Logical Reasoning - Chapter 13: Formal Logic

Example of how Formal Logic works using a few of those terms:

"Every author works long hours, and if you work long hours you are never happy. Some authors are female." Second, swiftly translate the statement into a set of symbols that represent the concepts and relationships: (where F = female; A = author; LH = long hours; and H = happy) F <-S-> A -> LH <-/-> H Inferences: F <-S-> LH F <-S-> /H A <-/-> H

What happens if the negative is in the middle?

"Intermediate" negativity will be bypassed. A <-s-> /B -> C. Inference is A <-s-> C.

What are the two rules of diagram creation?

1. Always combine common terms 2. There is no traditional direction in logic

What are the three components of Formal Logic diagrams?

1. Choosing Symbols to Represent Each Variable 2. Conditional Reasoning Terms and Diagrams 3. New Terms and Diagrams

What are the 11 Principles of Making Formal Logic Inferences?

1. Start by looking at the ends of the chain. 2. The vast majority of additive inferences require either an all or none statement somewhere in the chain. 3. When making inferences, do not start with a variable involved in a double-not arrow relationship and then try to "go across" the double-not arrow. 4. The Some Train 5. The Most Train 6. Arrows and double-not arrows 7. Use inherent inferences 8. Watch for the relevant negativity 9. Some and Most Combinations 10. Analyzing Compound Statements 11. Once an inference bridge is built, it does not need to be build again.

However, a problem involving two mosts that has appeared on the LSAT and does yield an inference is the following:

A <-m- B -m_> C Although from all appearances no inference can be made, the fact that most Bs are both As and Cs allows us to conclude that some As are Cs. For example, if there are five Bs, and three Bs are As, and three of the Bs are cs, then there must be an overlap of at least one A and C, and therefore we can conclude that some As are Cs. This inference can only occur when the two mosts each "lead away" from the middle variable.

The Logic Ladder Most/Some example

A close analysis of the Ladder explains the presence of inherent inferences in all and most statements. For example, if "All doctors are lawyers," then from the LL we know that "Some doctors are lawyers." And, because some is a reversible term, we know that "Some lawyers are doctors." The presence of inherent inferences in non-reversible such as all and most helps to make complex inferences easier to follow.

Formal Logic Diagrams 3. New Terms and Diagrams What are the three categories pertaining to this?

A. Relationships involving Some B. Relationships involving Most C. Contrapositives

Formal Logic Diagrams 2. Conditional Reasoning Terms and Diagrams What are the three diagrams pertaining to this?

A. The Single Arrow (->) B. The Double Arrow (<-->) C. The Double-Not Arrow (<-/->)

Formal Logic statements in relation to each other using a 0 to 100 unit scale:

All = 100 Most = 51 to 100 ("a majority") Some are not = 0 to 99 (also "Not All" Most are not = 0 to 49 Some = 1 to 100 ("at least one") None = 0

The basis for Formal Logic relationships are terms such as?

All, none, some and most

Do these create a formal logic relationship? None and All

Always. The inference can be a "some are not" or "all are not" inference depending on the direction of the All arrow.

Do these create a formal logic relationship? None and Most

Always. The inference can be a "some are not" or "most are not" inference depending on the direction of the Most arrow.

Do these create a formal logic relationship? None and Some

Always. The inference is always a "some are not" inference.

Is this reversible? Most

Answer: No (--M->)

Is this reversible? All

Answer: No (->)

Is this reversible? Double-arrow

Answer: Yes (<-->)

Is this reversible? None

Answer: Yes (<-/->)

Is this reversible? Some

Answer: Yes (<-S->)

The Logic Ladder: "Some"

At the lowest rung - some - no inherent inferences follow. From the definition of some we know that most and all are possible, but we cannot know for sure that they are true.

Rule #2. There is no traditional direction in logic

Because English speakers read from left to right, most people assume that inference-making patterns always travel from left to right. This assumption is false. For example, consider the following four diagrams: A -> B <-/-> C C <-/-> B <- A (and vertical but not going to type out) The four diagrams contain identical relationships - and produce identical inferences - but they look different because the variables are placed in different relative positions. Yet, the underlying relationships are the same in each instance: all As are Bs, and no Bs are Cs. Thus, you can create a diagram in any shape or direction because the physical placement of the variables is not critical as long as the relationships are properly represented.

11 Principles of Making Formal Logic Inferences 2. The vast majority of additive inferences require either an all or none statement somewhere in the chain.

Because all and none statements affect the entire group under discussion (for example, in A -> B, every single A must be a B), they are very restrictive, and when other variables are joined to these relationships then inferences often result. In fact, either all or none (or both) are present in almost every Formal Logic diagram that produces additive inferences.

11 Principles of Making Formal Logic Inferences 6. Arrows and double-not arrows

Because arrows and double-not arrows are so powerful, they almost always elicit additive inferences. Perhaps the most familiar inference is the following: A -> B <-/-> C This popular combination, which often appears in LG, yields the inference A <-/-> C Any combination of an arrow and a double-not arrow in succession will yield an inference (although inherent inferences may be needed to make the inference). Any combination of two arrows may yield an inference depending on the configuration. A combination of two double-not arrows in succession does not yield an inference.

The Special Case of "Some are Not"

Because some is a reversible term, some are not statements are also reversible. This is a typical some are not diagram: A<-S-> /B Starting from the A side, we know that some As are not Bs. Most students have no difficulty making that judgement. Trouble can arise when we look at the relationship from the other side. Correctly reversed, the relationship reads, "Some things that are not B are A." To most people this sounds strange and useless; nonetheless, that is the correct phrasing of the reversed statement. But, most students will incorrectly reverse the statement to read, "Some Bs are not As." That is not necessarily true! In effect, this incorrect reverse interpretation is the same as: B<-S-> /A The correct reversed diagram is: /B <-S-> A Thus, you can reverse a some are not statement but you must be careful when doing so in order to avoid accidentally moving the "not".

Formal Logic Diagrams 1. Choosing Symbols to Represent Each Variable

Choosing symbols to represent each group or idea is easy: simply choose the letter or letters that, to you, best represent the element. The exact letter you choose to represent each group are not critical; what is important is that you use those same letters to represent the group throughout your diagram and inferences. This is especially important when terms are negated. For example, if you represent "happy" with "H" as you begin your diagram, and later you are presented with a seemingly new element, "unhappy," do not create a new variable, "UH." Instead, simply negate "happy" and use /H.

The Logic Ladder: "Most" relationships

If "Most waiters like wine," then you automatically know that "Some waiters like wine," But, because most is below all on the Ladder, you do not know with certainty that "All waiters like wine" (it is possibly true, but not known for certain). This reveals a truth about the Logic Ladder: the upper rungs automatically imply the lower rungs, but the lower rungs do not imply the upper rungs. In other words, as you go down the rungs the lower relationships must be true, but as you go up the rungs the higher relationships might be true but are not certain.

11 Principles of Making Formal Logic Inferences 9. Some and Most Combinations

In general, two consecutive somes, two consecutive mosts, or a some and most in succession will not yield any inferences. Thus, a statement such as the following does not yield any additive inference: A <-s-> B <-S-> C Nor will this: A <-s-> B -m-> C Usually, two mosts in sequence do not yield an inference: A -m-> B -m-> C

Rule #1. Always combine common terms

In order to make complete and effective formal logic diagrams, you must always combine like terms through linkage. For example, most students tend to write statements separately like: A <-S-> B B -> C Instead, you should recognize that "B" is common to both diagrams, and combine the two diagrams into one linked diagram: A <-S-> B -> C Each variable should appear only one time; variables should not be duplicated if at all possible.

Mnemonic trick called Some Train

In the Some Train, each variable is considered a "station" and the relationship between each variable are "tracks." A successful "journey" (defined as a journey of at least two stops) yields an inference. An unsuccessful journey means no inference is present. A <-s-> B -> C We start at station A because A is the open variable in the some relationship. From A, we can ride over to station B because we get a free pass on the Some Train when we travel over some. Once at station B, we need a track "away" from B going to another station. Since the tracks are arrows, we need either an all arrow, a double arrow, or a double-not arrow (some and most arrows do no count because they do not necessarily include the entire group). In this case we have the all arrow, and thus we can ride over to C. We not have a successful journey between A and C. Now to make our inference, we look at two elements: 1. the weakest link in the chain 2. The presence of relevant negativity In our example, looking back on the journey, the weakest link is some, and there are no negative terms. Thus, our inference is A <- S -> C.

The Some Train wrong inference:

In the example above, the direction of the arrow is critical. Consider this example: A <-S-> B <- C. At first glance, you might think that this relationship will produce the same inference as the example above. There are three variables and the same internal relationship exist- some and all. But because the direction of the all arrow is different, ultimately we will not be able to make an inference from this diagram. Logically, the group of Bs could be so large that even though every C is a B, and some As are Bs, the groups of As and Cs do not overlap and thus no inference between A and C is present.

11 Principles of Making Formal Logic Inferences 3. When making inferences, do not start with a variable involved in a double-not arrow relationship and then try to "go across" the double-not arrow.

In the example diagram (A <-s-> B -> C <-/-> D), we know that the open ends are A and D, but, because D is involved in a double-not arrow relationship, we should not start at D and attempt to make an additive inference with B or A. Instead, you will find that starting at A or B will make it easier to create inferences involving D.

Relevant Negativity example: /A <-s-> B -> C

In this case, we start with /A as the fist term and then use the Some Train to connect to C. Inference in /A <-s-> C.

Some students fall into the trap of believing that the Some Train works only form left to right. Consider the following example: A<-B -> S -> C

In this diagram, following the first principle of making inferences, we begin at station C and then ride over to station B (remember, going "backwards" is acceptable since there is no true direction in logic, just relationships). Once we arrive at station B, we look for a track leading away, and we find one point to station A. thus we can ride from C to A, the weakest link is some, and our inference is A <- S -> C. The Some Train can work in any direction as long as it is following the rules discussed previously.

The Some Train Mnemonic wrong inference example A <-S -> B <- C

In this example, we again start at station A and ride over to station B. Once at station B, we need a track "away" from B going to another station, but there is no track away, only an incoming track. Thus, we are stopped and there is no inference that can be made in this example.

What is the inference of: A <-/-> B -m-> C

Inference C <-s-> /A

What is the inference of: A <-m- B <-/-> C

Inference: A <-s-> /C

What is the inference of: A <- B -m-> C

Inference: C <- S -> A

What is the inference of: A <-/-> B -> C

Inference: C <-s-> /A

What is the inference of: A <- B -> C

Inference: C <-s-> A (start at A or C and use the inherent inference some to ride the Some Train)

What is the inference of: A -m-> B <-m- C

Inference: None

What is the inference of: A -m-> B -m-> C

Inference: None

What are the two types of Formal Logic inferences?

Inherent and additive

Formal Logic Diagrams 2. Conditional Reasoning Terms and Diagrams B. The Double Arrow (<-->)

Introduced by "if and only if" or by situations where the author implies that the arrow goes "both ways," such as by adding "vice versa" after a conditional statement. Example: X if and only if Y Diagram X <--> Y These statements allow for only two possible outcomes: the two variables occur together, or neither of the two variables occur.

Formal Logic Diagrams 2. Conditional Reasoning Terms and Diagrams C. The Double-Not Arrow (<-/->)

Introduced by conditional statements where exactly one of the terms is negative, or by statements using words such as "no" and "non" that imply the two variables cannot "go together." Example: No Xs are Ys Diagram: X <-/-> Y Example: If you are a T, then you are not a V Diagram: T <-/-> V Remember, the statement above produces a diagram, T -> /V, which can more properly be diagrammed at T <-/-> V.

Formal Logic Diagrams 2. Conditional Reasoning Terms and Diagrams A. The Single Arrow (à)

Introduced by sufficient and necessary words such as: if...then, when, all, every, and only, where both elements are positive or both elements are negative. Example: All Xs are Ys (X and Y both positive) Diagram: X -> Y Example: If you are not T, then you are not V (T and V both negative) Diagram: /T -> /V Contrapositive: V -> T

Formal Logic defined:

Is simply a standard way of translating relationships into symbols and then making inferences from those symbolized relationships. And, because certain combinations always yield the same inference regardless of the underlying typic, a close study of the combinations that appear frequently on the test allows you to move quickly and confidently when attacking Formal Logic problems.

The Negative Logic Ladder

Is used for negative terms. None -> Most are not -> Some are not Like the last ladder, each term represents a "rung," and the upper rung terms automatically imply that the lower rung terms are known to be true. Thus, if you have a none relationship, you automatically known that the most are not and some are not relationships for the same statement are true. So, if a statement is made that "None of the waiters like wine," then you immediately know that "Most of the waiters do not like wine," and "Some of the waiters do no like wine." The same is true for most are not relationships as the lower rungs do not automatically imply the upper rungs. In other words, as you go down the rungs you automatically know that "Some waiters like wine," but not that "None of the waiters like wine." At the lowest rung - some are not- no inferences follow. From the definition of some are not we know that most are not and none are possible but we cannot know for sure that either is true.

What does every Formal Logic relationship feature?

It features at least two separate variables linked in a relationship. The variables represent groups or ideas. Like "A" to represent "authors" and other variables can be inked in relationships that were represented by the diagrammatic elements of "<-S->"

Reversibility in Formal Logic

Means that the relationship between the two variables has exactly the same meaning regardless of which "side" of the relationship is the starting point of your analysis. Statements that are non-reversible have a single "direction," that is, the relationship between the two variables is not the same.

The LSAT introduces the concept of most in a variety ways, including the following relationship indicators:

Most A majority More than half Almost all Usually Typically Likely More often than not

Do these create a formal logic relationship? None and None

Never

Do these create a formal logic relationship? Some and Most

Never

Do these create a formal logic relationship? Some and Some

Never

Are Logical Reasoning inferences useful in Logic Games section?

No, largely worthless because some is meaningless in LG.

What does "not all" phrase mean?

One of the most popular ways to introduce the idea that some are not is to use this phrase (not all), which is functionally equivalent to some are not. Example: Some Ws are not Zs Diagram W <- S -> /Z

11 Principles of Making Formal Logic Inferences 11. Once an inference bridge is built, it does not need to be built again

One question often asked by students is, "Does an inference have to be makeable from both sides in order to be valid?" The answer is No. Consider the following example: A <-B <-s-> C Starting at A, we can ride backwards across the arrow using the inherent inference, some. But, once we arrive at B, there is no all, non, or true double-arrow leading away, and thus we are stopping: no inference. Starting at C, we can ride some over to B. Once we arrive at B, there is a tack leading over to A, so we can ride the Some Train from C to A and make he inference C <-S-> A. Reviewing the example above, some students get confused. We cannot make an inference from A to C but we can make an inference from C to A? That sounds odd. Does this make the second inference invalid? No. If an inference can be made from one "side" of the relationship, that is enough to establish the inference even if the inference cannot be made from the other side. In this sense, making an inference is like building a bridge - you may start on one side, but once you build the bridge relationship is made between both sides.

The LSAT introduces the concept of some in a variety of ways, including the following relationship indicators:

Some At least some At least one A few A number Several Part of A portion Many

Formal Logic Diagrams 3. New Terms and Diagrams C. Contrapositives

Some students ask if there is a contrapositive for some and most statements. The answer is NO. Only the arrow statements like all have contrapositives; some and most do not because they do not necessarily encompass an entire group.

Do these create a formal logic relationship? Most and All

Sometimes. Most and All can produce an additive inference depending on the direction of the All arrow. For example, A -m-> B -> C produces an inference A -m-> C; A -m-> B <- C does not produce an inference.

Do these create a formal logic relationship? Some and All

Sometimes. Some and All can produce an additive inference depending on the direction of the arrow. For example, A <-s-> B <- C produces the inference A <-s-> C; A <-s-> B <- C does not produce an inference.

Do these create a formal logic relationship? All and All

Sometimes. The inference, when makeable, is either a "some" or "all" inference depending on the direction of the two All arrows.

Do these create a formal logic relationship? Most and Most

Sometimes. The only relationship with two Mosts that will produce an inference is A <-m- B -m-> C. The inference is A <-s-> C

Now, let us examine the Some Train when used with a double-not arrow. Example: A <-S -> B <-/-> C

Starting at station A, we ride over to station B. Once at station B, we need a track away from B going to another station. In this case we have the double-not arrow and therefore we can ride over to C. We now know we can travel from A to C, and to make our inference, we look again at two elements: 1. The weakest link in the chain 2. The presence of relevant negativity In our example, looking back on the journey, the weakest link is some, so we know that at least we have A some C. But, there is negativity between B and X and that negativity transfers to C and thus our inference is A <-s-> /C

Some Train Mnemonic Consider this example: A <-/-> B <- S -> C

Starting at station C, we ride over to station B. Once at station be, we need a track away from B going to another station. In this case we have the double-not arrow, and we can ride over to A. Thus, we know we can travel from C to A, adding the relevant negativity between B and A produces our inference: C <- s -> /A

A reversible relationship. "Some" is a classic example of a reversible statement. A <-S-> B

Starting from A yields "Some As are Bs" (A some B). Starting from B yields "Some Bs are As" (B some A). Because of the name of "some" these two statements are functionally identical (if some As are Bs, by definitions some Bs must also be As; alternatively, if some As are Bs, then somewhere in the world there is an AB pair, and thus somewhere a B is with an A and we can conclude some Bs are As). Reversible statements are easily identifiable because the relationship symbol is symmetrical and the arrow points in both directions. Non-reversible terms have arrows that point in only one direction.

Relationship that is not reversible: A -> B

Starting from the A side, we know that every single A is a B. If we start at B, does the relationship reverse? That is, is every single B an A? No- that would be a Mistaken Reversal. From B's side, we do not know if every B is an A. Instead, we only know that some Bs are As. Thus, the arrow between A and B in the diagram has a direction: the "all" travels only from A to B and it does not additionally travel from B to A. The relationship is therefore not reversible.

What is the difference between the Some Train and Most Train?

The critical difference between the Some Train and Most Train is that because most has direction, you can only follow the most-arrow to make a most inference. If you go "against" the arrow, the relationship will devolve to some, which is the inherent inference.

What happens when "most" appears as "most are not"?

The interpretation changes due to the not. "Most are not" can be defined as a majority are not, possibly all are not. Thus, if I say, "Most of my friends are not present," it could be true that none of my friends are present. When you diagram a statement involving most are not, simply place the word most between the two elements, place an arrow under the most pointing at the second element, and negate the second element. Example: Most Ws are not Zs Diagram: W <-M-> /Z

How can you determine the weakest link?

The weakest link is imply the least definite link, and so the major terms are in this order: 1. Some- this is the broadest term and the least definite; therefore it is the weakest. 2. Most this is more definite than Some 3. All and None - these two terms are the most definite and although they are polar opposites, they are equal in power.

Formal Logic Diagrams 3. New Terms and Diagrams B. Relationships involving Most

The word "most" can be defined as a majority, possibly all. Most includes the possibility of all. Thus, if I say, "Most of my friends graduated last week," using the definition above it could in fact be true that all my friends graduated last week. Example: Most Xs are Ys Diagram: X <-M-> Y

Formal Logic Diagrams 3. New Terms and Diagrams A. Relationships involving Some

The word "some" can be defined as at least one, possible all. Some includes the possibility of all. When you diagram statements involving some, simply place a double-arrow with the letter "S" between the two elements Example: Some Xs are Ys. Diagram: X <- S -> Y

Whare are Inherent inferences?

These follow from a single statement such as A -> B, and they are inferences that are known to be true simply from the relationship between the two variables. In the statement above we know that all As are Bs. Of course, if all As are Bs - these last two statements are inherently true because of the nature of the initial relationship. And, because we know from the last inference that some As are Bs, if we analyze the initial relationship from B's perspective (also called "coming backward" against the arrow) we can deduce that some Bs are As. This deduction can be incredibly useful when you are trying to attack complex problems and you need, in effect, to work "against" the arrow. The contrapositive is an example of this.

What are Additive inferences?

They result from combining multiple statements through a common term and then deducing a relationship that does not include the common term. Example: A -> B <-/-> C Individually, the two relationships have "B" in common. But, as we will discuss shortly, we can ultimately connect A and C in a relationship that drops B, and make the inference that A <-/-> C. This is an additive inference, so-called because it comes from "adding" two statements together to make the inference. In LR, these references are often the correct answer choice on Formal Logic problems.

The Logic Ladder

This details the relationship between all, most and some: All -> Most -> Some (arrows going down) In the ladder, each term represents a "rung" and the upper rung terms automatically imply that the lower rung terms are known to be true. Thus, if you have an all relationship, you automatically know that the most and some relationships for that same statement are true. So, if a statement is made that "All waiters like wine," then you immediately know that "Most waiters like wine," and "Some waiters like wine."

11 Principles of Making Formal Logic Inferences 5. The Most Train

This works in a very similar fashion to the some train, but because most is one step higher than some on the Logic Ladder, the Most Train produces stronger inferences. Consider the following example: A -m-> B -> C Starting at station A, we ride over to station B because we get a one-way pass on the Most Train when we travel over most. Once at station B, we need a track away from B going to another station. Since the tracks are arrows, we either need an all arrow, a double arrow or a double-not arrow (when we reach the second track, most arrows do not count because they do not necessarily include the entire group). In this case we have the all arrow, and thus we can ride over to C. Thus, we know we can travel from A to C. Now, just as in the Some Train, to make our inference we look at two elements: 1. The weakest link in the chain 2. The presence of relevant negativity Looking back the weakest link is Most and there are no negative terms. Thus, our inference is A -m-> C.

11 Principles of Making Formal Logic Inferences 4. The Some Train

To make an inference with a variable involved in a some relationship, either an all arrow, a none arrow, or a double-arrow "leading away" from the some relationship is required. Consider the first part of the example diagram: A <-s-> B -> C In this case, the arrow from B to C "leads away" from the some relationship. From this configuration, we can deduce the presence of an additive inference: A <-s-> B. Logically, if some As are Bs, and every single B is a C, then it must be true that some As are Cs.

What happens when you get two double-not arrows in a row?

Two double-not arrows in a row will not yield an inference. The same is true for two "somes" in a row.

11 Principles of Making Formal Logic Inferences? 1. Start by looking at the ends of the chain.

Variables that are linked in only one relationship are "open"; variables that are linked in two or more relationships are "closed." Because the ends of a chain are naturally open variables and are involved in fewer relationships, they are easier to analyze. Always begin your analysis by looking at the ends of the chain. Example: A <-s-> B -> C <-/-> D In the example diagram above, variables A and D are open; variables B and C are closed. To make inferences, first examine variables A and D, and thereafter examine variables B and C.

What words do not guarantee a majority?

What words do not guarantee a majority? "Frequently" and "often" do not appear on the previous list because when those terms- when used alone- do not guarantee a majority.

What does "some are not" mean?

When "some" appears as "some are not" (as in "Some Xs are not Ys), the interpretation changes due to the not. "Some are not" can be defined as at least one is not, possibly all are not. Thus, if I say, "Some of my friends are not present," then it could be true that none of my friends are present.

11 Principles of Making Formal Logic Inferences 8. Watch for the relevant negativity

When making inferences - especially with the Some and Most Trains - special care must be paid to identifying relevant negativity. The presence of relevant negativity is defined as the following: 1. Either the first or last term in the inference chain is negated, as in /A <-s-> B -> C Or 2. There is a double-not arrow in the chain (which will always appear just before the last station)

11 Principles of Making Formal Logic Inferences 10. Analyzing Compound Statements

When working with compound statements (statements where there are four or more variables), keep in mind the following guidelines: A. Recycle your inferences to see if they can be used to create further inferences. B. Make sure to check the closed variables. Compound Statements example: F <-s-> A -> LH <-/-> H The ends of the chain are F and H, but since H is at the end of a double-not arrow, we should begin our analysis at F. Starting at F, we can ride the Some Train through A to LH, yielding the additive inference F <-s-> LH. At this point, we must apply the guidelines given above. First we can recycle the F <-s-> LH inference we just made by adding LH <-/-> H to the end, creating the following chain: F <-s-> LH <-/-> H This diagram - which can easily be created mentally without drawing the diagram out physically - yields the additive reference F <-s-> /H We can now apply the second guideline, which is to examine the closed variables A and LH. Starting at LH does not yield an inference but starting at A, we can follow the arrow to LH and then to H, making the inference A <-/-> H Thus, by methodically attacking each part of the chain, we have made three separate inferences, representing all the possibly additive inferences: F <-s-> LH F <-s-> /H A <-/-> H

11 Principles of Making Formal Logic Inferences 7. Use inherent inferences

You will at times be forced to use an inherent inference to make an additive inference. Consider the following example: A <-/-> B -> C By using the principles we have established for making inferences, we should start at A or C. But, A is involved in a double-not arrow relationship, and thus we should not begin there. So, we should start at C. But, C is at the end of an all arrow - how can we go against the arrow? Remember, going "backwards" against the arrow is the inherent inference some, and thus we can use the Some Train. Starting at C, we can ride the Some "backwards" over to B. Once at B, we can take the arrow to A, and derive the inference C <-s-> /A. Thus, the Some Train does not require the explicit presence of some in a relationship. If the some relationship is implicit (as in the inherent inferences present in all and most), then Some Train can be used.