James B. Wilson

Implication for Process or Incident

"If --- then --- " when 

  • I care about process

  • I care about incidents

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

Weakening \[\frac{A \vdash B}{A,C\vdash B}\]

Contraction \[\frac{A,A \vdash B}{A\vdash B}\]

With implication \[\frac{\Gamma,P,\ldots,P \vdash Q}{\Gamma,P\vdash Q}\]

With implication \[\frac{\Gamma,P \vdash Q}{\Gamma,P,\ldots,P\vdash Q}\]

True for facts

False for scheduling (too many cooks in the kitchen)

True for facts

False for resources

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

"If --- then --- " when 

  • I care about process

  • I care about incidents

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)

Never allow "If True then False"

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

Allow

  • "If True then True"
  • "If False then False" 
  • "If False then True"

 

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

Summary

  • Process Implication:  Add weakening & contraction to reuse or reduce premises.  (Intuitionistic Logic)
  • Incidence Implication: Ignore reasons, just store truth values.  (Classical Logic)

 

Classical Logical Alchemy: Turn False into True.

If -- then -- 

If \(x>0\) then \(x^2>0\)

  • Issue 1: what is the context?  What is \(>\) and \(x^2\) and 0?
  • Issue 2: what is this saying?
    • Is it a process that takes evidence for \(x>0\) to evidence for \(x^2>0\)?
    • Is it an observation that \(x>0\) always precedes \(x^2>0\)?

Sentence (i.e. what we can attempt to judge  as true)

If -- then -- (Programs)

if list.length > 0 then print list.head

test

task

If -- then -- (Process based)

(L)ANGUAGE    <imp> ::= if <term> then <term>
                <imp> ::= <term> => <term>

   \(\langle imp\rangle ::= \langle term\rangle \Longrightarrow \langle term\rangle\)

\[\frac{P\vdash Q}{P\Rightarrow Q}\qquad (I_{\Rightarrow})\]

(I)NTRODUCTION  

 

\[\begin{array}{rl} & P\Rightarrow Q\\ & P \\ \hline  &  Q\end{array}\]

(E)LIMINATION