

In conclusion, we have differences between formal logic and natural language bit , I think, we have no contradiction.

The complex case is with the conditional ($\rightarrow$); in a "informal" argument, when a mathematician assert $A \rightarrow B$, he is using the conditional in a truth-functional way, i.e. he is denying the case when $A$ is true and $B$ is false.

This leave us with three possible cases: $A$ and $B$ both true and the two cases when $A$ is false.

But what is the context of the mathematician's assertion ? Usually, he is asserting a theorem, like (sorry for the triviality) "if $x$ is odd, then $x+1$ is even". In our "formal" way of using the conditional this sentence is true also when $x$ is even. But of course we are not really interested into this eventuality. What we are stating is that there is a condition ("$x$ is odd") that, when fulfilled, licenses us to assert the conclusion ("$x+1$ is even").

The crucial role is played by the act of asserting: in formal logic this "act" is symbolized with the $\vdash$, that plays a crucial role in the rule of inference of modus ponens :

from $\vdash A \rightarrow B$ and $\vdash A$, infer $\vdash B$.

In this case, we assert both $A \rightarrow B$ and $A$; what happens in this case ? The first assertion "exclude" the case when $A$ is true and $B$ false, while the second assertion ($\vdash A$) exclude the two cases when $A$ is false. What remains ? The case when both $A$ and $B$ are true; and this is what we want !

An introductory logic class may give the impression that there's only one propositional logic, but that's not accurate. Modern propositional logic is a formal system. It's a mathematical system that attempts to model the behavior of logical terms in a natural language, roughly speaking.

The natural language terms we are modeling do not have precise meanings. Formal logic is an attempt to make them precise. Different logics of "If ... then _" have been constructed, for example, corresponding to disagreements about the meaning of if-then.

Probably the most famous example of a disagreement about propositional logic is Intuitionism, in which negation does not behave the same as in 'classical' propositional logic. See: http://en.wikipedia.org/wiki/Intuitionistic_logic

This seems to be an example of "the same expression ... in our meta-language that seems to have two really distinct interpretations or meanings".

• This is a good point; but still the issue remains. There are books about Intuitionistic math (like Kleene & Vesley) that are written in a "classical setting", but with a constructive approach. The "underlying logic" is classical or intuitionistic ? There are mounting interest in relevant logic and paraconsistent one: but are there studies where the metamathematics of some math theory is "done" with relevant logic or paraconsistent one ? – Mauro ALLEGRANZA Mar 10 '14 at 10:12