A more appropriate heading for this would be “How a logical truth can be a lexical lie”, but hey, gotta have that clickbaity title. But nevertheless, I will frame this article as if I am addressing the title.

Apparently, sociologists/psychologists classify lies with a three tier system; primary, secondary, and tertiary. According to an article on Psychology Today, children as young as 2-3 tell have developed the ability to tell lies. And children of age 7-8 have developed the skill to tell what is dubbed “tertiary lies”, which are lies that are *“more consistent with known facts and follow-up statements”. *

But how does telling a lie relate to mathematics? And exactly what tools can you use for such?

There exists a branch of logic, where logic is a branch of math, called propositional logic. Propositional logic is all about combining statements. A statement is something you proclaim, that is either true or false. This can be an English sentence, or a mathematical expression.

Ideas in the field of logic are quite useful and have vast applications, such as creating electronic circuits called logic gates which power up pretty much almost every device that carries out computations like your smartphone, laptop and calculator. You also implicitly implement these ideas when you reason, e.g. when you convince yourself to stay in bed instead of attending your morning lecture/s. As an extension, you might’ve already been explicitly making use of these ideas in proof writing. e.g.

A simple form of proof by contradiction.

by way of contradiction

Here, P is a statement about all elements in a set and is set to be true.

Show

Therefore it follows

One important idea you need to take note from the above is the implication from the conditional statement. That a premise being true, implies that its conclusion is also true. This is called the rule of modus ponens. If the conclusion is false, then that implies the premise is false, this rule is called the modus tollens. In formal logic, this would look like this:

if true, then true (Modus Ponens)

if false, then false (Modus Tollens)

We then further define two other vacuously true conditional statements which are key to the lexical lie.

A vacuously true statement is a statement* “that asserts that all members of the empty set have a certain property”. *In formal logic we write

where

The truth table, which describes the possible combinations for the state of the statements (whether they’re true of false), looks something like this:

(source)

The first row applies Modus Ponens, the second, Modus Tollens and the last two are taken to be vacously true.

So, what does this mean?

Suppose you just built a time machine and you go back to the 1960s, you meet an optimistic young hippie who is aware of the time period you come from (2019), and makes the assumption that humanity has successfully landed on Mars by now. He then asks you, “so like… dude, what did the first man on Mars say”, and you respond, “one small step for man, and a killing for SpaceX”. The fact that his statement was “empty”, means that whatever you say in response to that statement is logically true, therefore, you’re logically asserting a true statement. In everyday English, we call this misleading.

A simple example is saying “my Porsche is parked down by Fuller”. Notice the statement is about where my Porsche is parked, not about its existence. Turns out I don’t have a Porsche, but my statement is logically true.

😉

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