After a inspiring catch-up dinner with the founder of Clinck, I’ve just resumed my week’s reading. I’m covering mathematics… the basics… conditional statements and that brings me to material conditional.
While looking at the $p \rightarrow q$ (if $p$ then $q$ or even better $p$ implies $q$) statements I noticed something curious – given a $false$ antecedent, the statement is always true.
From a computing point of view, one is used to formulating if/then statements as a means to describe causal relationships (simply stated: if $p$ happens to be true, make sure $q$ happens). But in terms of classical logic, we are not describing causal relationships but simply evaluating the truthfulness of the statement. We are basically saying that $p$ implies $q$.
Constantly ask yourself, is the statement true?
I have a silly case to demonstrate the evaluation of a statement. Assume we use the antecedent $p$ to represent that a creature has green blood and the consequent $q$ to represent that a creature is an alien simply read as if a creature has green blood, then it is an alien.
The Venn diagram has surrounded the subject we’re talking about (our antecedent $q$) with a white outline, call it the spotlight . Our consequent is represented by another circle which may overlap the antecedent wherever both conditions hold $true$. In case we have a truthful condition (because both conditions hold $true$) the entire area is white. False conditions within the spotlight maintain their non-white color and everything else simply falls beyond the scope of the statement we’re making (resides in darkness, outside the spotlight), after all we’re simply stating something about creatures with green blood.
When A Truth Implies Another Truth
If a creature has green blood $p=T$ and the creature happens to be an alien $q=T$ the statement is clearly true. We’re not bullshitting anyone when stating that (see the section labelled vicious goo-stuffed ET’s represented by the white area in the Venn diagram).
When A Truth Implies A Lie
Upon the discovery of a creature with green blood $p=T$ which somehow does not happen to be classified as an alien $q=F$ (perhaps some mutant venturing through Gotham) then the statement is suddenly discredited ($p\wedge\neg q$, read as $p$ and the inverse of $q$). The green-blooded non-alien lifeforms (captured by the green area in the Venn diagram) are represented by this statement.
When A False Statement Implies Anything
If the creature has blue blood and happens to be an alien, the statement is still true because we initially only said something about the green-blooded critters which can not be discredited by this unrelated observation (unrelated because blue-blooded creatures have no business in a discussion about green-blooded ones). Remember that the our statement really reads green blood implies alien. Without green blood we’re not even having this discussion.
If the creature has blue blood and does not happen to be an alien, the statement is still true because yet again our statement only said something about that which bleeds green. Take the dark blue area in the Venn diagram. It covers all non green-blooded creatures that are not extra-terrestrial (humans fall into this category). Why are we bringing this up? Out of scope.
In short, anything with a false antecedent $q=F$ will end up being a truthful statement because we have formulated our statement as such to only describe the cases where $q=T$ (think spotlight), therefore the making of other claims does not result in any logical discrepancies.
To stick with the blood color theme, stating something about green-blooded creatures can not possibly be discredited by showing up with a non-green blooded speciment, regardless of the peculiarities one wishes to demonstrate. The claim about green-blooded creatures still stands.
The interesting behavior of material condition becomes more natural when looking at it’s logical equivalent $\neg(p\wedge \neg q)$. One can clearly see in the logical equivalent that the falseness of $p$ will always result to a truthful outcome of the statement. Basically a truthful statement will require a $false$ within the parentheses in order to evaluate to $true$. This may help in constructing truth tables, but really… thinking about green-blooded aliens will too.