James B. Wilson
Logical Alchemy: Turn False into True.
Chemical Alchemy: Turn Straw into Gold.
Logical Alchemy: Turn False into True.
(L)anguage, (I)ntroduction, & (E)liminiation of \(\Rightarrow\)
If \(x\geq y\) then \(x^2\geq y^2\)
\(x\geq y\quad \Rightarrow \quad x^2\geq y^2\)
if x >= y then
do something!Lets deal with truth and leave computation "if -- then --" untill later,
Rule breaking
\(x^2\geq y^2\), if \(x\geq y\)
When \(x\geq y\) it follows that \(x^2\geq y^2\)
if = when= because ....
then = implies = follows = thus...
Non-implications that use the same language
A triangle is equilateral if all its sides have equal length.
Term being defined
"If" used to separate equivalent clause
Definitions are names/symbols to substitute often lengthy or technical expressions, e.g. \(x:=\sqrt{17}^{\log 541}\)
(L)anguage, (I)ntroduction, & (E)liminiation of \(\Rightarrow\)
If it rains then the ground is wet.
\[\begin{array}{rl} \Gamma & \vdash P\Rightarrow Q\\ \Gamma & \vdash P \\ \hline \Gamma & \vdash Q\end{array}\qquad (E_{\Rightarrow})\]
(E)LIMINATION
"Modus Ponens"
It rained.
Therefore the ground is wet.
Premise
Conclusion
\[\begin{array}{rl} & P\Rightarrow Q\\ & P \\ \hline & Q\end{array}\qquad (E_{\Rightarrow})\]
(L)anguage, (I)ntroduction, & (E)liminiation of \(\Rightarrow\)
If \(x\geq y\) then \(x^2\geq y^2\)
\[\frac{P\vdash Q}{ P\Rightarrow Q}\qquad (I_{\Rightarrow})\]
(I)NTRODUCTION
\(\Gamma\): Natural numbers \(m\geq n\) means there is a \(k\) where \(m=n+k\)
\(m\geq n\) means there is a \(k\) with \(m=n+k\)
P
Q
Process states \(\Gamma,P\vdash Q\)
If \(x\geq y\) then \(x^2\geq y^2\)
CONTEXT MATTERS
If \(-3\geq -4\) then
\((-3)^2=9\geq 16=(-4)^2\)
In the natural numbers, if \(x\geq y\) then \(x^2\geq y^2\)
In the integers, \(-3\geq -4\) but
\((-3)^2=9\geq 16=(-4)^2\)
\[\frac{\Gamma, P \vdash Q}{\Gamma \vdash P\Rightarrow Q}\]
\(\Gamma\)
\(\Gamma\)
If -- then --
<imp> ::= if <term> then <term>
<imp> ::= <term> => <term>
\(\langle imp\rangle ::= \langle term\rangle \Longrightarrow \langle term\rangle\)
\[\frac{\Gamma, P\vdash Q}{\Gamma \vdash P\Rightarrow Q}\qquad (I_{\Rightarrow})\]
(I)NTRODUCTION
(naive version)
\[\begin{array}{rl} \Gamma & \vdash P\Rightarrow Q\\ \Gamma & \vdash P \\ \hline \Gamma & \vdash Q\end{array}\qquad (E_{\Rightarrow})\]
(E)LIMINATION
(naive version/ modus ponens)
(L)ANGUAGE
Logical Alchemy: Turn False into True.
If -- then -- (Vagueness)
If pigs can fly then pigs can get bird flu.
This is a valid statement because pigs can can get bird flu.
However, pigs cannot fly.
What explains this situation?
If -- then -- (Vagueness)
If pigs can fly then pigs can get bird flu.
It is seems illogical to argue that a false premise leads to a true conclusion
What explains this situation?
False Premise P
True conclusion Q
However context is always available as well!
Context (maybe implicit) may be responsible for true conclusion, i.e. \(\Gamma \vdash P\)
Classical & Intuitionistic context even have \(\bot \vdash Q\)
If -- then -- (Vagueness)
False Premise P
True conclusion Q
\(\Rightarrow \)
This is a confusing abbreviation of reality.
Instead read this as:
Context, or
False Premise P
\(\Rightarrow \)
True conclusion Q
Since context is often implicit never forget to ask if context is responsible for implications.
Summary
- False \(\not\Rightarrow\) True,
- but Context implies True \(\Gamma \vdash \top\) and context is always around for implication.
- This is still a naive take on implication, several things are still to consider
- Process (Intuitionistic) vs. Incidence (Classical)
- Resource counting (Linear logic)