Process

• I care why things work
• But not how long they take or how many resources they consume.

So premises must be present but their number does not matter.

Resource Free rules

Intuitionistic: If -- then -- 

(I)NTRODUCTION$$\frac{\Gamma, P\vdash Q}{\Gamma \vdash P\Rightarrow Q}\qquad (I_{\Rightarrow})$$

(naive version)

(E)LIMINATION $$\begin{array}{rl} \Gamma & \vdash P\Rightarrow Q\\ \Gamma & \vdash P \\ \hline \Gamma & \vdash Q\end{array}\qquad (E_{\Rightarrow})$$

(Modus Ponens)

(L)ANGUAGE.        

$\langle imp\rangle ::= \langle term\rangle \Longrightarrow \langle term\rangle$

+ Weakening & Contraction

Incidence

• Care that things work.
• Don't care how long, how much it cost, or why it works.

Premise leads to conclusion for any reason is enough,

Same as simply replacing actual statements (sequents) with True or False

If -- then -- (Incidence based)

Classical: If -- then -- 

$$\begin{array}{|cc|c|} \hline P & Q & P\Rightarrow Q\\ \hline\hline \top & \top & \top\\ \top & \bot & \bot\\ \bot & \top & \top \\ \bot & \bot & \top\\ \hline \end{array}$$

Because of implicit context, classical logic seems to allow a meaningless "Formal" implication