Formal Logic

Subset

Sufficiency (Subset membership is sufficient for superset membership)

Superset

Necessary (Superset membership is necessary for subset membership)

If, then

A → B

Contrapositive

/B → /A

If

Sufficient

When

Sufficient

Where

Sufficient

All

Sufficient

Every

Sufficient

Any

Sufficient

Only

Necessary

Only if

Necessary

Only when

Necessary

Only where

Necessary

Always

Necessary

Must

Necessary

Or

Negate, Sufficient

Unless

Negate, sufficient

Until

Negate, sufficient

Without

Negate, sufficient

No

Negate Necessary

None

Negate Necessary

Not both

Negate necessary

Cannot

Negate necessary

Never

Negate necessary

De Morgan’s Laws

M → N and O
CP: /N or /O → /M

Rule + Exception Framework

Treat everything before the unless as the rule and everything after as the exception
R → prohibit
Unless = purpose

Joint Sufficient Condition Framework

Translate claim with group 1 and group 3 rules
/purpose → (resident → prohibit)
Extract embedded sufficient condition
/purpose and resident → prohibit

Domain + Rule Framework

Kick outside sufficient condition into the domain
Domain: /purpose
Rule: resident → prohibit

Bi-Conditional

Conjunction of uni conditional claim and its converse
Ex: Luke will become a Jedi if Yoda trains him, and Luke will become a Jedi only if Yoda trains him
Yoda ←→ Luke Jedi

If and only if

Bi-conditional

If but only if

Bi-conditional

Then and only then

Bi-conditional

Or…but not both

Bi-conditional

If…then…but not otherwise

Bi-conditional

Some

Must include at least one but could go up to include as many as all

Most

Most starts with more than half and could include as many as all

Many

Some; Must include at least one but could go up to include as many as all

Few

Some are and most are not
Few X are Y
X -s- Y; X -m→/Y

Negate All Statement

Not all A are B negated is Some A are /B
A → B
A -s- /B

Negate Conditional Statement

A → B
A and /B
Negated: A can occur and B not occur

Negate “Some” Statements

A -s- B
A → /B
Some A are B negates to no A are B

Negating “Most” Statements

Most A are B negates to “It’s not the case that most A are B”
A -m→ B
/(A -m→ B)

Some Before All

VALID
A -s- B

B → C  

A -s- C

Most Before All

VALID
A -m→ B → C
____________
A -m→ C

Two Mosts

VALID; the two most statements need to share the same sufficient condition
A -m→ B
A -m→ C
________
B -s- C

All Before Most

FLAW; In this flaw “all A are B” and “most B are C” is mistakenly thought to imply that “some A are C”
A → B —m→ C  

_________  

A ←s→ C NOT VALID

All Before Some

FLAW; In this flaw, “all A are B” and “some B are C” is mistakenly thought to imply that “some A are C”
NYP → PV

PV -s- /Good

____________

NYP -s- /Good INVALID

Most Before Most

FLAW, “most A are B” and “most B are C” is mistakenly thought to imply that “some A are C”
Chaining two most statements does not yield a some statement
A -m-> California

California -m-> EB

___________________

A -s- EB

*The shared term is the necessary, not the sufficient making it invalid

Some Before Some

FLAW, In this flaw, “some A are B” and “some B are C” is mistakenly thought to imply that “some A are C”
When you see a logic chain with two some arrow, there are no valid conclusions to be drawn.