Formal Logic 101 – Part 22: Identity and Definite Descriptions (Part 1)

Identity:

We’ve already learned about statements that involve individuals. For example, if I say “Mark Twain wrote Letters From The Earth” (Wml), Mark Twain is an individual. We’re not merely stating “someone wrote Letters From the Earth” ((∃x)Wxl), we’re stating that it was one specific person. Individuals will often be recognizable because they will be proper names, however they need not be proper names. For instance, consider the phrase “The thirteenth president of the US was a member of the Whig party.” In this “the thirteenth president of the US” denotes one specific person so this person counts as an individual.

I stated in a previous post that we can generally identify the subject and predicate of a sentence because the subject is the person, place, or thing being spoken about, and the predicate answers the question “what about them?” As an example:

Me: You know Mark Twain?
You: Yeah, what about him?
Me: He wrote Letters from the Earth.

Me: You know the 13th President of the US?
You: Yeah, what about him?
Me: They were a member of the Whig Party.

Now, with “Identity statements” we are not saying something about what they did, so much as we are speaking about who they are. For instance:
“Mark Twain is Samuel Clemens.” Or
“The 13th President of the US was Millard Fillmore.”
In identity statements both the subject and the predicate are individuals, and we are saying that these individuals are one and the same. We symbolize these sorts of statements, simply enough, with an equal sign.

So “Mark Twain is Samuel Clemens.” becomes “m=s”.

Similarly, we can have negative identity statements which state that two individuals are not the same person, such as “The 13th President of the US was not George H.W. Bush” (p≠g).

Ok great, but what do we gain from this? First of all, we gain the ability to make logical sense of the word “except”. Consider the phrase “If everyone except John failed the test, then the test was poorly constructed.” Without identity statements we could symbolize “everyone failed the test” (x)Fx, and we could symbolize “the test was poorly constructed (Pt), but we don’t have any way of carving out that the test was still poorly constructed even though John passed. Using identity statements we can make sense of the statement as follows

1. (x)[(Fx ^ x≠j) ^ ~Fj) → Pt
Translating this back to English, we get “For all x, if x failed the test, and x isn’t John, and John did not fail the test, then the test was poorly constructed”, which is functionally the same as “if everyone except John failed the test”.
 
We can also include multiple individuals as exceptions. For instance: “Everyone except for John and Tim understood the question” could be symbolized as
(x)[(x≠j ^ x≠t)→Uxq] ^ (~Ujq ^ ~Utq)

Translated back we have “For all x, if x is not John and x is not Tim, then x understood the question; also John did not understand the question and Tom did not understand the question.” So, everyone except those two understood, and those two did not.

Lastly, “only” works the same way as “except”, for instance “John was the only cashier to get a raise” could be symbolized:
(x)[(Cx ^ x≠j) → ~Rx) ^ (Cj ^ Rj)
Or “For all x, if x is a cashier and not John, then x did not get a raise; also, John is a cashier and got a raise.”
Note that this is the exact same symbolism as we would use for the phrase “every cashier except john was denied a raise.” So “except” and “only” are two words that function identically in logic.

Superlatives

Identity statements also allow us to make use of superlatives such as “Tallest”, “Poorest”, “Best”, “Heaviest” etc. Consider the statement:

“Alex was the shortest person on the team.”

Without using identity statements, the closest we can come to that is:
1. (x)(Tx → Sax) ^ Ta P
This comes to “For all x, if x is on the Team, then Alex is shorter than x; also, Alex is on the team.”

Now at first glance this may seem perfectly fine. However, consider the following steps:

2. (x)(Tx → Sax) 1 simp
3. Ta → Saa 2 UI
4. Ta 1 simp
5. Saa 3,4 MP

In 4 simple steps, I have established that Alex is shorter than themself, which is just flat out impossible. To avoid crafting impossible conclusions from perfectly reasonable premises, we would want to modify our premise as follows:

1. (x)[(Tx ^ x≠a) → Sax] ^ Ta P

Or “For all x, if x is on the team, and x isn’t Alex, then Alex is shorter than them; also Alex is on the team. This way we establish that Alex is the shortest person on the team, without implying that they are even shorter than themself.

*****
I think that will suffice for our intro to the idea of Identity statements. We’ll be picking up here next time with learning about to specify multiple individuals.

Comments

Post a Comment

Popular posts from this blog

Formal Logic 101 – Part 3: Intro to Symbolization

So far, we’ve seen fairly short three-line proofs. For those proofs, writing them out longhand is fine. However, proofs can get much longer. Consider: Argument 1 1. If Clara will come to the party then we’ll have a good time. premise 2. Clara will come to the party. premise 3. If we have a good time then we’ll come home after midnight. premise 4. If we come home after midnight then we’ll sleep in late. premise 5. We’ll have a good time. from 1 and 2 6. We’ll come home after midnight. from 3 and 5 7. We’ll sleep in late. from 4 and 6 That’s only 7 lines with fairly simple sentences, and 3 applications of modus ponens, and we’re rapidly approaching the point where I think a lot of people – if they had to do i...
Read more

Formal Logic 101 – Part 19: Proofs in Predicate Logic (Part 2)

Last time we covered two of our final 4 inference rules (Universal Instantiation and Existential Generalization). Today we’ll be wrapping up the last of our new rules. Beyond this I have a couple posts planned for less-common situations, and then the series will be done. So let’s get to it. Existential Instantiation (EI) If you remember our last past, you might have a guess about how this next rule is going to work. EG let us go from Dm to (∃x)Dx, so EI is probably going to let us go from (∃x)Dx to Dm. If that was your guess, you’re basically right, however, EI is going to have some additional restrictions on how it can be used. To understand the need for these restrictions, I will present the following example where I will INCORRECTLY use EI by treating it exactly the same way as UI. Consider the following premises: Some numbers are odd. Some numbers are even. Let us say that “Nx” is “x is a Number”, “Ox” is “x is Odd”, and “Ex” is “x is Even”. Observe the following argument: 1. (∃x)(...
Read more

Formal Logic 101 – Part 11: Replacement Rules (Part 2)

Welcome back. Today we’ll be picking up where we left off with more replacement rules. Commutation (Comm.) Commutation is a handy step for dealing with “And” or “Or” statements, where the statement you have is close to what you would need to complete another step (such Modus Ponens, for example) but is not quite exactly right. I think the easiest way to explain this would be by example. Consider the premises: If the Sun is very Large but looks very Small, then it must be Far away. The Sun looks very Small, but it is very Large. Our symbolism would be as follows: 1. (L^S)→ F                Premise (P) 2. S^L                            P Now, instinctively, I think we understand that “L^S” and “S^L” mean the same thing, but as was stated in the introduction, formal logic requires you to show your work. Every step must be documented. It’s not enough to assume that th...
Read more