- True or False? If monkeys can fly, then 1 + 1 = 3.
- What is logically equivalent to all x (p(x) + ~q(x))?
For the first one I think it is False.
There are two questions.
The antecedent of the conditional, "monkeys can fly" is false. So is the consequent, "1 + 1 = 3". In classical truth-functional logic the conditional connecting these two sentences also has a truth-value. Wikipedia describes this "material conditional" as follows:
The material conditional (also known as material implication, material consequence, or simply implication, implies, or conditional) is a logical connective (or a binary operator) that is often symbolized by a forward arrow "→".
This binary operator would return true, based on the truth table Wikipedia provides. This is because the material conditional is defined to be false only when the antecedent is true and the consequent is false. Otherwise, even when the conditional doesn't make sense, it is defined as true.
Because the antecedent and the consequent are unrelated there may be objections to assigning a truth-value to such a statement or to assigning the value true. See the Wikipedia entry for relevance logic or the SEP article on relevance logic for a discussion.
I assume the statement is ∀x(P(x) v ¬Q(x).
Using Wikipedia's list of logical equivalences suggests a conditional that would be logically equivalent to the disjunction.
p ⟹ q ≡ ¬p ∨ q
Since ¬p ∨ q is logically equivalent to p ⟹ q, p v ¬q should be logically equivalent to ¬p ⟹ ¬q. This suggest that ∀x(P(x) v ¬Q(x) is logically equivalent to ∀x(¬P(x) ⟹ ¬Q(x).
For added confirmation, one can use a tree proof generator to see if one could derive such an equivalence:
Wikipedia contributors. (2019, May 27). Material conditional. In Wikipedia, The Free Encyclopedia. Retrieved 18:44, August 11, 2019, from https://en.wikipedia.org/w/index.php?title=Material_conditional&oldid=898972669
Tree Proof Generator. https://www.umsu.de/logik/trees/
Please note that my answer is entirely based on my personal expertise, and therefore not necessarily always representative of all logicians from Aristotle to the Scholastic, and definitely not of mathematical logic.
True or False? If monkeys can fly, then 1 + 1 = 3.
This argument is not formally valid (see note).
Validity is assessed on form. Nothing in the form of this argument could help us decide whether the antecedent "monkeys can fly" is true or false and whether the consequent "1 + 1 = 3" is true or false. Thus, the four basic combinations of a true or false antecedent and a true or false consequent are all a priori possible.
In particular, it is formally possible that the antecedent is true and the consequent is false, which makes the implication If monkeys can fly, then 1 + 1 = 3 false in this case, and thus not valid.
You could make the antecedent formally false, as was probably your intention, by modifying it as follows:
It is false that monkeys can fly;
It is true that monkeys can fly;
Therefore, 1 + 1 = 3.
As rephrased, the argument now has contradictory premises, making the conjunction of the premises false, and this irrespective of whether monkeys can or cannot in fact fly.
According to most methods in mathematical logic, arguments with contradictory premises (or a false antecedent) are all valid.
According to my own experience, however, it is clear that most people who are not trained in mathematical logic will say that the argument is not valid. Many people will even say that the argument doesn't make sense, which is somewhat stronger since it implies that it is also not valid.
So, you may want to keep in mind that the theoretical definition of validity used in mathematical logic isn't equivalent to what most people have in mind.
I would certainly recommend not to walk into a bar on a Saturday night to use arguments with contradictory premises to try and convince big guys with tattoos there that their logic is just wrong because mathematical logic says so.
On the other hand, if you plan to have a career in mathematics, may be you can try to see the good side of the definition of validity as used in mathematical logic.
Note on "formally valid"
Formally valid just means what it says, i.e. valid according to the outward form, or ostensive structure, of the argument, in contrast to the semantic content, or meaning, of the argument.