Follow up to: Why Is No One Making a Chatbot?

In a previous post, I mentioned that it might be worthwhile to create a computer program capable of making syllogistic inferences — that is understanding new information from inputted information via syllogistic logic. Such programs already exist, but I would like to use my amateur skills to add something.

Wikipedia’s article on syllogisms is good, but not entirely complete or easy to follow — some of the syllogisms listed also seem to not make sense. Thus, to aid my own understanding and add yet another esoteric essay, I would like to completely categorize all possible syllogisms, so that I would have a benchmark for a program. If a program can take all of this information and make the proper conclusions, it will properly understand syllogistic logic (and can therefore move on to something more difficult).

The names of the syllogisms are also in latin, and seem crafty, but otherwise arbitrary. Therefore, I will arrogantly rename all the syllogisms to make more sense.

The Categories and The Basics

Syllogisms are broken into four different categories of facts: “All” statements, “Some” statements, “Some Not” statements, and “All Not”, statements.

For examples,
All humans are mortal. (Each and every human is also a mortal.)
Some humans are male. (As few as one and as many as all humans are male.)
Some humans are not firefighters. (As few as one and as many as all humans are not firefighters.)
All humans are not lizards. (Each and every human is not a lizard.)

Some means that

#1: Some From All

These statements can be taken together and additional data can be drawn. For example, the most basic is the “Some from All”. If we know all X are Y, we can correctly infer that some Y are X. It also may or may not be true that all Y are X.

For example, if all humans are mortal we can therefore infer that some mortals are human.

#2: Some From Some

We can also do the same with a some statement. If we know some humans are male, we know that some males are human.

Unfortunately, the reverse does not happen for “some not”. Just because some humans are not firefighters does not mean some firefighters are not human.

#3: All Not From All Not

Perhaps the biggest amount of additional information we can get comes from the “all not” statements. If we know all humans are not lizards, then we know that all lizards are not humans, which is a wide sweeping fact about lizards.

Counterfactuals

All of these statements imply something else, by virtue of being true. Consider these three statements:

1. All humans are mortal
2. All humans are not mortal
3. Some humans are mortal
4. Some humans are not mortal

If 1 is true, then 2 and 4 not true; as they contradict. Any “all” statement will contradict the opposite “all not” and “some not” statements. Likewise, if 2 is true, then 1 and 3 are false.

If 3 is true, then 2 is false. It can be possible for 1 and 3 to simultaneously be true — 1 is just a more powerful version of 3.

Lastly, if 4 is true, then 1 has to be false, but all other statements can be true.

Combinatorial Inferences

We can get even more information when two statements are both known. For example, take the following two statements:

1. All humans are people
2. All people are mortal

From the “all to some” inference, we know that:
C1: Some people are humans
C2: Some mortals are people

However, we can also connect “people” and realize that all humans must therefore be mortal. This is (#4) the All + All = All inference.

Reversing the statement is also true.

1. All people are mortal
2. All humans are people

C1: All humans are mortal

#5: All + Some = Some

Now let’s combine an all statement and a some statement.

1. All humans are people
2. Some mortals are humans

This allows us to connect via the term “humans” and see that some mortals are also people (since all the human mortals are also people mortals).

Again, the reverse is also true:

1. Some mortals are humans
2. All humans are people
C1: Some mortals are people

This means that All + Some = Some, and Some + All = Some. This reversal of two statements is true for every combination, so we won’t count both of them as separate inference methods.

#6: All + Some Not = Some Not

Now for an all statement and a some not statement.

1. All humans are people.
2. Some people are not Socrates.

This allows us to infer that “C1: some humans are also not Socrates“.

#7: All + All Not = All Not

1. All humans are people.
2. All people are not skyscrapers.

C1: All humans are not skyscrapers.

#8: Some + Some Not = Some Not

1. Some humans are philosophers
2. Some philosophers are not Socrates

C2: Some humans are not Socrates

#9: Some + All Not = Some Not

1. Some humans are philosophers
2. All philosophers are not skyscrapers

C1: Some humans are not skyscrapers

#10: Some Not + Some Not = Some Not

1. Some humans are not philosophers
2. Some philosophers are not firefighters

C1: Some humans are not philosophers

#11: Some Not + All Not = Some Not

1. Some humans are not philosophers
2. All philosophers are not lizards

C1: Some humans are not lizards

A Dud: Some + Some Tells Us Nothing

Some + Some can occasionally work:

1. Some humans are philosophers
2. Some philosophers are Socrates

C1: Some humans are Socrates

However, it can also occasionally fail:

1. Some humans are mammals
2. Some mammals are dogs

C1: Some humans are dogs obviously does not follow.

A Second Dud: All Not + All Not Tells Us Nothing

The last possible combination, an all not + all not, tells us absolutely nothing of use.

1. All humans are not lizards
2. All lizards are not firefighters

There are no conclusions we can draw from this that we couldn’t through a pre-existing rule.

Multi-Step Inferences

Those eleven methods are all you really need to know, however you can get even more information by adding an additional step. All of the previous inference methods are self contained — the conclusion results immediately from the two premises with no additional work needed.

However, this is not always the case. Consider:

1. All humans are people
2. All humans are mortal

We can’t immediately connect people to mortals via the previous ” All + All = All” right now. However, we can use (#1) Some From All to conclude that:
C1: Some people are humans
C2: Some mortals are humans

and then use All + Some = Some on (C1:) and (2.) to conclude that:
C3: Some people are mortal

Multi-Premise Inferences

Additionally, we are not limited to just two pieces of information. The most simple is a chain:

1. All Socrates are philosophers
2. All philosophers are people
3. All people are human
4. All humans are mortal

C1: All Socrates are people (via All (1.) + All (2.) = All)
C2: All Socrates are human (via All (3.) + All (C1:) = All)
C3: All Socrates are mortal (via All (4.) + All (C2:) = All)

But really, you could have any combination or rules in any order:

1. All humans are people
2. Some “Barack Obama”s are presidents
3. All presidents are humans

C1: Some people are humans (via Some From All)
C2: Some presidents are “Barack Obama”s (via Some From All)
C3: Some humans are presidents (via Some From All)

C4: All presidents are people (via All (3.) + All (1.) = All)
C5: Some “Barack Obama”s are humans (via All (3.) + Some (2.) = Some)

C6: Some “Barack Obama”s are people (via All (1.) + Some (C5:) = Some)

The End: For Now

With the eleven syllogisms (some from all, some from some, all not from all not, all + some = some, all + some not = some not, all + all not = all not, some + some not = some not, some + all not = some not, some not + some not = some not, and some not + all not = some not), you can do some serious logic.

And hopefully computer programs can too. I’ll keep you updated on my progress.

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