Additive Inference Drill #3 and #4

[jtwl54]

Hi,

I need explanation regarding two questions in Formal Logic Additive Inference Drill (p.334) on the same issue.

#3.
No E's are F's
All F's are G's
All G's are H's

#4.
Some T are U's
All U's are V's
All T's are S's

In the drill answer key, both questions contain one inference that is "made by recycling either one of the two previous inferences." But I can't understand it for the following reason.

#3.
I agree with the answer key's mention that I can make inference H<--s-->/E via F--->H.
But how can I make the same inference by G<--s-->/E?
In that case, two 'some's appear in succession (H<--s-->G<--s-->/E), which cannot make any inference.

#4 has exactly the same problem.
Making inference S<--s-->V by recycling either T<--s-->V or U<--s-->S is impossible since two 'some's come together, from which no inferences can be drawn.

I'd like to pick your brain on it.


I have two more questions:

Is the additive inference bound to adding only "2 statements"? not more than 2?

What is the difference between conditional reasoning and formal logic?

[Jon Denning]

Hey jtwl,

Thanks for the questions and welcome to the Forum! A couple of tricky ideas here, but hopefully I can help you sort things out.

The "recycling a previous inference" simply means that sometimes when you make a discovery about a variable (or variable connection/pair), you can then take that new information back and use it to make further inferences. Let's see how that works for both #3 and #4:

3. I wonder if perhaps you've diagrammed something backwards here, although I can't figure out what since you correctly identified F ---> H...

Regardless, take the contrapositive of the first statement, and you can form one long chain of connections as follows:

H <--- G <--- F ---> E

Forget the "some" for a moment, and look at the absolute inference: F ---> H. That's the inference we can "recycle," so to speak, here, as it let's us connect both G and H to the NOT E.

So now we can make two 3-part linkages, both involving F:

G <--- F ---> E

H <--- F ---> E

That gives two "some" inferences once we replace the "<---" with , as shown by these chains:

G F ---> E , so G E

H F ---> E , so H E

4. This one does the same thing, although with an arguably easier chain:

S <--- T U ---> V

First, make your two "some" inferences:

T V

U S

But remember, the T and the U both have absolutes attached to them from the original diagram, so "recycle" one of the some inferences (let's use T V, although they both work to produce the same outcome) back into the first diagram:

S <--- T V

Now we get a final inference: S V

At no point in either example were two "some" statements combined; rather, it was always a "some" followed by an absolute.


Finally, let's look at your last two questions, which I've numbered for convenience:

1. Additive inferences really just mean truths that result from the combination (the addition) of multiple statements, so it could be 2, or it could be more. Usually we tend to just look at two statements at a time for inferences though (like A --> B, and B --> C...what results?), so that's typically how you'll see them addressed.

2. There's some blurriness between conditional reasoning and formal logic, certainly, but the difference to me is really this: conditional reasoning is simply the absolute nature of a relationship between two or more variables/terms/conditions that can be used to potentially determine truths about their behavior (like "if A, then B"), whereas Formal Logic is almost a mathematical process that tends to incorporate conditional reasoning with other , less absolute relationships ("some," "most," etc) and asks you to specifically make inferences based on a series of connections. In other words, conditional reasoning says "these things are related in this strict way," while formal logic says "here's everything we know about a bunch of variables and the numerous types of relationships...what can you deduce?"

As a result, conditional reasoning happens all over the place--13% or so of LR questions, more than 50% of Games--and Formal Logic is extremely rare (maybe 2-3% of LR and that's it).


I hope that helps clear things up!

Jon

[Steve10297]

Hi Jon,

In the below forum, you said "Regardless, take the contrapositive of the first statement, and you can form one long chain of connections as follows," are you talking about the "No E's are F's" statement? If so, how do you take a contrapositve of a double not arrow ( )? Could you show step by step?

Thank you,

Steve

[Jon Denning]

Hi Steve,

Thanks for the questions and welcome to the Forum!

To your first question about "No E's are F's," yes, that's the one whose contrapositive is useful here. Originally it's:

E F

But the contrapositive is more useful since NOT F doesn't point to G, regular F does. The contra then:

F E

And now we can form a chain as I show above using the shared variable, F.

Now, to your broader question about the Double Not Arrow. Man. We get a lot of questions about this, as you can probably imagine, especially for people in the early stages of conditional reasoning. So my first piece of advice is, until you're really comfortable with diagramming conditional statements, taking contrapositives, forming chains, and making inferences, don't worry about the Double Not. Just show the pair of statements that make the individually.

Here's what I mean.

Imagine that same sentence, "No E's are F's." That can be diagrammed really concisely with a Double Not since we know that E and F cannot go together:

E F

But that's tough for some people because they want to use it to make chains, like connecting F G, when you can't! Why not? Because E doesn't tell you F, E tells you NOT F. That's the Double Not at work.

So instead of E F, just show the two individual components of that single diagram separately:

E F , and F E

You'll note that those are contrapositives of one another, so they're really the same idea, and they tell you the same thing the Double Not does: E and F must be kept apart.

Those two diagrams also make proper chains easier to see and create, since you know at all times exactly what you're dealing with (F or F, for instance).

I hope that helps!

Jon