Intuitionistic Logic

Last time, we discussed Dummett's motivation for insisting that each logical operator must be independently defined (molecularism required for a justification of the coherence of our reasoning), and independently defined in terms, not of truth, but of verification, conditions, which, in the case of mathematics, means proof conditions (use thesis).

So, to say what the logical particles "mean" it is necessary and sufficient to produce a proof system in which each connective is defined independently of the others and in which each connective is a conservative extension of the others: That is supposed to mean that no rules, for example, for negation, allow us to deduce any new formulas not involving negation.

Here goes:
$A,B\vdash A\land B$, $A\land B\vdash A$, $A\land B\vdash B$
$A\vdash A\lor B$, $B\vdash A\lor B$, if $A\vdash C$ and $B\vdash C$ then $A\lor B\vdash C$
Our language does not have a symbol for negation, instead, it has a symbol $\perp$ for falsehood. We then will use $A\rightarrow\perp$ as a replacement for $\lnot A$. There is only one rule for $\perp$: $\perp\vdash A$.
If $A\vdash B$ then $\vdash A\rightarrow B$, $A,A\rightarrow B\vdash B$.
$A(\tau )\vdash\exists xA(x)$, $\exists xA(x)\vdash A(c)$, where $c$ is new.
$A(x)\vdash\forall xA(x)$, where $x$ is new. $\forall xA(x)\vdash A(\tau )$.

Classical logic can be obtained from this system by adding any one of the following rules:
$\vdash A\lor\lnot A$, $\lnot\lnot A\vdash A$, if $\lnot A\vdash\perp$ then $A$.

Motivations for taking this logic to be right and classical logic to be wrong. We considered Dummett's favored argument above, but it requires that intuitionistic logic be the only correct logic everywhere. Thus, considerations that have nothing directly to do with mathematics could torpedo the thing.

So, is there an argument that just applies to mathematics. Here is what Dummett proposes: Suppose there is no such thing as a mathematical object. Then, there are no truth conditions (and no holist considerations relating them to other stuff). The only possibility, then, is verification conditions, and hence intuitionistic logic.

-- ProfessorShaughanLavine? - 28 Jan 2005