Quick n Dirty: if and only if

[_MrStakhanovite]

In my last installment, I introduced the basic concept of propositional logic and conditional statements. We did briefly touch on how to draw inferences, but my goal here is not to teach propositional logic, but to borrow some ideas to help structure arguments. Our next step is to understand necessity and sufficiency.

To say that X is necessary for Y, what we are really saying is that the absence of X will guarantee the absence of Y, you can’t have Y without X. For example, it is necessary to be born in the United States if one wants to become the President, but there must be other components to becoming the President because not every natural born citizen is President.

To say that X is sufficient for Y, what we are really saying is that X guarantees the presence of Y, if you have X you must have Y. For example, being a mother is sufficient for being a woman.

Now there are four ways the logical components X and Y can relate to each other:

1: X is necessary but not sufficient for Y (having three sides is necessary but not sufficient for being a triangle.)

2: X is sufficient but not necessary for Y (having a daughter is sufficient for being a mother, but not necessary)

*3*: X is both necessary and sufficient for Y (being a bachelor is both necessary and sufficient for being an unmarried man)

4 X is neither necessary nor sufficient for Y (being an atheist is neither necessary nor sufficient for being intelligent)

Out of these four relations, I want to focus in on the third, because it is an essential component in understanding modern epistemology. Number 3 is called a “bi-conditional statement” and is related to the conditional statements we learned about last time.

Here are a few ways English uses bi-conditional statements:

X exactly if Y

X just in case Y

X is a necessary and sufficient condition for Y

X, but only if Y

To symbolize a bi-conditional statement, we’ll use this: X <~> Y

As the name suggests, a bi-conditional statement is telling us two things. One is we can derive X ~> Y (if X then Y) and second, we can also derive Y ~> X (if Y then X). Another way symbolize a bi-conditional statement that is very common in Philosophical literature is “iff”, which stands for, “ if and only if”. For example, let’s consider this statement:

(P1) Brade is a parent iff he has a Child.
(B <~> C)

The two sentences that we can logically derive from this statement is:

If Brade is a parent then he has a Child
( B ~> C)

If he has a Child then Brade is a parent
( C ~> B)

One way to look at ‘iff’ and ‘<~>‘ is to think of an equal sign:

Brade is a parent = he has a Child.

So if I wanted to prove P1, I would need to find some way to satisfy the left or right side of the equation and I’d establish the truth of P1. Let’s say I’m able to get a copy of Brade’s tax return and I see the dependents he’s claimed, after that, I interview all the children in his house to see if Brade is their dad. After getting all that confirmation, I assert C (which stands for: he has a child). Now I can show how a bi-conditional works:

(P1) B <~> C beginning assumption
(P2) C beginning assumption
(P3) C ~> B derived from P1
(P4) B inferred from P2 and P3
(P5) B ~> C derived from P1
(P6) C inferred from P4 and P5

What the above shows is how a bi-conditionals has two sides that are essentially equal, and meeting the conditions on the left side will meet the conditions on the right side (and vice versa).

Thankfully, we’ve covered all the concepts we need to know to finally move on to epistemology, which will be the topic of my next installment.