Question regarding validity of argument

Question

In Hausman, et. al. (2013), we are given an example of an unsound, invalid argument from two premises that are considered to be true:

P1 Either George Washington was an American President, or Abraham Lincoln was an American President
P2 George Washington was an American President
C Abraham Lincoln was an American President

I understand that both premises are true, but I still do not understand how this is an invalid argument. Is it because of the "or" connective? Considering a statement with "or" connective is true when one or more of its operands are true.

What conclusion and how is it that we can change this argument to be valid?

Answer 1

P1 Either George Washington was an American President, or Abraham Lincoln was an American President

That premise isn't actually true. Unlike TKoL's example of:

Obama was an american president or Arnold Schwarzenegger was an American president.

The operand isn't an OR or logic disjunction, but an "either ... or" or XOR or logical inequality. And for A XOR B, if either A or B is true, the other MUST be false for the operation to be true. So A and B must not have the same truth value.

So the valid statement would be:

P1 Either George Washington was an American President, or Abraham Lincoln was an American President

P2 George Washington was an American President

C Abraham Lincoln was NOT an American President

(Under the assumption that these names refer to definitive dead people who's status cannot change over time), this would be valid. If the premises would be all true and only one of them could be president and you know which one it was, then the other cannot have been president.

Now it's still not sound because we know that both have been presidents so the first statement is false.

The other case of how true premises can produce an invalid conclusion is shown by causative and TKoL, in that if you use an OR rather than an XOR and show that one of the conditions of the OR is True, then the other could be anything and the statement would still be true. Though as it could be anything you can't conclude that it must be true as well.

Edit: Though if you of course had no idea of who George Washington and Abraham Lincoln are and just known them as A and B placeholder you might be able to argue that the premises are true because so far you'd not have a contradiction. (A XOR B) AND A could still fit and would imply ¬B though not B, so yeah that would be invalid (and as validity is a requirement of soundness, also obviously unsound) because it's the opposite of what it would actually imply.

Answer 2

Consider that not all arguments in the exact same format are valid and with a true conclusion, even if their premises are true.

Obama was an american president or Arnold Schwarzenegger was an American president.

Obama was an American president.

Therefore, Schwarzenegger was an American president.

Think of classical logic as a set of allowable operations, like mathematics. A or B, A, therefore B is not one of the allowable operations, nor is it derivable from any allowable operations.

how is it that we can change this argument to be valid?

That really depends, the way you asked this question implies you think there's some kind of valid argument in there.

Here's one possible valid argument.

A and B. Therefore B. If you change the or to an and, the conclusion follows.