1. True or False? If monkeys can fly, then 1 + 1 = 3.
2. What is logically equivalent to all x (p(x) + ~q(x))?

For the first one I think it is False.

• 1) is true. Think about true conditionals as promises that were upheld. I promise you that if monkeys can fly, then 1+1 is equal to 3. Did I violate my promise? No, because I would violate it only if monkeys could fly, but 1+1 would still not be equal to 3. But this didn't happen. Aug 11 '19 at 18:28
• Thank you. I changed it to True before I turned it in, Aug 11 '19 at 21:18
• @user161005 - Are you assuming the "if A then B" formulation in English is equivalent to a logical proposition of the form A -> B? If this is a homework problem then probably that was the intent, but conditionals can also be interpreted in terms of modal logic which deals with possible worlds, David Lewis apparently developed a modal logic analysis of conditionals...it isn't true that in a possible world where monkeys can fly, 1+1=3, so I assume the statement wouldn't be true under this sort of translation. Aug 11 '19 at 22:27
• What needs to be understood here is that this is a math thing & a bunch of people are getting confused about. When the average Speaker of English hears the word true or false they THINK IN REALITY. So they are confused with the “only in this math class scenario” which is NOT EXPLICITLY STATED as distinct from true in reality. So what the math person MEANS is TRUE OR FALSE BY TRUTH TABLE only. had that been explicitly explained so many human being would not be getting confused. Tell the student to study truth tables, then draw a truth table for the following statements. That would be so easy. Aug 12 '19 at 3:16
• @Logikal Decisions made by Aristotelians and Scholastics are not 'a math thing'. Ex Falso Quodlibet has a long and honored history outside math, or it wouldn't have a Medieval Latin name. The Classical rules like this one, the Excluded Middle, and the rest of these untranslated or strangely translated named rules exist to simplify reasoning, and they focus on avoiding contradiction over preserving trust. But they did so long before logic was a branch of mathematics. And people argued about them then, too.
– user9166
Aug 12 '19 at 17:26

There are two questions.

1. True or False? If monkeys can fly, then 1 + 1 = 3.

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.

1. What is logically equivalent to all x (p(x) + ~q(x))?

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.

• And argument with one false premise is always valid according to Ex Falso Quodlibet, a longtime fixture in classical logic. So something is lacking in this notion that validity is related only to form, or the notion of what constitutes form. You can omit any premise you don't use, and False implies anything. Again, present for Aristotle to talk about, so much, much older than the related math.
– user9166
Aug 12 '19 at 17:33
• @jobermark Sure, it is clear something is lacking to formal logic as a whole, and this after 2,500 years of erudite investigation. But, I think there are very good reasons for that. Try to articulate a method to produce the same result as any of our thought processes... And we don't even know if human logic isn't somehow fundamentally flawed, making any "perfect" model irredeemably incorrect. Aug 12 '19 at 17:51
• @We kind of do. Russel's paradox exists. And I am not sure what your comment has to do with my objection. Your first sentence is false, the argument is formally valid in traditional systems. It is not formally valid in more modern ones, but those generally don't use the term 'valid' in a formal sense anyway.
– user9166
Aug 12 '19 at 17:55
• I actually don't think this answer deserves being downvoted--the OP wasn't clear whether or not to interpret question 1 as an argument or a statement. As a statement evaluated in the real world, it is formally true, but as Speakpigeon explains, as an argument is formally invalid.
– jhch
Aug 12 '19 at 18:20
• @Speakpigeon No, but I think that when you use a language they should not deviate severely from either the most common, or the documented standard usage. The people who say 'formally valid' most often don't mean by it what you mean, nor is it the norm in documents. It is like deciding dog means horse and going with it because it works for you. Go ahead, but don't promote your solecisms as if you are an expert.
– user9166
Aug 13 '19 at 20:46