Propositional Logic

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Propositional Logic

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Title: Propositional Logic

1
Propositional Logic

In categorical syllogism, we deal with terms that
are related by quantifiers and copula in four
different manners.
In propositional logic, we deal with propositions
as the basic units of meaning.
All complicated propositions or arguments are
constructed from propositions that are basic or
simple.

Simple/basic propositions are those that are the
simplest and contain within themselves no other
simple propositions.
Compound propositions are composed of a number of
simple propositions.
Imagine that when you think, write or speak, you
are actually constructing a model with the basic
units.

3
Simple/basic propositions

John is a man.
Mary is beautiful.
Paul stole the watch.

4
Compound propositions

It is not the case that John is clever.
If you give me the money, then I will be rich.
Either Mary is a liar or John misunderstands Mary.

5
Symbols

We use capitalized letters to represent
propositions.
So if the proposition John is clever is A and
the proposition Mary is happy is B, then the
compound proposition John is clever and Mary is
happy is AB.

6
Logical operators

We have five logical operators
1) negation (not, it is not the case that)
2) conjunction (and, but, also)
3) v disjunction (or, unless)
4) ? implication (if then, only if)
5) equivalence (if and only if)

In some books, or ? is used for and ? is
used for not ? is used for imply ? is used
for equivalent.

Logical operators are logical connectives that
join the simple propositions into a logically
meaningful compound proposition.
Example
It is not the case that E E
B and C B C
Either C or D C v D
If A then B A ? B
H if and only D H D

Note that A ? B is a conditional proposition
expressing the relation of material implication.
Generally, it may be read as our normal If
then. Exceptional cases will be encountered
later.
A is the antecedent and B is the consequent.
is called the material equivalence.

Only is placed in front of an expression.
Other operators are placed between expressions.

11
Main Operator

The main operator is that operator in a compound
proposition that governs the largest component(s)
in the proposition. It is also the last operator
to be dealt with in finding out the truth value
(T or F) of a compound proposition.
Compare to our use of , ?, ?, ?.
In 4 ? (3 2), ? is the main mathematical
operator.

The main operator of the following propositions
is
B
(A ? B)
((A F) (D E))

Main operator is .
K L
(E v F) (G v H)
((R ? T) v (S ? U)) ((W X) v (Y Z))

The main operator is v.
C v D
(F H) v (K L)
(S (T ? U)) v (X (Y Z))

The main operator is ?.
H ? J
(A v C) ? (D E)
(K v (S T)) ? (F v (M O))

The main operator is .
M T
(B v D) (A C)
(K v (F ? I)) (L (G v H))

17
Actual translation

It is sometimes difficult to read the operators
as normal English words.
Therefore, some cases must be specified with
precision.
Remember that our aim is to obtain accurate
meaning.
The meaning of a complicated proposition
completely depends on the information of its
composing units.

18
Negation

Negation is not a problematic symbol.

19
Conjunction

Conjunction means the obtaining of truth for both
conjuncts. Therefore, but in English is also
read as a conjunction.
John left early but Mary stayed means it is
true that John left early and Mary stayed.
(J M)

20
Disjunction

Disjunction should be read as the inclusive or.
That means both disjuncts can be true or either
one of the disjuncts is true. The only impossible
case is where both disjuncts are false.
Unless is also considered a disjunction.
Unless you work hard, you will fail is
equivalent to Either you work hard, or you will
fail.

21
Material implication

Material implication is similar to the English
If then.
The antecedent always follows If, provided
that, on condition that, etc.
The consequent always follows then, only if,
and implies that, etc.

22
Material implication

If John walks fast, then Mary eats early.
J ? M
John walks fast if Mary eats early.
M ? J
John walks fast only if Mary eats early.
J ? M

23
Material implication

John walks fast provided that Mary eats early.
M ? J
John walks fast on condition that Mary eats
early.
M ? J
Johns walking fast implies that Mary eats early.
J ? M

24
Material implication

Although (A B) is logically equivalent to (B
A) and (A v B) is also logically equivalent to (B
v A), (A ? B) is not logically equivalent to (B ?
A).
We must therefore distinguish between sufficient
condition and necessary condition.

25
Material implication

A necessary condition F is one that must occur in
order for the condition G to occur.
In other words, without F, G cannot occur.
For example, water is a necessary condition for
bathing.

26
Material implication

A sufficient condition F is one with which that G
can occur.
But note that there may be many other ways for G
to occur.
For example, to have a good A-Level result is a
sufficient condition to have a university place
but you may also have a university place by
having good GPA.

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Material implication

Remember that
Sufficient condition ? Necessary condition

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Material Equivalence

Also remember that K F means the truth
conditions for K and F must be identical.
That means K F is logically equivalent to (K ?
F) (F ? K).
You will become familiar with such
transformations in the next topic on natural
deduction.

29
Parentheses

Parentheses are sometimes required to avoid
ambiguity.
For example, A B v C may mean (A B) v C or A
(B v C).
Similar to the use of brackets in mathematics.

John cooks dinner and Dick reads novels, or Mary
drinks wine.
(J D) v M
John cooks dinner, and Dick reads novels or Mary
drinks wine.
J (D v M)

Either John cooks dinner and Dick reads novels or
Mary drinks wine.
(J D) v M
John cooks dinner and either Dick reads novels or
Mary drinks wine.
J (D v M)

John cooks dinner or both Dick reads novels and
Mary drinks wine.
J v (D M)
John cooks dinner or Dick and Mary drink wine.
J v (D M)

If John cooks dinner, then if Dick reads novels,
then Mary drinks wine.
J ? (D ? M)
If Johns cooking dinner implies that Dick reads
novels, then Mary drinks wine.
(J ? D) ? M
If either John cooks dinner and Dick reads novels
or Billy sleeps late, then Mary drinks wine.
((J D) v B) ? M

A single letter is a well-formed formula (WFF).
Any combination of letters by the legitimate uses
of the logical operators and parentheses are also
WFFs.
Therefore, v B ? C is not a WFF.

35
Truth Tables

The behaviors of the 5 logical operators are
completely determined by their respective truth
tables.
That is why we can determine the truth-values of
any compound propositions.

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Question

Each one of the 4 cards below has a letter on one
side and a number on the other. Tell me which
card(s) you definitely need to turn over, and
only that (those) card(s), in order to determine
whether the cards are following the rule to the
effect that if a card has vowel on one side, it
has an even number on the other side.

E
T
7
4
41
Conditional Statement

Please note that when P is F, whether Q is T or
F, P ? Q is T.
E.g. The proposition If Paul is bald, then he
is rich is true when Paul is not bald.
However, there are some conditional propositions,
we call counterfactuals, expressing lawful
relation between the antecedent and the
consequent.

42
Conditional Statement

E.g. If I threw a rock to the window, the
window would break.
The fact that I do not throw a rock to the window
does not make the above conditional true.
Otherwise, the fact that I do not throw a rock to
the window will also make the following
proposition true If I threw a rock to the
window, the rock will become a marshmallow and
rebound from the window.

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Applications

We use the five tables to compute the
truth-values of any compound propositions.
Steps
1) Assign letters to basic propositions (avoid
duplication).
2) Translate by joining letters with the logical
operators.
3) Insert parentheses as required.

4) Assign truth-values to individual letters.
5) Compute the truth-values of the next-smallest
compound proposition, and so on.
6) Compute the truth-value under the main
operator.

46
Example 1

(A v D) ? E
Given A is true and D, E false, then
(T v F) ? F
T ? F
F

47
Example 2

Given A, B, C true and D, E, F false
((D v F) (B v A)) ? (F ? C)
((F v F) (T v T)) ? (F ? T)
(( F ) ( T v F)) ? (F ? F)
(T ( T )) ? (T)
T ? F
F

48
Complete Truth Tables for Propositions

The previous examples work because the truth
values of individual letters are given in
advance. That corresponds to how normal people
understand a complicated saying by checking the
truth and falsity of the basic propositions
first.
Now we want to exhaust all the possible
combinations of truth-values in order to know the
properties of the compound propositions. This may
save us time to check the truth and falsity of
the individual propositions.

We have a way to know the number of rows of a
complete truth table for any given propositions.
It depends on the number of letters involved.
If there are 2 letters, there are 4 rows 3
letters, 8 rows and so on.
Therefore, for n Letters, there are 2n rows.

For example, we can draw the truth-table for (A v
B) ? B by constructing 4 rows.
Then we assign the truth values to individual
letters.
The rule is to alternate the truth-values for the
rightmost column, then alternate pairs for
truth-values for the next column and so on. In
this way, we need not worry about omission or
repetition.

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Then we put in the truth value of the conclusion.

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Then we compute the truth-value of the shortest
compound statement.

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Then we move on to do the same for v and finally
we compute the truth-values under the main
connective ?.

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A B (A v B) ? B
T T T T FT T T
T F T T TF F F
F T F F FT T T
F F F T TF F F

The previous slide shows the most convenient way
of drawing a complete truth table.
1) Draw a cross that partitions the space into
four areas.
2) Put the problem to be solved in the upper
right quadrant.
3) Extract the letters for the basic propositions
in any order, and list them out in the left upper
quadrant.
4) List out the truth-values of the basic
propositions line by line in the lower left
quadrant.
5) Compute the truth values of the logical
operators one by one in the right lower quadrant.

In the above example, we notice that the
truth-values of the compound proposition only
depend on the truth-values of B.
That means the compound proposition is a
needlessly complicated way of saying B, if we are
only interested in the truth and falsity of a
proposition in all possible circumstances.

As a summary, drawing the truth-tables can enable
us to see certain important features of a
compound proposition.
There are three important features.

62
Tautology

A compound proposition is a tautology or
logically true if it is true regardless of the
truth-values of its components.

63
Self-contradiction

A compound proposition is said to be
self-contradictory or logically false if it is
false regardless of the truth-values of its
components.

64
Contingency

A compound proposition is said to be contingent
if its truth-values depend on the truth-values of
its components.
Most of the things we learn are contingent
propositions.
That is why we need to check the information of
the simple things before we can determine the
information of the more complicated things.

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Comparing different propositions

With the above ideas in mind, we can compare
different propositions.
Two propositions are logically equivalent if they
have the same truth-values regardless of the
truth-values of the components.
Two propositions are logically contradictory if
they have opposite truth-values regardless of the
truth-values of their components.

Two propositions are consistent if there is at
least one line in the truth table on which both
of them are true.
Two propositions are inconsistent if there is no
line on which both of them are true.

68
Logically equivalent

K ? L L ? K
T T T F T T F T
T F F T F F F T
F T T F T T T F
F T F T F T T F

69
Logically contradictory

K ? L K L
T T T T F F T
T F F T T T F
F T T F F F T
F T F F F T F

70
Logically consistent

K V L K L
T T T T T T
T T F T F F
F T T F F T
F F F F F F

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Logically inconsistent

K L K L
T T T T F F
T
T F F T T T
F
F F T F F F
T
F T F F F T
F

Any pair of propositions can only be either
consistent or inconsistent.
If two propositions are logically contradictory,
they are also logically inconsistent.
But if two propositions are logically equivalent,
they may not be logically consistent (two
self-contradictions are both logically equivalent
and logically inconsistent).

73
Truth tables for arguments

1) Follow the previous rules.
2) Write out the symbolized argument with single
slash between premises and double slash between
the last premise and the conclusion.
3) Look for a line in the table in which the main
operators of all of the premises are true and the
main operator of the conclusion false. If there
is, the argument is invalid. Otherwise, the
argument is valid.

4) Any argument having mutually inconsistent
premises is valid regardless of what the
conclusion may be.
5) Any argument having a tautologous conclusion
is valid regardless of what its premises may be.

75
Example

If Bin Laden is responsible for the 911
holocaust, then bombing Afghanistan is a right
action.
Bin Laden is not responsible for the 911
holocaust.
Therefore, bombing Afghanistan is not a right
action.

Use letter L, B to symbolize the two
propositions.
L ? B
L
---------
B

L B L ? B / L // B
T T T T T F T F T
T F T F F F T T F
F T F T T T F F T
F F F T F T F T F

78
Indirect truth tables

Drawing a large table is troublesome.
We need a faster way to spot invalidity.
We do it by assuming that the argument is
invalid.
Then we work backward to obtain truth-values for
all the components.
If no contradiction is seen, then the argument is
indeed invalid.

79
Testing for Invalidity

A ? (B v C)
B
------------------
C ? A

A ? (B v C) / B // C ? A
Then assign false to the conclusion and truth to
the premises.
A ? (B v C) / B // C ? A
T T F
F F
T F T F T T T F T F F

The previous slide shows how each step is
obtained. But in practice we need not solve a
problem by showing each step with different
lines. The below is a one-line proof of the
arguments invalidity, the red T/F highlighting
the assumptions.
A ? (B v C) / B // C ? A
T F T F T T T F T F F

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Testing for consistency

Similar to before except that we have no
conclusion.
Instead we assume that all the propositions are
consistent by assigning T to the main operator of
each.
If there is a contradiction, then the
propositions are inconsistent, otherwise
consistent.

83
Example