More bonus, optional material. The question is this–What happens when you have an “unnecessary” premise in a deductive argument? The answer is–for **deductively valid** arguments–**nothing**!

For instance:

Argument A

- (1) If Iceland is an island, then Iceland is surrounded by water.
- (2) Iceland is an island.
- Therefore: (3) Iceland is surrounded by water. (valid; modus ponens)

To say that the argument is valid is to say that the premises deductively support the conclusion–you can’t consistently accept the premises and deny the conclusion at the same time. Adding other–redundant, useless–premises won’t change this fact. For instance:

Argument B

- (1) If Iceland is an island, then Iceland is surrounded by water.
(1a) If Iceland is surrounded by water, then Iceland is an island. <<added- (2) Iceland is an island.
- Therefore: (3) Iceland is surrounded by water. (still valid)

Adding (1a) doesn’t change the fact that (1) and (2) already validly entail (3). Nor does the fact that (1a), (2) won’t validly entail (3) (fallacy of affirming the consequent) matter. Just repeat to youself: “(1) and (2) already validly entail (3)!”

Now, you might wonder if we are just being lucky with (1a). Does the observation above depend on what has been inserted? Could there be weird additions that will break valid arguments? Turns out the answer is “no”, though showing this requires a somewhat longer explanation.

Let me begin by invoking a more technical way to state the definition of a valid argument:

An argument is valid if and only if it is logically impossible for all the premises to be true and the conclusion false.

Or put another way, given a valid argument, the collection of statements that consists of all the premises plus the *negation* of the conclusion cannot all be true at the same time.

Now let’s introduce an outrageous additional premise to the otherwise perfectly valid Argument A. How outrageous? How about a premise that **contradicts the conclusion**?

Argument C

- (1) If Iceland is an island, then Iceland is surrounded by water.
- (2) Iceland is an island.
(2a) Iceland is not surrounded by water. <<added; contradicts (3)- Therefore: (3) Iceland is surrounded by water. (valid)

Since Argument A is valid, the collection–(1), (2), plus the negation of (3)–can’t all the true at the same time. Now we add to it (2a). Since (2a) contradicts (3), we are really talking about the negation of (3) again. But that just means that it’s as if we have expanded the collection to–(1), (2), plus the negation of (3), plus the negation of (3) again. If the original collection can’t all the true at the same time, this one won’t be any different. Therefore, Argument C is valid as well.

What if the additional premise **contradicts an existing premise**? Now things get even stranger–

Argument D

- (1) If Iceland is an island, then Iceland is surrounded by water.
- (2) Iceland is an island.
(2b) Iceland is not an island. <<added; contradicts (2)- Therefore: (3) Iceland is surrounded by water. (valid)

Since Argument A is valid, the collection–(1), (2), plus the negation of (3)–can’t all the true at the same time. But what about the new expanded collection? The collection–(1), (2), (2b), plus the negation of (3)–likewise can’t all be true at the same time because it contains the contradiction (2) & (2b)! Therefore, Argument D is valid.

In case you haven’t noticed, if the contradicting premise is added to an argument that wasn’t valid earlier, the new one is now valid. Consider:

Argument E

- (1) If Iceland is an island, then Iceland is surrounded by water.
- (2b) Iceland is not an island.
- Therefore: (3a) Iceland is not surrounded by water. (Invalid; denying the antecedent)

Since Argument E is invalid, the collection–(1), (2b), plus the negation of (3a)–**can** all be true at the same time. Now add a new premise, one that contradicts an existing premise:

Argument F

- (1) If Iceland is an island, then Iceland is surrounded by water.
(2) Iceland is an island. <<added; contradicts (2b)- (2b) Iceland is not an island.
- Therefore: (3a) Iceland is not surrounded by water. (valid)

The collection–(1), (2), (2b), plus the negation of (3a)–**can’t** all be true at the same time–because (2) contradicts (2b)!

The more general lesson is this: **any collection of premises that includes a contradiction always get a free pass when it comes to the issue of validity**.

Yes, it’s very weird. But guess what, this is actually an idea that is quite deeply embedded in standard logic. And a little bit of it will will make a cameo appearance in the Problem of Evil topic, if you know how to spot it…

Thankfully, arguments in the style of Argument D/F cannot be **sound**–since the premises contain a contradiction, they can’t all be true. What about arguments in the style of Argument C, where a premise contradicts the conclusion? To recall:

Argument C

- (1) If Iceland is an island, then Iceland is surrounded by water.
- (2) Iceland is an island.
(2a) Iceland is not surrounded by water. <<added; contradicts (3)- Therefore: (3) Iceland is surrounded by water. (valid)

Since the underlying Argument A is valid, the collection–(1), (2), plus the negation of (3)–can’t all be true. But the additional premise (2a) is equivalent to the negation of (3). It will thus follow that the collection–(1), (2), plus (2a)–can’t all be true. Therefore, again, arguments in the style of Argument C cannot be **sound.**

I guess that’s the silver lining in all this weirdness…