Formal Logic

Front 1

Not all T’s are V’s

Back 1

T some ~V

Front 2

Some W’s are not Z’s

Back 2

W some ~Z

Front 3

All X’s are Y’s

Back 3

X --->Y

Front 4

If you are not T, then you are not V

Back 4

~T ---> ~V

The contrapositive of the above is:

V ---> T

Front 5

X if and only if Y

Back 5

X <---> Y

Front 6

All W’s are Z’s, and all Z’s are W’s

Back 6

W <---> Z

Front 7

No X’s are Y’s

Back 7

X <--I--> Y

Front 8

If you are a T, then you are not a V

Back 8

T <--I--> V

or less efficiently expressed as

T ---> ~V

Front 9

Some X’s are Y’s

Back 9

X some Y

Front 10

synonyms for "some"

Back 10

some
at least some
at least one
a few
a number
several
part of
a portion

...it could even mean "all"

Front 11

All

Back 11

100/100

Front 12

meaning of "most"

Back 12

a majority, possibly all

Front 13

Most X’s are Y’s

Back 13

X ----most---> Y

Front 14

Most/a majority

Back 14

51 to 100/100

Front 15

Most W’s are not Z’s

Back 15

W ----most---> ~Z

Front 16

synonyms of "most"

Back 16

most
a majority
more than half
almost all
usually
typically

Front 17

Reversible Relationships

Back 17

None ( <---I---> ),
Some (some),
Double-arrow ( <------->)

Front 18

Contrapositives

Back 18

The contrapositive of

A ---> B is ~B ---> ~A

and both statements have identical truth values to one another

Statements containing "most" or "some" do NOT have contrapositives.

Front 19

Some are not/Not all are

Back 19

0 to 99/100

Front 20

Non Reversible Relationships

Back 20

All ( ---> ),
Most (----most--->)

Front 21

Most are not

Back 21

0 to 49/100

Front 22

Some/At Least One

Back 22

1 to 100/100

Front 23

Introductory words for:

X ----> Y

Back 23

if...then, when, all, every, only

Front 24

Introductory words for:

X <----> Y

Back 24

both ways, vice versa, all...and all

Front 25

Introductory words for:

X <--I--> Y

Back 25

no, none, cannot go together

Front 26

None

Back 26

0/100

Front 27

"All" implies...

Back 27

"most" and "some"

Front 28

"Most" implies...

Back 28

"some"

Front 29

"None" implies...

Back 29

"most are not" and "some are not"

Front 30

"Most are not" implies

Back 30

"some are not"

Front 31

Additive Inference Strategy

Back 31

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 looking to make inferences, do not start with a variable involved in a double-not arrow relationship and then try to “go across” the double-not arrow.

Front 32

The "Some" Train

Back 32

To make our inference, we look at two elements:

1. The weakest link in the chain
2. The presence of relevant negativity

To make an inference with a variable involved in a some relationship, an arrow “leading away” from the "some" relationship is required.

(A some B ----> C) implies (A some C)

(A some B <---- C) implies nothing

Get "free pass" on the train starting at the "some" stop, and seek tracks (arrows) leading AWAY.

(A some B <--I--> C) implies (A some ~C)

(A some ~B ---> C) implies (A some C)

Front 33

The "Most" Train

Back 33

To make our inference we look at two elements:

1. The weakest link in the chain
2. The presence of relevant negativity

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.

Front 34

Weakest Link

Back 34

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 though they are polar opposites, they
are equal in power.

Front 35

Relevant Negativity

Back 35

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 is negated (as in ~A some B ----> C), or
2. There is a double-not arrow in the chain (will always appear just before the last station)

Front 36

Inference of:

A <--most-- B --most--> C

Back 36

A some C

Front 37

Inference of:

A ----> B -----> C

Back 37

A ----> C

Front 38

Inference of:

A ----> B <--I--> C

Back 38

A <--I--> C

Front 39

Inference of:

A <--I--> B <--I--> C

Back 39

none

Front 40

Inference of:

A ----> B <----> C

Back 40

A <----> C

Front 41

Inference of:

A --most--> B --most--> C

Back 41

none

Front 42

Inference of:

A --most--> B <--most-- C

Back 42

none

Front 43

Inference of:

A some B some C

Back 43

none

Front 44

Inference of:

A some B --most--> C

Back 44

none

Front 45

Inference of:

A some B <--most-- C

Back 45

none

Front 46

Inference of:

A some B <---- C

Back 46

none

Front 47

Inference of:

A --most--> B ----> C

Back 47

A --most--> C

Front 48

Inference of:

A --most--> B <---- C

Back 48

none

Front 49

Inference of:

A <---- B ----> C

Back 49

A some C

Front 50

Inference of:

A ----> B <---- C

Back 50

none

Front 51

Inference of:

A <--I--> B ----> C

Back 51

C some ~A