Logic Workshop
Expectations?
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What do you expect to get out of this
workshop?
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What do you currently know about logic?
Schedule
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Day One
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Introduction to logic (what is it? why is it important?)
Historical connections within the context of philosophy
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Major philosophers and their contributions to logic
The building blocks of logic
Day Two
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Applications of logic
Symbolic logic
Paradoxes and other interesting philosophical questions
Logic 101
λόγος
Greek

λόγος – logos
 From Classical Greek
 Essentially means “thought, idea, argument, account, reason, or principle”
 A very important term in philosophy (as well as in analytic psychology,
rhetoric, & religion)
 Derives from
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Logic is about just this (logos) – it’s about developing an
understanding of the methods and principles used to distinguish
between correct and incorrect reasoning
Traditionally considered a branch of philosophy
Today it has also spread to other fields such as mathematics (e.g.
mathematical logic) and computer science (e.g. programming,
writing code)
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Everyday Use
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We use “logic” everyday
Arguments, justifications, reasons, most all rational
functions
 Listen to just how many logical arguments (with
premises and conclusions) you hear on TV shows,
the radio, etc.
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Therefore it’s important to develop our logical
knowledge…
…because if we don’t
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then we might be the victim of Bad Reasoning
But, as the Stanford Encyclopedia of
Philosophy notes, logic “does not, however, cover
good reasoning as a whole. That is the job of
the theory of rationality. Rather it deals with
inferences whose validity can be traced back to
the formal features of the representations that
are involved in that inference, be they linguistic,
mental, or other representations”
http://www.youtube.com/watch?v=G40OEBuIZdM
Brief Introduction to Logic
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Logic: the organized body of knowledge, or
science, that evaluates arguments
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Our aim then is to develop system of methods to
use as criteria for evaluating arguments of others
and for constructing our own; to determine good
arguments from bad arguments
Logic Evaluates Arguments
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Argument: a group of statements, one or more of
which (premises) are claimed to provide support
for, or reasons to believe, one of the others
(conclusions)
Good argument: premises support the conclusion
 Bad argument: premises do not support conclusion
(even if they claim to)
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Syllogism: a kind of logical argument in which
one proposition (the conclusion) is inferred
from at least two others (the premises) of a
certain form
Arguments are Made up of Statements
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Statement: a sentence that is either true (T) or false (F)
 Melatonin helps relieve jet lag. (T)
 No wives ever cheat on their husbands. (F)
and is made up of…
 1. Premises
 Statements that set forth the reasons or evidence
 2. Conclusions
 Statements that the evidence is claimed to support or
imply (claimed to follow from the premises)
Propositions
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Proposition: the meaning or information content of a statement
(the ‘material of our reasoning’)
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Inference: a process of linking propositions by
affirming one proposition on the basis of one or
more other propositions
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“Proposition=Statement”
 Can be simple (making only one assertion) or compound (containing
two or more “simple” propositions
“Inference=Argument”
Many sentences, unlike statements, cannot be said to be T or F
 Questions (Where is Tom?); Proposals (Let’s go to a movie.); Suggestions
(I suggest you get contact lenses.); Commands (Turn off the TV.);
Exclamations (Wow!)
Good vs. Bad Arguments
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Good Argument:
1. All film stars are celebrities. (Premise 1)
2. Halle Berry is a film star. (Premise 2)
C. Therefore, Halle Berry is a celebrity. (Conclusion)
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Bad Argument:
1. Some film stars are men.
2. Cameron Diaz is a film star.
C. Therefore, Cameron Diaz is a man.
Example
The space program deserves increased
expenditures in the years ahead. Not only does
the national defense depend upon it, but the
program will more than pay for itself in terms
of technological spinoffs. Furthermore, at
current funding levels the program cannot fulfill
its anticipated potential.
Argument Reconstruction
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Break up compound statements
List premises first, then conclusions:
P1: The national defense is dependent upon the space program.
P2: The space program will more than pay for itself in terms of
technological spinoffs.
P3: At current funding levels the space program cannot fulfill its
anticipated potential.
C: The space program deserves increased expenditures in the
years ahead.
Classes of Arguments
1.
Inductive Argument:
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2.
Claim that the premises support the conclusion with a certain
degree of probability
If the premises are true, then it’s likely the conclusion is true
Ex. This cat is black. That cat is black. A third cat is black.
Therefore all cats are black.
Deductive Argument:
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Claim that the premises support the conclusion conclusively –
there is no probability involved, but logical certainty
If the premises are true, then it’s impossible the conclusion is false
Ex. All men are mortal. Joe is a man. Therefore Joe is mortal.
Deductive Arguments
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Look for…
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Validity: if all premises are true, then the conclusion must be
true (follows from the structure, not the content - no matter
how ridiculous it may seem - of an argument)
Soundness: if the argument is valid, and the premises are true,
then the argument is sound
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All sound arguments are valid, but not all valid arguments are sound
Cogency: if the argument is valid and sound, is it cogent
(convincing) to a community of listeners
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If all sound arguments are valid, must all cogent arguments be sound?
Belvedere
by M.C. Escher
•A structure in which the relations of the
base to the middle and upper portions
are not rational
•A deductive argument rests upon
premises that serve as its foundation
•To succeed, its parts must be held
firmly in place by the reasoning that
connections those premises to all
that is built upon them – this way
the argument can stand
Waterfall
by M.C. Escher
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Doesn’t make sense: the
water flows away, and going
away comes closer
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Flows downward, and
going down it comes up,
returning to the point from
which it began
As perception may be
tricked by a clever picture,
our thinking may be tricked
by a clever argument
In the picture we confront
disorder in seeing, and then
with scrutiny detect its cause
In logic we confront many
bad arguments, and then
with scrutiny learn what
makes them bad
History
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Logic as a discipline started with the
transition from the unreflective use of
logical methods and argument patterns to
the reflection on and inquiry into these and
their elements, including the syntax and
semantics of sentences
Mesopotamia (11th century BC):
– Medical diagnostic handbook containing
numerous axioms and assumptions (i.e.
through examination and inspection of the
symptoms of a patient, it is possible to
determine the patient's disease, its aetiology
and future development, and the chances of
the patient's recovery)
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Babylon (8th – 7th centuries BC):
– Babylonian astronomers began employing an
internal logic within their predictive planetary
systems, which was an important
contribution to logic and the philosophy of
science.
– Babylonian thought had a considerable
influence on early Greek thought
Greece (4th century BC)
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Logic as a fully systematic discipline begins with
Aristotle (384-322 BC)
He systematized much of the logical inquiries of his
predecessors
Main achievements:
–
“Term Logic”
 Fundamental elements of his syllogistic logic
are “terms,” which are not true or false in
themselves, such as ‘human being,’ ‘animal,’
‘white’ (i.e. H, A, and W)
 For Aristotle, a term is simply a “thing,” a part
of a proposition; and a proposition is just a
particular kind of sentence, in which the
subject and predicate are combined so as to
assert something true or false
 The essential feature of term logic is that, of
the four terms in the two premises, one must
occur twice. Thus
– All Greeks are men
– All men are mortal.
 This system became known as the “categorical
syllogistic”
Another System Emerges
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Chrysippus (279-206 BC)
– Was a stoic
 {Stoicism: teaches the development of selfcontrol and fortitude as a means of
overcoming destructive emotions; holds that
becoming a clear and unbiased thinker allows
one to understand the universal reason, logos}
– Main achievements:
 “Propositional Logic”
– Fundamental elements of logic are “whole
propositions” - something over and above the
terms are true
– Studies ways of joining and/or modifying entire
propositions, statements, or sentences to form
more complicated ones
– Also looks at the logical properties derived from
these methods of combining or altering
statements (e.g. logical connectives in symbolic
logic).
– Does not study those logical properties that
depend upon parts of statements that are not
themselves statements on their own, such as
the subject and predicate of a statement
– This system became known as the
“hypothetical syllogistic”
1300-year Intermission
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200 BC – 1100 AD
– Lack of logical innovations
– Mostly just commentators either preserving
Aristotelian logic, such as Boethius (480-524
AD), or criticizing it, such as Islamic
philosopher Avicenna (980-1037 AD)
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Peter Abelard (1079-1142 AD)
– Theory of universals in the mind rather than natures
outside the mind (as Aristotle argued)
 Distinguishes between arguments valid in form and
arguments valid in content
 Abelard observes that the same propositional content can be
expressed with different force in different contexts:
– the content that Socrates is in the house is expressed in an
assertion in “Socrates is in the house”; in a question in “Is
Socrates in the house?”; in a wish in “If only Socrates were in
the house!” and so on.
– Hence Abelard can distinguish the assertive force of a sentence
from its propositional content, a distinction that allows him to
point out that the component sentences in a conditional
statement are not asserted, though they have the same
content they do when asserted
 Ex. "If Socrates is in the kitchen, then Socrates is in the
house" does not assert that Socrates is in the kitchen or
that he is in the house
 Likewise, the distinction allows Abelard to define negation,
and other propositional connectives, purely truthfunctionally in terms of content, so that negation, for
Gottfried Leibniz
(1646 – 1716 AD)
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Leibniz is the most important logician
between Aristotle and 1847, when
George Boole and Augustus De Morgan
each published books that began
modern formal logic
• Leibniz enunciated the principal properties of what we now call conjunction,
disjunction, negation, identity, set inclusion, and the empty set, essentially
building the foundation for symbolic logic as we know it today
• Two main principles:
1. All our ideas are compounded from a very small number of simple ideas,
which form the alphabet of human thought.
2. Complex ideas proceed from these simple ideas by a uniform and
symmetrical combination, much like arithmetical multiplication.
This world is the best of all possible worlds. - Leibniz
Symbolic Logic AP
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Is a technique for analysis of deductive arguments
English (or any) language can make any argument
appear vague, ambiguous; especially with use of
things like metaphors, idioms, emotional appeals,
etc.
We want to avoid these difficulties to move into
logical heart of argument: use symbolic language
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Now can formulate an argument with precision
Symbols facilitate our thinking about an argument
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These are called “logical connectives”
The Logical Connectives
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1.
We can translate arguments from sentences and simple or
compound propositions/statements into symbolic logical form
using:
Conjunction: “(conjunct 1) and (conjunct 2)”
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2.
pvq
Negation: “It is not the case that…”
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4.
p&q
Disjunction: “(disjunct 1) or (disjunct 2)”
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3.
p•q
~p
Conditional: “If (antecedent), then (consequent)”
p
∩
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q
pq
Basic Abbreviation & Translation Rules
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“Charlie’s neat and Charlie’s sweet.”
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N•S
Dictionary: N=“Charlie’s neat” S=“Charlie’s
sweet”
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Can choose any letter to symbolize each
statement (in this case, two conjuncts), but it is
best to choose one relating to the content of that
conjunct to make it easier to remember
Punctuation
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As in mathematics, it is important to correctly punctuate
logical parts of an argument
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Ex. (2x3)+6 = 12 whereas 2x(3+6)= 18
Ex. p • q v r (this is ambiguous)
To avoid ambiguity and make meaning clear
Make sure to order sets of parentheses according to how the
argument reads:
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{ A • [(B v C) • (C v D)] } • ~E
would be a different read than
[(A • B) v (C • C)] v [D • ~E]
Example
“There were three people involved in the accident,
and no one was injured.”
(Steps: Translate into logic by making a dictionary
then arrange statements correctly in logical form)
(Hint: Note: When symbolizing statements, always
make the statement a positive one. If you have a
negative statement in the sentence, put its
positive in the dictionary – then when you
translate, simply negate that sentence)
Solution
There were three people involved in the
accident, and no one was injured.
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T • ~O
T=Three people were involved in the accident.
O=Someone was injured.
One more…
“Either you are male or female but not both.”
Solution
“Either you are male or female but not both.”
(M v F) • ~(M • F)
M=You are male.
F=You are female.
Inference Rules
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There are many different systems of formal logic,
each one with its own set of well-formed formulas,
rules of inference, and even semantics (e.g.
temporal logic, modal logic, intuitionistic logic,
quantum logic)
These rules are like shortcuts for us when doing
logical proofs
Take a look at the handout you have on Inference
Rules
Nine Basic Inference Rules
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These nine rules of inference correspond
to elementary argument forms whose
validity is easily established by truth
tables. With their use, formal proofs of
validity can be constructed for a wide
range of more complicated arguments.
Let’s put this all together now:
First, translate the following argument from
English into symbolic logic:
If Anderson was nominated, the she went to
Boston.
2. If she went to Boston, then she campaigned
there.
3. If she campaigned there, she met Douglas.
4. Anderson did not meet Douglas.
5. Either Anderson was nominated or someone
more eligible was selected.
C. Therefore: someone more eligible was
selected.
1.
Solution (Part 1)
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A B
B C
C D
~D
AvE
Therefore E
∩ ∩ ∩
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Now let’s try to prove the conclusion
from only the given premises:
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This is now where our inference rules may
(or may not) be of help to us
Once translated…
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Write the premises and the statements that
we deduce from them in a single column –
to the right of this column, for each
statement, its “justification” is written (e.g.
the reason why we include that statement
in the proof)
Here’s what it looks like:
Formal Proof
2.
3.
4.
5.
7.
8.
9.
∩ ∩
6.
A B
B C
C D
~D
AvE
A C
A D
~A
E
∩ ∩ ∩
1.
Premise
Premise
Premise
Premise
Premise
1,2 H.S.
6,3 H.S.
7,4 M.T.
5,8 D.S.
The justification for each
statement (the right most
column) consists of the
numbers of the preceding
statements from which that
line is inferred, together with
the abbreviation for the rule of
inference used to get it
Proofs
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There are many other examples of simpler
(and more difficult) proofs, but we don’t
have time to get into the rules associated
with proving even the most basic of them,
so we will skip over that for now
The main point here is to show the
application of symbolic logic in the context
of symbolic translations and in constructing
formal proofs
Paradoxes and Questions
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Paradox – seemingly sound piece of reasoning based
on seemingly true assumptions that leads to a
contradiction or another obviously false conclusion
Zeno's paradoxes are a set of problems devised by
Zeno of Elea to support Parmenides' doctrine that "all
is one" and that, contrary to the evidence of our senses,
the belief in plurality and change is mistaken, and in
particular that motion is nothing but an illusion
Zeno’s Paradoxes
Story of a race between Achilles
and the tortoise
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Achilles can run faster so tortoise given
big head start
1.
2.
3.
4.
When race starts, Achilles’s first goal is
to get to the point where the tortoise
started
By the time he gets there, the tortoise
has moved again (but only a little) – so
Achilles must now get to that spot
No matter how many times Achilles
reaches the tortoise’s prior location,
even if he does it an infinite number of
times, he’ll never catch up with the
tortoise, although he’ll get awfully close
Tortoise just has to not stop then in
order to win
“In a race, the
quickest runner can
never overtake the
slowest, since the
pursuer must first
reach the point
whence the pursued
started, so that the
slower must always
hold a lead.”
-Aristotle
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Racetrack paradox:
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In order to get to the end
of a racetrack, a runner
must first complete an
infinite number of
journeys
1.
2.
3.
4.
He must run to the
midpoint
Then he must run to the
midpoint of the remaining
distance
Then to the midpoint of
the still remaining distance,
etc etc
Therefore he can never get
to the end of the track
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Arrow paradox
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1.
2.
3.
4.
5.
6.
If everything, when it occupies an equal space, is at rest, and if that which is in
locomotion is always occupying such a space at any moment, the flying arrow is
therefore motionless.
- Aristotle
Imagine an arrow in flight
Divide up time into a series of indivisible moments
At any given moment, if we look at the arrow it has an exact location, so
it is not moving
Yet movement has to happen in the present (it can't be that there's no
movement in the present yet movement in the past or future)
So throughout all time, the arrow is at rest
Thus motion cannot happen
Zen Koans
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A kōan is a story, dialogue,
question, or statement in the
history and lore of Zen
Buddhism, generally containing
aspects that are inaccessible to
rational understanding (yet may
be accessible to intuition)
The Gateless Gate
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Joshu’s Dog
A monk asked Joshu, a Chinese Zen master:
“Has a dog Buddha-nature or not?”
Joshu answered: “Mu.”
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Mumon’s comment: Enlightenment always comes after the
road of thinking is blocked. ‘Mu’ is not nothingness,
just concentrate your whole energy into this Mu, and
do not allow any discontinuation.
The Gateless Gate
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Wakuan complained when he saw a picture of bearded
Bodhidharma: “Why hasn’t that fellow a beard?”
Daibai asked Baso: “What is Buddha?” Baso said: “This
mind is Buddha.”
A monk asked Baso: “What is Buddha?” Baso said: “This
mind is not Buddha.”
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Two hands clapping make a sound. What is the sound
of one hand clapping?
Thank you!
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Were your expectations met? If not, why?
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What do you now know about logic?