# Is there a uniform way of differentiating sufficient and necessary conditions?

I am struggling to formulate symbolic conditional logic rules from basic sentences (studying for the LSAT).

It seems that subtle differences in syntax are throwing me off. Is the conditional relationship between antecedent and consequence uniformly fixed by syntactical rules, such that I might be able to consistently identify which one is which? Or is it a free-for-all and I have to somehow intuit the distinction.

An example might read as follows:

Raimundo appears in every photograph that Yakira does not appear in

which apparently translates to

Not Y → R.

From Jan von Plato, Elements of Logical Reasoning (Cambridge UP, 2013), pag 10 :

The two sentences "if A, then B" and "B if A" seem to express the same thing. Natural language seems to have a host of ways of expressing a conditional sentence that is written A → B in the logical notation. Consider the following list :

From A, B follows; A is a sufficient condition for B; A entails B; A implies B; B provided that A; B is a necessary condition for A; A only if B.

The last two require some thought. The equivalence of A and B, A ↔ B in logical notation, can be read as A if and only if B, also A is a necessary and sufficient condition for B. Sufficiency of a condition as well as the 'if' direction being clear, the remaining direction is the opposite one. So A only if B means A → B and so does B is a necessary condition for A.

It sound a bit strange to say that B is a necessary condition for A means A → B. When one thinks of conditions as in A → B, usually A would be a cause of B in some sense or other, and causes must precede their effects. A necessary condition is instead something that necessary follows, therefore not a condition in the causal sense.

• Thank you for this comprehensive response. So in relation to my question, am I correct in concluding that the conditional relationship between antecedent and consequence is not uniformly fixed by syntactical rules, such that I might be able to consistently identify which one is which? These examples have proven quite helpful for my understanding of these relationships. Thanks. Commented Sep 30, 2015 at 5:40

There are innumerable ways to express conditionals, so even if there are standard ways to translate some syntactic structures into logical notation (e.g., "If A, then B" = "A only if B" = A → B), you can only get so far with these.

Also, not every sentence that seems to be a conditional is actually a conditional. Consider the difference between "Men are grumpy when they're hungry" and "Men are bachelors when they're unmarried" which have the same syntax but different logical form.

So you need 2 things. First, a way to tell if a sentence is really conditional, and second, a way to tell which of the conditions in a conditional sentence is sufficient and which is necessary.

The easiest way to do this is to draw a truth table for your sentence. Basically, you have to ask if the sentence you're considering is true or false for every possible distribution of truth values for it's clauses. For this sentence:

Raimundo appears in every photograph that Yakira does not appear in.

You have to consider what happens if:

R appears and Y does not = T (Both clauses obtain)

R appears and Y appears = T (1st clause obtains but second does not)

R does not appear and neither does = F (2nd clause obtains and 1st one does not)

R does not appear and Y does = T (Both clauses are false)

When only one of these possibilities makes your sentence false, and it is one in which one of the clauses is false and the other is true, you know that you're dealing with a conditional sentence. The false clause in that possible world is your necessary condition, the true one is a sufficient condition.