Logic · Units 1–6 and 9 · Quick reference

Logic

Grades 8–12+8 pages

Terms and definitions; categorical propositions, the square of opposition and immediate inferences; syllogisms with the six rules and Venn tests; informal fallacies; propositional logic with truth tables, valid forms and short proofs; the classical tradition.

1What logic is2Terms3A E I O4Square5Immediate inferences6Syllogisms7Six rules8Venn tests9Named forms10Relevance11Presumption12Ambiguity13Truth tables14Conditionals15Testing validity16Valid forms17Proofs18Machines19Tradition20Dialectic21Argument maps22Practice

1What Logic Isthe art of right reasoning

The three acts of the mind
Act of the mind	Product	Ask of it
Simple apprehension: grasping what a thing is	Term (a word or phrase)	Is it clear?
Judgment: affirming or denying	Proposition (a statement)	Is it true?
Reasoning: moving from known to new	Argument	Does it follow?

Reading an argument

Premises give the reasons; the conclusion is the claim they support.
Premise indicators: because, since, for, given that.
Conclusion indicators: therefore, so, thus, hence.
No reasons offered means no argument, only an assertion.

Two kinds of argument
	Deduction	Induction
Claims	the conclusion follows necessarily	the conclusion is probable
Good form	valid	strong
Good form + true premises	sound	cogent

Valid but unsound Worked

All fish can fly. A salmon is a fish. So a salmon can fly.
the form is valid: if both premises were true, the conclusion would beform
premise 1 is false, so the argument is unsoundfacts

Why reason well

“Come now, and let us reason together, saith the LORD” (Isaiah 1:18, KJV).

First principles of thought

Identity: a thing is what it is.
Non-contradiction: nothing can be and not be at the same time and in the same respect.
Excluded middle: a statement is either true or false; there is no third option.

Truth and validity
Form	Premises	Conclusion
Valid	all true	must be true (sound)
Valid	one or more false	true or false
Invalid	any	true or false; nothing guaranteed

Form first, facts second

Ask “Does it follow?” before “Is it true?” Valid plus true premises gives sound.

2Terms & Definitionsfirst act of the mind

How a term is used
Use	Meaning	Example
Univocal	one sense every time	“triangle” in any geometry book
Equivocal	unrelated senses	“pitcher” on a mound, “pitcher” of lemonade
Analogical	different but related senses	“bright” lamp, “bright” student

Five rules of a good definition

Convertible: not too broad, not too narrow.
States the essence, by genus and difference.
Not circular.
Not negative when it can be positive.
Not obscure or figurative.

Rules of division

One principle at each step.
Parts do not overlap and leave nothing out.
No leaps: divide into the next kinds down.

Extension and intension

Extension: the members a term covers. Intension: what the term means.
Add to the meaning and the class shrinks: vehicle, car, electric car.

Porphyry’s five predicables
Predicable	Says of a thing	For “square”
Genus	the wider kind	rectangle
Difference	what sets it apart in the genus	with four equal sides
Species	genus + difference	square
Property	follows from the essence, not part of it	diagonals that are equal and perpendicular
Accident	may come and go	drawn in red

Genus, then difference

Name the family, then the trait no other member of the family has.

Aristotle’s ten categories

Substance, quantity, quality, relation, place, time, position, state, action, passion.

Testing a definition Worked

“A pencil is a writing tool.”
pens and crayons fit too: too broadcounterexample
“A pencil is a writing tool with a graphite core in a wooden case.”genus + difference
mechanical pencils are left out: too narrow; drop “in a wooden case”fix

A faulty division Worked

Books: fiction, nonfiction, paperbacks
two principles at once (content and binding): cross-divisionparts overlap
fix: by content, fiction and nonfiction; by binding, hardcover and paperbackone principle each

LogicLogic · Units 1–6 and 9 · continued

3Categorical Propositionsthe four forms

A, E, I, O
Form	Standard form	Quantity and quality	Distributed
A	All S are P	universal affirmative	S
E	No S are P	universal negative	S and P
I	Some S are P	particular affirmative	neither
O	Some S are not P	particular negative	P

“All … are not”

“All birds are not flightless” could mean E or O. Rewrite it before you test it.

Parts and distribution

Quantifier + subject + copula (are, are not) + predicate.
Distributed: the statement speaks of every member. Universals distribute the subject; negatives, the predicate.

Putting sentences in standard form Worked

“Not every bus is on time.”
Some buses are not on-time vehicles (O)not every = some … not
“Only ticket holders may enter.”
All who enter are ticket holders (A)only reverses the terms

Shading = empty; X = at least one member

AffIrmo, nEgO (Traditional)

From the Latin: A and I affirm, E and O deny.

4Square of Oppositionhow the four forms relate

Same subject and predicate at all four corners

The modern (Boolean) square

Universals do not say anything exists, so only the contradictories survive.
With no dragons in the castle, “All dragons in the castle are asleep” is true; “Some … are asleep” is false.

The four relations
Relation	Pair	Rule
Contradictories	A–O, E–I	exactly one is true
Contraries	A–E	can both be false, never both true
Subcontraries	I–O	can both be true, never both false
Subalternation	A→I, E→O	truth flows down; falsity flows up

Reading the square Worked

Given: “All the eggs in the carton are cracked” is false.A false
O, “Some eggs in the carton are not cracked,” is truecontradictory
E and I stay undetermined: maybe none cracked, maybe some?

What follows (? = undetermined)
Given	A	E	I	O
A true	T	F	T	F
A false	F	?	?	T
E true	F	T	F	T
E false	?	F	T	?
I true	?	F	T	?
I false	F	T	F	T
O true	F	?	?	T
O false	T	F	T	F

Diagonals always disagree

Corners joined by a diagonal never share a truth value.

5Immediate Inferencesone premise, new proposition

Three immediate inferences
Inference	How	Valid for
Conversion	swap S and P	E, I; A only by limitation (to I)
Obversion	change the quality; P becomes non-P	all four
Contraposition	swap S and P; make both non-	A, O; E only by limitation (to O)

Illicit conversion

“Some birds are not penguins” is true; its converse “Some penguins are not birds” is false. Never convert O.

The forms side by side
Given	Obverse	Converse	Contrapositive
A All S are P	No S are non-P	Some P are S (limit.)	All non-P are non-S
E No S are P	All S are non-P	No P are S	Some non-P are not non-S (limit.)
I Some S are P	Some S are not non-P	Some P are S	not valid
O Some S are not P	Some S are non-P	not valid	Some non-P are not non-S

A string-instrument test Worked

All violins are string instruments.A
convert: “All string instruments are violins” is false (guitars)invalid for A
by limitation: Some string instruments are violinsvalid
contrapose: All non-string instruments are non-violinsvalid for A

Obvert all, convert EI

Obvert anything. Convert E and I. Contrapose A and O.

LogicLogic · Units 1–6 and 9 · continued

6Categorical Syllogismsthree terms, three propositions

The three terms

Major (P): predicate of the conclusion; in the major premise, written first.
Minor (S): subject of the conclusion. Middle (M): in both premises only.

Mood

The mood is the three letters (major, minor, conclusion): EAE, AII, …
64 moods × 4 figures = 256 forms.
15 are valid on any reading; the traditional reading adds 9 more, for 24.

The four figures
Figure	Major premise	Minor premise	Middle term is
1	M–P	S–M	subject, then predicate
2	P–M	S–M	predicate twice
3	M–P	M–S	subject twice
4	P–M	M–S	predicate, then subject

Find the conclusion first

Its predicate is P and its subject is S. The term left over is M.

Naming mood and figure Worked

No planets are stars. All gas giants are planets. So no gas giants are stars.
P = stars, S = gas giants, M = planetsfrom the conclusion
major: No M are P (E); minor: All S are M (A); conclusion Emood EAE
M is subject, then predicate: figure 1EAE-1, Celarent: valid

Standard order

Major premise, then minor premise, then conclusion. Rewrite before naming.

7Rules of Validitysix rules, named faults

The six rules
#	Rule	Breaking it
1	Exactly three terms, each in one sense	four terms (often an equivocation)
2	The middle term is distributed at least once	undistributed middle
3	A term distributed in the conclusion is distributed in its premise	illicit major or illicit minor
4	Not both premises negative	exclusive premises
5	A negative premise requires a negative conclusion, and the reverse	negative and affirmative mismatched
6	Two universal premises cannot yield a particular conclusion	existential fallacy (Boolean)

Undistributed middle Worked

All poets are dreamers. Some dreamers are night owls. So some poets are night owls.
form IAI-1: Some M are P; All S are M; Some S are P
“dreamers” (M) is distributed in neither premiserule 2 broken

Illicit major Worked

All cardinals are birds. No robins are cardinals. So no robins are birds.AEE-1

the conclusion (E) distributes “birds”; the major premise (A) does notrule 3 broken

Four terms Worked

Every bank holds money. The grassy edge of the creek is a bank. So it holds money.

“bank” means a business, then a riverside: four termsrule 1 broken

Passing the rules

A form that breaks none of rules 2 to 6 is valid; the rules and Venn diagrams always agree.
Mark each term d or u before testing.

Middle once, nothing extra

Spread the middle term at least once. The conclusion may not claim more than the premises gave.

8Venn Diagram Teststhree circles, drawn by rule

The method

Draw three overlapping circles: S and P on top, M below.
Diagram the universal premise first (shade what is empty), then the particular (X).
If an X could go in either of two regions, put it on the line.
Valid when the conclusion is already drawn. Never draw the conclusion itself.

Testing on the traditional reading

Put an X in the one open region of the class assumed to have members, then read again.
Darapti: the X lands in the center, so “Some S are P” shows.

Left: Barbara. Right: Darapti

Left: undistributed middle. Right: Festino

Reading the diagrams Worked

AAA-1: all of S outside P is shadedvalid
AAI-3: M is shaded except the center, but nothing says the center has a membervalid only if M exists
AII-2: the X sits on the P line, so it may be outside Pinvalid
EIO-2: the X lands in S outside P: “Some S are not P”valid

Shade first, X second

Universals shade, particulars mark. An X with two possible homes goes on the line.

LogicLogic · Units 1–6 and 9 · continued

9Named Forms & Chainsthe medieval names

The 24 named forms
Fig.	Valid on any reading	Only if a class has members
1	Barbara, Celarent, Darii, Ferio	Barbari (S), Celaront (S)
2	Cesare, Camestres, Festino, Baroco	Cesaro (S), Camestrop (S)
3	Disamis, Datisi, Bocardo, Ferison	Darapti (M), Felapton (M)
4	Camenes, Dimaris, Fresison	Bamalip (P), Fesapo (M), Camenop (S)

Carroll’s puzzles

Lewis Carroll’s Symbolic Logic (1896) turned sorites into playful puzzles: line up the premises so each predicate meets the next subject.

Reading a name

Its first three vowels give the mood: Celarent = EAE.
The letter in parentheses is the class that must have members.

Supplying an enthymeme Worked

“The bakery is closed, since today is a holiday.”
missing: “Every holiday is a day the bakery is closed.”unstated major premise
completed: All M are P; today is M; so today is PBarbara, valid

The memory verse (Traditional)

“Barbara, Celarent, Darii, Ferioque prioris”: the four forms of the first figure.

A sorites (chain) Worked

All tigers are cats. All cats are mammals. All mammals are animals.

link 1: so all tigers are mammals; link 2: so all tigers are animalseach link AAA-1

10Fallacies of Relevancetrue but beside the point

Kinds of fallacy

A fallacy only seems to support its conclusion. Formal flaws lie in the shape (§16); informal ones in the content: relevance, presumption, ambiguity.

Attacking the person
Fallacy	Original example
Abusive ad hominem: attacks the person	“Don’t take Mia’s advice on bike repair; her room is a mess.”
Circumstantial ad hominem: blames the situation	“Of course the coach says drills help; drills are his job.”
Tu quoque (“you too”)	“Why should I floss? You skipped it last night.”

Aim at the argument

Ask what the reasons prove, not who gave them. A real expert in the field is fair evidence.

Judging the source
Fallacy	Original example
Poisoning the well: discredits in advance	“Before Sam describes the field trip, remember he exaggerates.”
Genetic fallacy: judges by origin	“That recipe came off a cereal box, so it can’t be good.”
Appeal to authority outside the field	“A famous goalkeeper says this vitamin cures colds.”

The crowd and the calendar
Fallacy	Original example
Ad populum (bandwagon)	“Everyone in class has these sneakers, so they must be the best.”
Appeal to tradition or novelty	“We have always mowed on Saturday.” “It’s the newest case, so it protects better.”

Appeals to feeling and force
Fallacy	Original example
Appeal to pity	“I deserve an A on this quiz; I was up all night.”
Appeal to fear	“Skip this brand of sunscreen and your skin will be ruined forever.”
Appeal to force (ad baculum)	“Say my drawing is best, or you can’t sit with us.”

Dodging the point
Fallacy	Original example
Red herring: changes the subject	Asked why the dishes aren’t done: “That kitchen light has flickered for weeks.”
Straw man: refutes a weaker version	“You want a shorter recess? So you think kids shouldn’t play at all.”

11Fallacies of Presumptiona hidden, unproved assumption

Assuming too much
Fallacy	Original example
Begging the question: assumes the conclusion	“This is the most exciting book, because no book is more thrilling.”
Question-begging word: a loaded label	“We should drop this useless spelling rule.”
Complex question: hides an assumption	“Have you stopped leaving your bike out in the rain?”
False dilemma: only two options offered	“Either you join the soccer team or you don’t care about fitness.”

Generalizing badly
Fallacy	Original example
Hasty generalization: too few cases	“Two mangoes from this store were sour, so all its fruit is bad.”
Accident: a rule pushed past its purpose	“Running is healthy, so Grandpa should jog on his sprained ankle.”
Slippery slope: an unproved chain	“Let the puppy on the couch once and soon he’ll eat at the table.”

Answering a presumption

Split a complex question; name a third option; look for a common cause.

Mistaking causes
Fallacy	Original example
Post hoc: after, so because	“I wore my blue socks and we won, so the socks won the game.”
False cause: ignores a common cause	“On days the pool sells more lemonade, more swimmers get sunburned, so lemonade causes sunburn.” (Hot sun causes both.)

What is assumed here?

Before you answer, find the claim the speaker slipped in without proof.

LogicLogic · Units 1–6 and 9 · continued

12Ambiguity & Fair Playwords that shift meaning

Shifting words
Fallacy	Original example
Equivocation: one word, two senses	“The sign says ‘Fine for parking here,’ so parking here must be fine.”
Amphiboly: loose grammar	“Grandma saw the hiker with binoculars.” Who had them?
Accent: stress or quoting out of context	A review said “surprisingly good for a cafeteria”; the poster quotes “surprisingly good.”

Parts and wholes
Fallacy	Original example
Composition: parts to whole	“Every brick is light, so the wall is light.”
Division: whole to parts	“The orchestra is loud, so each violinist plays loudly.”

Collective or distributive?

Distributive: said of each member (“The players are tall”).
Collective: said of the group as a whole (“The players ate a mountain of pasta”).
Composition and division slide from one to the other.

Naming fallacies fairly

Charity: read an argument in its most reasonable sense first.
Answer the strongest form of the other side.
The fallacy fallacy: a bad argument does not make its conclusion false.
Avoid fallacy hunting; aim to understand, then repair.

In advertising

Weasel words: “helps fight,” “up to,” “may.”
Unfinished comparison: “Now crunchier!” Than what?
Puffery: “simply the best,” which no one can test.

Repairing a weak argument Worked

“Everyone loves this trail, so it’s the best hike.”ad populum

repair: “This trail has shade, water and views of the falls, so it’s the best hike for summer.”reasons, not a crowd

Pin the word down

If a key word could mean two things, define it before you argue.

13Connectives & Truth Tablesbuilding compound statements

The five connectives
Connective	Symbol	Read	True when
Negation	~p	not p	p is false
Conjunction	p • q	p and q	both are true
Disjunction	p ∨ q	p or q (or both)	at least one is true
Conditional	p ⊃ q	if p, then q	not (p true, q false)
Biconditional	p ≡ q	p if and only if q	both match

Parentheses matter

~(p • q) means “not both.” ~p • q means “not p, and q.”

Truth table for every connective
p	q	~p	p • q	p ∨ q	p ⊃ q	p ≡ q
T	T	F	T	T	T	T
T	F	F	F	T	F	F
F	T	T	F	T	T	F
F	F	T	F	F	T	T

Evaluating a compound Worked

p true, q false, r true: (p ∨ q) ⊃ ~r
p ∨ q = T; ~r = Finside first
T ⊃ F = Fmain connective ⊃

Building a table

Rows = 2 to the number of letters: 2 letters, 4 rows; 3 letters, 8 rows.
First letter: half T, then half F; each next letter halves the run.
Work from the inside out; the main connective is computed last.
Other books write ¬, ∧, → and ↔ for ~, •, ⊃ and ≡.

The conditional fails once

p ⊃ q is false in one row only: p true and q false.

14Conditionals & Equivalenceif-then and its relatives

Four related conditionals
Name	Form	Same as p ⊃ q?
Conditional	p ⊃ q	(itself)
Converse	q ⊃ p	no
Inverse	~p ⊃ ~q	no (it matches the converse)
Contrapositive	~q ⊃ ~p	yes

A geometry conditional Worked

“If a figure is a square, then it is a rectangle.”true
converse: if a rectangle, then a squarefalse
contrapositive: if not a rectangle, then not a squaretrue

Sufficient and necessary

In p ⊃ q, p is sufficient for q, and q is necessary for p.
“p only if q” is p ⊃ q. “p unless q” is ~q ⊃ p, the same as p ∨ q.

Equivalences checked by truth table
Equivalence	Statement
Double negation	~~p ≡ p
De Morgan	~(p • q) ≡ ~p ∨ ~q; ~(p ∨ q) ≡ ~p • ~q
Material implication	p ⊃ q ≡ ~p ∨ q
Contraposition	p ⊃ q ≡ ~q ⊃ ~p
Biconditional	p ≡ q is (p ⊃ q) • (q ⊃ p)
Exportation	(p • q) ⊃ r ≡ p ⊃ (q ⊃ r)

De Morgan’s laws: matching columns
p	q	~(p • q)	~p ∨ ~q	~(p ∨ q)	~p • ~q
T	T	F	F	F	F
T	F	T	T	F	F
F	T	T	T	F	F
F	F	T	T	T	T

Classifying a statement

Tautology: true in every row (p ∨ ~p).
Contradiction: false in every row (p • ~p).
Contingent: some of each (p ⊃ q).

Flip and negate both

The contrapositive always agrees with the conditional. Flip only, or negate only, and it may not.

LogicLogic · Units 1–6 and 9 · continued

15Testing Validityhunt for a bad row

The full method

Give each premise and the conclusion a column.
An argument is invalid if some row has every premise T and the conclusion F; otherwise valid.

The shortcut method Worked

test p ⊃ q, q ⊃ r ∴ p ⊃ r
make the conclusion F: p = T, r = Fassume the bad row
q ⊃ r true forces q = F; p ⊃ q true forces q = Tcontradiction
no bad row can exist: valid

Testing p ⊃ q, q ∴ p
p	q	p ⊃ q	q (premise)	p (concl.)
T	T	T	T	T
T	F	F	F	T
F	T	T	T	F
F	F	T	F	F

Reading the table Worked

row 3: p ⊃ q is T, q is T, p is Fpremises true, conclusion false

so the argument is invalid: affirming the consequentone bad row is enough

Hunt for one bad row

True premises with a false conclusion in any single row sinks the argument.

Full test of p ⊃ q, q ⊃ r ∴ p ⊃ r: no bad row
p	q	r	p ⊃ q	q ⊃ r	p ⊃ r
T	T	T	T	T	T
T	T	F	T	F	F
T	F	T	F	T	T
T	F	F	F	T	F
F	T	T	T	T	T
F	T	F	T	F	T
F	F	T	T	T	T
F	F	F	T	T	T

“Or” means “or both”

p ∨ q, p ∴ ~q is invalid: in row 1 both are true. Only the wording “but not both” allows it.

16Valid Forms & Formal Fallaciespatterns worth memorizing

The conditional forms
Form	Pattern	Original example
Modus ponens (MP)	p ⊃ q, p ∴ q	“If the kettle whistles, the water is boiling. It whistles. So the water is boiling.”
Modus tollens (MT)	p ⊃ q, ~q ∴ ~p	“If the bread is done, the crust is brown. The crust is not brown. So the bread is not done.”

The dilemma
Form	Pattern	Original example
Constructive dilemma (CD)	(p ⊃ q) • (r ⊃ s), p ∨ r ∴ q ∨ s	“If we take the bus, we wait; if we walk, we get wet. We take the bus or walk. So we wait or get wet.”

Chains and choices
Form	Pattern	Original example
Hypothetical syllogism (HS)	p ⊃ q, q ⊃ r ∴ p ⊃ r	“If it snows, school closes; if school closes, we go sledding. So if it snows, we go sledding.”
Disjunctive syllogism (DS)	p ∨ q, ~p ∴ q	“The key is in my coat or my backpack. It is not in my coat. So it is in my backpack.”

Why the fallacies fail

A conditional says p is enough for q, not that p is the only route to q.
So learning q, or learning ~p, settles nothing about the other letter.

Two formal fallacies
Fallacy	Pattern	Original example
Affirming the consequent	p ⊃ q, q ∴ p	“If it rained, the grass is wet. The grass is wet. So it rained.” (Sprinklers?)
Denying the antecedent	p ⊃ q, ~p ∴ ~q	“If it rained, the grass is wet. It did not rain. So the grass is dry.” (Sprinklers again.)

Test what you hear

“Prove all things; hold fast that which is good” (1 Thessalonians 5:21, KJV).

Affirm the front, deny the back

Valid: affirm the antecedent (MP) or deny the consequent (MT). The reverse moves are fallacies.

17Short Proofsstep by valid step

How a proof works

List the premises and number every line.
Each new line follows from earlier lines by a named rule; cite the line numbers.
Rules apply only to whole lines.

Rules of inference
Rule	Pattern
MP, MT, HS, DS, CD	the five valid forms (§16)
Simplification (Simp)	p • q ∴ p
Conjunction (Conj)	p, q ∴ p • q
Addition (Add)	p ∴ p ∨ q

Proof 1: prove r ∨ t
#	Line	Reason
1	p ⊃ q	premise
2	q ⊃ r	premise
3	p • s	premise
4	p	3, Simp
5	p ⊃ r	1, 2, HS
6	r	5, 4, MP
7	r ∨ t	6, Add

Cite every line

If you cannot name the rule and the lines, the step is not yet proved.

Proof 2, in words and symbols Worked

The cat is in the attic (p) or the basement (q). If in the basement, the laundry basket is tipped (r). It is not tipped.
p ∨ q; 2. q ⊃ r; 3. ~rpremises
~q2, 3, MT
p: the cat is in the attic1, 4, DS

Proof strategy

Work backward: what line would give the conclusion?
Split conjunctions with Simp; build a needed disjunction with Add.

LogicLogic · Units 1–6 and 9 · continued

18Logic & Machinestruth tables in circuits

Boole’s algebra

True = 1, false = 0. AND behaves like multiplication: 1 × 0 = 0.
OR is x + y − xy; NOT x is 1 − x.

Series AND, parallel OR

Switches in series act like AND: all must close. Switches in parallel act like OR: one is enough.

Gates in 1s and 0s
A	B	A AND B	A OR B	NOT A
1	1	1	1	0
1	0	0	1	0
0	1	0	1	1
0	0	0	0	1

Binary numerals Worked

1101 in binary

8 + 4 + 0 + 1 = 13each place doubles

Each gate applies one connective; wires carry 1 or 0

19The Classical Traditionfrom Aristotle to computers

The trivium

Trivium (“where three roads meet”): grammar, logic, rhetoric. Then the quadrivium: arithmetic, geometry, music, astronomy.
Logic sits in the middle: grammar supplies its words; rhetoric needs it so persuasion rests on truth.

Schoolmen to Watts
Who	When	Contribution
Peter of Spain	1200s	Summulae Logicales, a standard textbook
Thomas Aquinas	1225–1274	commentaries on Aristotle’s logic
Port-Royal Logic	1662	conceiving, judging, reasoning, method
Isaac Watts	1724	a logic textbook used for generations

Ancient and early medieval
Who	When	Contribution
Aristotle	384–322 B.C.	founded logic; the syllogism; six works gathered as the Organon (“the instrument”)
Chrysippus, Stoics	about 279–206 B.C.	whole propositions joined by if, and, or
Boethius	about 480–524	translated Aristotle’s logic into Latin
Abelard	1079–1142	Sic et Non (“Yes and No”)

Words, judgments, arguments

Aristotle’s order still runs the course: terms, then propositions, then arguments.

Toward symbols and machines
Who	When	Contribution
Leibniz	1646–1716	dreamed of reasoning done like arithmetic
George Boole	1854	The Laws of Thought: an algebra of logic
Gottlob Frege	1879	Begriffsschrift, the start of modern logic
John Venn	1880	overlapping-circle diagrams
Claude Shannon	1937	switching circuits behave like Boole’s algebra

The Organon

Categories (terms), On Interpretation (propositions), Prior Analytics (the syllogism), Posterior Analytics (proof), Topics (probable reasoning), Sophistical Refutations (fallacies).

20Dialectic & Rhetoriclogic in live argument

The Socratic method

Ask for a definition; test it with counterexamples and consequences.
A definition that fails is revised or dropped; the puzzled end is aporia (“no way through”).

A short disputed question Worked

Whether a tomato is a vegetable?
Objection: it is cooked and served with vegetables.
I answer that: by the botanist’s definition it is a fruit (it holds seeds); by the cook’s, a vegetable.distinguo
Reply: the objection is true in the kitchen sense only.

The disputed question (Aquinas)
Part	What it does
Whether …?	a yes-or-no question
Objections	the other side, stated fairly
On the contrary (sed contra)	a brief point toward the answer
I answer that (respondeo)	define, distinguish, give the reasoned answer
Replies	answer each objection in turn

Answering a premise

Concedo: I grant it. Nego: I deny it. Distinguo: true in one sense, false in another.

Fair before forceful

State the other side so well its holders would sign it; then answer it.

Aristotle’s three appeals
Appeal	Persuades through
Ethos	the speaker’s character: good sense, good character, goodwill
Pathos	fitting feelings in the hearers
Logos	the argument itself

Finding the real question (stasis)

Fact: did it happen? Definition: what should we call it?
Quality: was it good or bad? Procedure: who should decide, and how?

Four-step refutation

They say (state it fairly), but (your answer), because (your reason), therefore (why it matters).

LogicLogic · Units 1–6 and 9 · continued

21Argument Mapsseeing an argument’s structure

Drawing a map

Number each claim; draw arrows from premises to the claims they support.
The main conclusion is the claim no arrow leaves.
Joint premises work only together: bracket them with +.
Independent premises each support the claim alone: give each its own arrow.

Chains and convergence

Chain: 1 → 2 → 3, each claim supporting the next.
Convergent: several independent reasons meet at one conclusion.

Hidden premises

A bridge premise joins what is stated to the conclusion: “All A are B” or “If p, then q.”
Supply the most reasonable one (charity), then judge it.

A passage to map Worked

(1) The museum closes soon. (2) The drive takes most of an hour. So (3) we would arrive after closing. Besides, (4) we already have tickets for Saturday. So (5) we should go Saturday.

1 + 2 → 3 (joint); 3 → 5 and 4 → 5 (independent)3 is a sub-conclusion

Reading a map aloud

Read from the bottom up: “5, because 3 and 4; and 3, because 1 and 2 together.”

Finding the bridge Worked

“Maya will be tired tomorrow, since she runs the race today.”
bridge: anyone who runs the race is tired the next dayunstated major premise
now ask whether the bridge is truejudge it fairly

The museum argument as a map

The removal test

Cross out one premise. If the other still gives some support, they are independent; if it gives none, they are joint.

No arrow out: the point

On a finished map, the claim with no arrow leaving it is what the speaker is proving.

22Practice with Answerscheck yourself

Syllogisms and the square
Problem	Answer
1. Some athletes are musicians. All musicians are early risers. So some early risers are athletes. Mood, figure, verdict?	IAI-4, Dimaris: valid
2. No fish are mammals. Some pets are not fish. So some pets are mammals.	EOI-1: exclusive premises, invalid
3. Contradictory of “Some songs are not catchy”?	“All songs are catchy”

Inferences and a proof
Problem	Answer
8. Convert “Some teachers are coaches.”	“Some coaches are teachers” (valid)
9. Prove q from p ⊃ q, r ⊃ p, r.	4. p (2, 3, MP); 5. q (1, 4, MP)
10. “Some stamps are rare” is false. What follows?	E true; A false; O true

Forms, rules and definitions
Problem	Answer
15. Test: All P are M; No S are M; so No S are P.	AEE-2, Camestres: valid
16. Which rule does AAA-2 break?	undistributed middle
17. “A clock is something that ticks.” Fault?	too broad (metronomes) and too narrow (digital clocks)

Truth tables and inferences
Problem	Answer
4. Classify (p ⊃ q) ∨ (q ⊃ p).	tautology
5. Valid? p ⊃ q, ~p ∴ ~q	invalid: row 3 (denying the antecedent)
6. Obvert “All cars are vehicles.”	“No cars are non-vehicles”
7. Equivalent to “If it’s a holiday, the library is not open”?	“If the library is open, it is not a holiday”

Connectives and conditionals
Problem	Answer
18. p true, q false: value of ~(p • q)?	T
19. Is p ∨ ~q equivalent to q ⊃ p?	yes (material implication)
20. Contrapositive of “If it snows, the game is canceled”?	“If the game is not canceled, it did not snow”

Problems 15 and 16, drawn

Name the fallacy
Problem	Answer
11. “This shampoo must be great; it outsells every other brand.”	ad populum
12. “Our neighbor’s dog barked, then the power went out, so the barking cut the power.”	post hoc
13. “Is your cat still scratching the new couch?”	complex question
14. “Each player is excellent, so the team must be excellent.”	composition

Name the fallacy, part 2
Problem	Answer
21. “My uncle is a dentist, so his advice on car engines must be right.”	appeal to authority
22. “Don’t trust Leo’s math answer; he’s always late.”	abusive ad hominem
23. “Either we paint the fence blue or it will look terrible.”	false dilemma
24. “Skip one practice and you’ll quit the team and never play again.”	slippery slope

Name it, then explain it

A label is not an answer. Say which rule or relation applies and why.