The Language of Truth-functional Logic
The Plan:
LTFL: The Language of Truth-functional Logic
I We will develop a formal language, the Language of Truth-functional
Logic, or LTFL.
I We can then symbolize English sentences (or whole arguments) into
LTFL to reveal the truth-functional form of the sentence/argument.
I Then we will develop a semantics for LTFL, which will let us define
what it is for an argument to be truth-functionally valid.
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The Lexicon of LTFL
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Expressions and Sentences
I An expression of TFL is any string of symbols of TFL.
The lexicon of LTFL contains the following symbols:
I Of course, many expressions of TFL will be total gibberish.
We want to know when an expression of TFL amounts to a sentence.
I Atomic sentences: ‘A’, ‘B’, ‘C’, .. , ‘A1 ’, ‘B1 ’, .. , ‘A2 ’, ‘B2 ’ .. etc.
I Connectives: ‘∼’ (tilde), ‘&’ (ampersand), ‘∨’ (vee), ‘→’ (right
arrow), ‘↔’ (double arrow).
I Punctuation marks: ‘(’ and ‘)’
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I A sentence of TFL is a string of symbols of LTFL constructed
according to the Syntax of LFTL. Such a string is called a
well-formed formula (wff).
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Syntax of LTFL
Exercise
We can now recursively define what counts as a sentence (or a
“well-formed formula”) of LTFL:
Exercise 2c: which of the following are sentences of LTFL? If it’s not a
sentence, how would you rewrite it to make it well-formed?
1. Any atomic sentence (‘A’, ‘B’, ‘C’ ... etc.) is a sentence.
1.
2.
3.
4.
5.
6.
7.
2. If p is a sentence, then ’∼p’ is a sentence.
3. If p and q are sentences, then ’(p&q)’ is a sentence.
4. If p and q are sentences, then ’(p∨q)’is a sentence.
5. If p and q are sentences, then ’(p→q)’ is a sentence.
6. If p and q are sentences, then ’(p↔q)’ is a sentence.
7. Nothing else is a sentence.
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Bracketing Conventions
(A ∨ B)
(∼P&Q)
((∼A) → B)
((∼P → Q ∨ R)
(R ∨ (p&Q))
(A ∨ B ∨ F ∨ G )
∼(P → Q)
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Main Connective
I We allow ourselves to omit the outermost brackets of a sentence.
I It can be difficult to read sentences with many nested pairs of
brackets. To make things a bit easier on the eyes, we allow ourselves
to use square brackets instead of the round one.
I Combining these two conventions, we can rewrite the unwieldy
sentence
(((H → I ) ∨ (I → H))&(J ∨ K ))
I Definition: If p is a sentence, then the main connective of p is the
last connective added to the sentence in the process of building it up
from its parts.
I Question: What is the main connective in each of the following
sentences?
1. (∼A&B)
2. ∼(A&B)
rather more simply as follows:
[(H → I ) ∨ (I → H)]&(J ∨ K )
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Terminology
Main Connective
I A sentence of the form ’∼p’ is called a negation; p is called the
negated sentence.
I A sentence of the form ’(p&q)’ is called a conjunction; p and q are
its left and right conjuncts.
I A sentence of the form ’(p∨q)’ is called a disjunction; p and q are its
left and right disjuncts.
I A sentence of the form ’(p → q)’ is called a conditional; p is its
antecedent and q its consequent.
I A sentence of the form ’(p ↔ q)’ is called a biconditional; p is its
lefthand side and q its righthand side.
I A sentence with no connectives is called an atomic sentence; a
sentence with connectives is called a compound sentence.
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The main connective of a sentence tells us what kind of sentence it is.
For example, (A& B) is a conjunction, and ∼(A&B) is a negation.
Exercise 2f: What kind of sentence each of the following sentences is?
I (B ∨ (A&∼B))
I ((B ∨ A)&∼B)
I (∼(B ∨ A) → (∼B&∼A))
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Symbolization: Connectives
Roughly speaking, our LTFL connectives represent the following English
connectives:
Basic Symbolization
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English Connective
’not’, ’it is not the case that’
’and’
‘or’
‘if ... then’
‘if and only if’
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LTFL Connective
∼
&
∨
→
↔
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Symbolization: Atomic Sentences
Symbolization: Connectives
English sentences with just one connective are easy to symbolize:
I It isn’t cold
Atomic English sentences, ones with no connectives, are symbolized by
atomic sentences of LTFL, i.e. a single sentence-letter:
∼C
I Syracuse is in New York and Syracuse is in the US.
I It is cold : C
I It is raining outside : R
I Syracuse is in New York : N
(N&U)
I It is raining outside or it is sunny.
(R ∨ S)
Remember: always give the symbolization that captures as much of the
logical structure of the English sentence as possible. Only atomic English
sentences are symbolized as single atomic sentences.
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Symbolization Guide
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Expanding to Full Truth-functional Form
When you offer a symbolization of an English sentence in LTFL, you
always have to provide a symbolization guide (or “dictionary”) that says
what English sentences each atomic sentence is meant to represent!
English sentences are often “condensed.” Such sentences have to be
expanded into their full truth-functional form, i.e. into a form where the
English connectives connect complete statements.
This will help you identify the atomic English sentences you should put in
your symbolization guide. For example:
I ‘Plato has a beard and Socrates is not tall.’
I Bill and Ted met Socrates.
I Bill met Socrates and Ted met Socrates.
Symbolization Guide:
B: Plato has a beard.
S: Socrates is tall.
Symbolization Guide
B: Bill met Socrates.
T: Ted met Socrates.
Symbolization: (B&∼S)
(B & T)
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Expanding to Full Truth-functional Form
Symbolization with ‘&’
You also don’t want to leave any pronouns in your symbolization guide:
I Meg went to Disney World, and she didn’t enjoy it.
I Meg went to Disney World, but she didn’t enjoy it.
I Meg went to Disney World, yet she didn’t enjoy it.
I Meg went to Disney World, however she didn’t enjoy it.
I Meg went to Disney World, although she didn’t enjoy it.
I Meg went to Disney World and she didn’t enjoy it.
Symbolization Guide
D: Meg went to Disney World.
E: She enjoyed it. ⇐= No!
Symbolization Guide
D: Meg went to Disney World.
E: Meg enjoyed Disney World.
I Meg went to Disney World; furthermore she didn’t enjoy it.
I Meg went to Disney World; moreover she didn’t enjoy it.
These differ in nuances of meaning, but these differences are not of
logical importance (i.e. don’t matter to the validity of arguments). All
can be symbolized as ‘(D&∼E )’.
Symbolization: (D&∼E )
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We’ll use ‘&’ to symbolize a variety of English connectives besides ‘and’:
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Exercise
Exercise 2e: symbolize the following in LTFL (remember to give a
symbolization guide):
Grouping
1. Simon and his sister went shopping.
2. Melissa gave her dog a biscuit but he didn’t like it.
3. That tower is either small or far away.
4. Bill’s coat was inexpensive even though it’s of good quality.
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Grouping
Grouping
English sentences with multiple connectives can get trickier.
I Either it’s sunny, or it’s cold and icy.
I Either it’s sunny, or it’s cold and icy.
I S: It’s sunny.
C: It’s cold.
Y: It’s icy.
I Symbolization: S ∨ C &Y ⇐= No!
The English sentence has the overall form of a disjunction (with a
conjunction as the right-hand disjunct), which is also the structure of
‘S∨(C&Y)’. So that’s the correct symbolization.
Which English sentence would ‘(S∨C)&Y’ be a proper symbolization of?
I It’s either sunny or cold, and it’s icy.
‘S ∨ C &Y ’ is not a well formed formulas of LTFL. Our two options are
(S ∨ C )&Y and S ∨ (C &Y ). Which is the correct symoblization?
(Notice that even ‘(S ∨ C )&Y ’ and ‘S ∨ (C &Y )’ are missing the
outermost parentheses. It’s ok to leave off outermost parentheses to aid
in readability, but any other parentheses have to be included.)
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This means something different from the original sentence. Moral:
parentheses matter — ‘(S∨C)&Y’ and ‘S ∨ (C &Y )’don’t mean the same
thing.
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Exercises
Exercise 2g: provide symbolizations of the following, making sure to get
their structure right.
1. Simon either went downtown and ate a sandwich, or he’s sleeping at
home.
Interactions of ‘∼’ with ‘∨’ and ‘&’
2. Monica didn’t bring her raincoat, but she did bring both an umbrella
and a hat.
3. Peter is either in the Adirondacks or the Catskills, although he
doesn’t like hiking.
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Interactions of ‘∼’ with ‘∨’ and ‘&’
Interactions of ‘∼’ with ‘∨’ and ‘&’
Negation interacts in interesting ways with disjunction and conjunction.
I Socrates wasn’t both tall and handsome.
I The Holy Roman Empire was neither holy nor roman.
T: Socrates was tall.
H: Socrates was handsome.
H: The Holy Roman Empire was holy.
R: The Holy Roman Empire was roman.
I It’s not the case that both T and H.
I ∼(T & H)
I Neither H nor R.
I ∼H&∼R
Can you think of another way to symbolize this into LTFL?
Can you think of another way to symbolize this into LTFL?
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DeMorgan’s Laws
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Exercises
Exercise 2h: provide symbolizations of the following.
1. Socrates was neither tall nor handsome.
2. The book was both intelligent and funny, but the movie was neither.
Logish
LTFL
pNeither p nor qq
p(∼p & ∼q)q
p∼(p ∨ q)q
pNot both p and qq
p∼(p & q)q
p(∼p ∨ ∼q)q
3. Liz isn’t flying to both San Francisco and New York today, though
she is flying to one of them.
We’ll see these DeMorgan’s Laws again and again as we continue.
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Conditional
Conditionals are ‘if ... then ...’ statements, which we symbolize with ‘→’.
1. If I finish my reading, then I will watch a movie.
R→M
Symbolizing Conditionals: →
Notice that the ‘then’ can be omitted, and that the ‘if’ clause can come
second in the English sentence:
2. If I finish my reading, I will watch a movie :: R → M
3. I will watch a movie if I finish my reading :: R → M
In general: the antecedent is the part that’s preceded by ‘if’ in English,
and the consequent is the part that isn’t preceded by ‘if’.
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Exercises
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If v. Only If
It’s very important to distinguish ‘if’ from ‘only if’ statements.
Exercise 2i: symbolize the following.
1. Meg will get the job if she submits an application.
2. Meg will get the job only if she submits an application.
1. If Megan doesn’t go to the party, Liz won’t go either.
2. Liz will go to the party if Megan and Ben both go.
3. If Megan goes to the party, Ben will go if Liz does too.
4. If it’s the case that Ben will go if Megan doesn’t, then Liz won’t go.
These mean different things: (1) says that submitting an application is
sufficient for getting the job (probably false), whereas (2) says that it’s
necessary for getting the job (probably true).
I (1) is symbolized: S → J
I (2) is symbolized: J → S
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Necessary and Sufficient Conditions
Necessary and Sufficient Conditions
With → sentences, the antecedent is always the sufficient condition, and
the consequent is always the necessary condition.
Exercise 2j: true or false:
1. Me being in Syracuse is sufficient for me to be in the US.
p→q
2. Meg’s attending class is sufficient for her to do well on exams.
3. Fluffy’s being a cat is necessary for Fluffy to be a mammal.
Sufficient Condition
Necessary Condition
In the following, what is the necessary condition and what is the
sufficient condition? How would you symbolize each?
I ‘If A then B’ or ‘B if A’ say that A is sufficient for B. So they’re
symbolized A → B.
I ‘B only if A’ says that A is necessary for B. So it’s symbolized
B → A.
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5. If you startle Sam, he’ll jump out of his seat.
6. The toaster will work only if it’s plugged in.
7. Ted will go to the movies if he finishes his logic homework on time.
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Necessary and Sufficient Conditions
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Symbolization with ‘→’
As with ‘&’, we use ‘→’ to symbolize many English connectives.
Notice that since being in Syracuse is sufficient for being in the US, it
can’t be the case that I am in Syracuse but not in the US. This illustrates
the following general principle:
I pp → qq means the same as p∼(p&∼q)q.
The following say that p is sufficient for q, and get symbolized pp → qq:
I pif p then qq
I pp implies qq
I pq if pq
I pq provided that pq
I pq whenever pq
I pq is the case as long as p is the caseq
Similarly: since being in Syracuse is sufficient for being in the US, it
follows that if I’m not in the US, I’m also not in Syracuse. This
illustrates the following general principle (called contraposition):
I pp → qq means the same as p∼q → ∼pq
The following say that p is necessary for q, and get symbolized pq → pq:
I pq only if pq
I pq is contingent on pq
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A Tricky Case: ‘Unless’
A Tricky Case: ‘Unless’
One particularly tricky case involving ‘→’ is ‘unless’. Consider:
Later in the course we’ll see that p∼q→pq is equivalent to (i.e. means the
same as) pq∨pq. This means that we can also translate ‘unless’ by ‘∨’:
I You will catch cold unless you wear a jacket.
I Unless you wear a jacket, you will catch cold.
I pp unless qq can be symbolized pp ∨ qq.
Both sentences mean that if you do not wear a jacket, then you will
catch cold. So it gets symbolized:
I You will catch cold unless you wear a jacket .
I C ∨J
I ∼(wear a jacket) → you will catch cold
I ∼J → C
Notice: we are claiming that this sentence doesn’t say that if you do wear
a jacket, you won’t catch cold, just that you will catch cold if you do not
wear a jacket..
In general:
I pp unless qq means pp if not qq, so
I pp unless qq gets symbolized as p∼q→pq.
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The Biconditional: ↔
Exercises
The following are translated by pp↔qq
Exercise 2k: symbolize the following in LTFL:
I pp if and only if qq
I pp if, but only if, qq
I pp just in case qq
1. Bob’s getting a raise is contingent on his not getting fired first.
2. Dan won’t take Biology unless his friends do too.
3. Annie will mow the grass if her sister does the dishes, provided that
there are dishes to be done.
4. Liz will go hiking as long as Meg comes along, unless the weather
turns bad — in that case she’ll go on a bike ride.
Notice that pp if and only if qq means the same thing as
pp if q and p only if qq. So:
I pp ↔ qq is equivalent to p(q → p) & (p → q)q
Biconditionals state necessary and sufficient conditions.
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Exercises
Symbolizing Arguments
Exercise 2l: symbolize the following:
Ultimately, what we’re interested in is the validity of arguments. But
symbolizing an argument is not just a matter of symbolizing the
statements that compose it.
1. If Lisa gets paid, then she will get a ride to the mall if and only if
she either buys a CD or a DVD or a pair of shoes.
2. Neither Sweden nor Ireland will attend the summit, provided Russia
and China don’t both attend.
3. Sarah isn’t going unless Richard and Pam are both going, and Tim
is going if, and only if, neither Pam nor Quincy are going.
4. If Canada subsidizes exports, then the US will raise tariffs if Mexico
opens new factories.
5. If evolutionary biology is correct, higher life forms arose by chance,
and if that’s so, then there isn’t any design in nature and divine
intervention is false.
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Example
Arguments have structure to them: we need to identify which statement
is the argument’s conclusion, and we need to spot reappearances of the
same statement in different parts of the argument.
1. Indicate the conclusion by putting a ‘∴’ symbol in front of it.
2. Use a separate line for each premise.
3. Identify reappearances of the same statement, even if the wording
changes slightly. Use the same atomic sentence to symbolize
reappearances of the same statement.
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Example
Argument: Suppose fewer than two contestants enter; then there will be
no contest. No contest means no winner. Suppose all contestants
perform equally well. Still no winner. There won’t be a winner unless
there’s a loser. And conversely. Therefore, there will be a loser only if at
least two contestants enter and not all contestants perform equally well.
Step 3: Should we now use the following additional atomic statements
to symbolize the premises?
F: Fewer than two contestants enter.
C: There will be a contest.
W: There will be a winner.
Step 1: The conclusion is: “There will be there will be a loser only if at
least two contestants enter and not all contestants perform equally well.”
Answer: No! If we use ‘F’ in addition to our earlier ‘T’, we won’t get a
valid argument form!
Step 2: We can symbolize this using the following atomic statements:
‘Fewer than two contestants enter’ is the negation of ‘At least two
contestants enter’. So we can symbolize it as ‘∼T ’.
L: There will be a loser.
T: At least two contestants enter.
E: All contestants perform equally well.
I ∴ L → (T & ∼E )
Step 4: now we use our atomic sentences to symbolize the premises.
Premise 1: Suppose fewer than two contestants enter; then there will be
no contest.
I ∼T → ∼C
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Example
Exercises
Premise 2: ‘No contest means no winner.’
I ∼C → ∼W
Exercise 2m: symbolize the following argument.
Premise 3: ‘Suppose all contestants perform equally well. Still no winner.’
I E → ∼W
Premise 4: ‘There won’t be a winner unless there’s a loser. And
conversely.’
2. If God exists then there is no evil in the world, unless God is unjust,
or not omnipotent, or not omniscient. But if God exists, then He is
none of these, and there is evil in the world. So we must conclude
that God does not exist.”
I (∼L → ∼W ) & (∼W → ∼L)
Step 5: The whole (valid) argument is then symbolized like this:
∼T → ∼C
∼C → ∼W
E → ∼W
(∼L → ∼W ) & (∼W → ∼L)
∴ L → (T & ∼E )
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1. If Cathy plays the piano in the morning, then Roger wakes up
cranky. Cathy plays piano in the morning unless she is distracted. So
if Roger does not wake up cranky, then Cathy must be distracted.
(This is called “the argument from evil.”)
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