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PHIL-2200
Formal Logic
(Diploma III Semester)
2009-2010
By
Dr.V.Maheswara Rao
Department of Information Technology
College of Technology, Salalah
Sultanate of Oman
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CONTENT
Chapter
Title
1
Introduction to logic,
Nature of Logic,
Sentences and
Arguments,Ambiguity
2
Validity, Logical truth,
consistency and sound
3
Sentential logic,
Translating into
Natural Language,
Formalization in
sentential logic
4
Truth table, Logical
Properties and
Relations
Topics
Introduction
Principles of logic
Nature of logic
Types of logic
Informal logic
Formal logic
Sentences
Arguments
Ambiguity
Lexical ambiguity
Referential ambiguity
Syntactical ambiguity
Validity
Deductive validity
Inductive validity
Logical notations
Truth values
Logical truth
Logical equivalence
Consistency
Sound
Formal systems of logic
Advantage of formal systems of logic
Language of Sentential Logic
Sentence letter
Sentential connectives
Open and closed brackets
Rules of formation for SL
Grammatical Categories
Translating into Natural Language
Connectives
Negation
Conjunction
Disjunction
Conditional
Biconditional
Formalization in Sentential Logic
Introduction
Negation
Conjunction
Disjunction
Conditional
Biconditional
Logical Status
Tautology
Inconsistency
Contingency
2
Pages
4
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Full Truth table method
Some informal comments on decidability:
Indirect method
The limitations of Sentential Logic
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CHAPTER-1
INTRODUCTION TO LOGIC
Introduction:
The term "logic" from the Greek word logos(reasoning ,discourse).The branch of
philosophy that deals with forms of reasoning and thinking, especially inference or
argument and scientific method. Logic may be defined as the study of the principles of
correct reasoning.
The principles of logic:
There are many principles of reasoning that are part of logic, but the main thing that we
study in logic are principles governing the validity of arguments - whether certain
conclusions follow from some given assumptions.
For example,

If it is noon, then it is day time.
It is noon.
Therefore, it is day time.

If it is raining, then there are clouds.
It is raining.
Therefore, there are clouds.

If you studied for the exam, you are not going to fail.
You studied for the exam.
Therefore, you are not going to fail.

If a > 10, then a > 3.
a > 10.
Therefore a > 3.
These arguments are obviously good arguments in the sense that their conclusions
follow from the assumptions. If the assumptions of the argument are true, the
conclusion of the argument must also be true.
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Nature of logic:
The concept of logical form is central to logic; it being held that the validity of an
argument is determined by its logical form, not by its content.
Types of logic:
There are mainly two types of logic which are formal logic and informal logic.
Informal logic:
Informal logic is the study of natural language arguments. The study of fallacies is an
especially important branch of informal logic. Most of the work in informal logic focuses
on the reasoning and argument (in the premise-conclusion sense) one finds in personal
exchange, advertising, political debate, legal argument, and the social commentary that
characterizes newspapers, television, the World Wide Web and other forms of mass
media.
Formal logic:
Formal logic is the study of inference with purely formal content, where that content is
made explicit. An inference possesses a purely formal content if it can be expressed as a
particular application of a wholly abstract rule, that is, a rule that is not about any
particular thing or property.
There are many reasons for studying formal logic. Formal logic helps us to identify
patterns of good reasoning and patterns of bad reasoning, so we know which to follow
and which to avoid. Formal logic helps us to improve critical thinking. Formal systems
of logic are also used by linguists to study natural languages. Computer scientists also
employ formal systems of logic in research relating to Aritificial Intelligence. Finally,
many philosophers also like to use formal logic when dealing with complicated
philosophical problems, in order to make their reasoning more explicit and precise.
Statements:
Logic deals with the logical properties of statements and how one statement is related
to another.
There are three main sentence types in English:


Declarative sentences are used for assertions, e.g. "He is here."

Interrogative sentences are used to ask questions, e.g. "Is he here?"
Imperative sentences are used for making requests or issuing commands, e.g.
"Come here!"
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For present purposes, we shall take a statement to be any declarative sentence. A
declarative sentence is a complete and grammatical sentence that makes a claim. So
here are some examples of statements in English:

Snow is white.

Sun rises in east.

Everyone is here.

Whatever will be, will be.
As you can see, statements can be true or false, and they can be simple or complex.
But they must be grammatical and complete sentences. So these are not statements:

The United Nations [ A proper name, but not a sentence ]

Bridge over troubled waters. [ Not a complete sentence ]

Come here right now! [ A command that is not a complete
sentence making a claim ]

will you be available on Tuesday or Wednesday? [ A question ]

JGS&*^@#JHGKJAS*&^*!@GJHGAA*&S [ Ungrammatical ]
Arguments:
An inference or argument is any list of statements divided into premises
and conclusions. When analyzing an argument, the first thing to do is to
separate the premises from the conclusion. Word like these are clue to what
the argument is supposed to be, especially if – in the argument as given- the
conclusion comes at the beginning or in the middle of the argument.
Premises indicators: since, because, given that
Conclusion indicators: therefore, hence, thus, then, so
To be perfectly general, we can define an argument as a series of sentences.
The sentences at the beginning of the series are premises. The final sentence in
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the series is conclusion. If the premises are true and argument is a good one,
then you have a reason to accept the conclusion.
For example:

Either this man’s dead or my watch has stopped.

This man is not dead.

Therefore, my watch has stopped.
Here statements 1 and 2 are premises, and 3 is conclusion.
Ambiguity (Two ways that can go wrong):
To communicate meaning precisely, it is important to avoid ambiguity. An expression
is said to be ambiguous when it has more than one interpretation or meaning. There
are three main sources of ambiguity:
1 Lexical ambiguity:
If a word has more than one meaning in a language, then it is lexically ambiguous.

For example, the word "deep" can mean profundity ("What you have said is
very deep."), or it can be used to describe physical depth ("This hole is very
deep").
 Similarly for words like "young" (inexperienced or young of age), "bank" (river
bank or financial institution), etc..
2 Referential ambiguity:
If it is unclear what a referring expression is referring to, then the expression is
referentially ambiguous. For example, a pronoun is a referring expression such as "it",
"he", "they", etc. You might point to a famous basketball player and say "he is rich",
and here "he" refers to the player. But if it is not clear who you are pointing to, then
we might not know to whom the pronoun refers, and so might not be able to determine
whether you are saying something true. Referential ambiguity can also arise if you are
talking about a group using an expression such as "every".
3 Syntactic ambiguity:
Syntactic ambiguity applies to a complex expression when it has more than one
meaning because there is more than one way to understand its grammatical structure.
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This can happen even when the individual words of the expression are not lexically
ambiguous. Here are some examples :

We shall be discussing violence on TV.
This statement is syntactically ambiguous because it has two meanings, depending on
how we understand its grammatical structure :


We shall be discussing violence on TV.
We shall be discussing violence on TV.
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CHAPTER 2
VALIDITY
Validity:
An argument is formally valid if its form is one such that for each
interpretation under which the premises are all true also the conclusion is
true.There are two types of validities
Deductive validity:
An argument is deductive validity if and only if it is impossible for the premises to be
true or false and the conclusion false.
The crucial thing about a valid argument is that it is impossible for premises to be true
at same time that the conclusion is false.
For example,
 Oranges are either fruits or musical instrument.

Oranges are not fruits

Oranges are musical instruments.
The conclusion of this argument is ridiculous. Nevertheless, it follows validly from the
premises. This is a valid argument. If both premises were true, then conclusion would
necessarily be true.
This shows that deductively valid argument does not need to have true premises or a
true conclusion. Conversely, having true premises and a true conclusion is not enough
to make an argument valid.
For example,
 London is in England
 Salalah is in oman
 Dubai is in UAE
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The premises and conclusion of this argument are all true. This is a terrible argument.
however, because the premises have nothing to do with the conclusion. Image what
would happen if Salalah declared independence from the rest of the Oman. Then the
conclusion would be false, even though the premises would both still be true. Thus it is
logically possible for the premises of this argument to be true and conclusion false.
The argument is invalid.
The important thing to remember is that valid is not about the actual true or falsity of
the sentences in the argument. Instead, it is about the form of the argument: The truth
of the premises is incompatible with the falsity of the conclusion.
Inductive validity:
There can be good arguments which nevertheless fail to be deductively valid.
Consider this one:
 In February 2003, rained in Salalah
 In February 2004, rained in Salalah
 In February 2005, rained in Salalah
 In February 2006, rained in Salalah
Therefore, every year February rains in Salalah.
This is an inductive argument, because it generalizes from many cases to conclusion
about all cases.
Certainly, the argument could be made stronger by adding additional premises: In
February 2007, it rained in Salalah. In February 2008,…and so on. Regardless how
many premises we add, however, the argument will still not be deductively valid. It is
possible, although unlike, that it will fail to rain next February in Salalah. More over
we know that the weather can be fickle. No amount of evidence should convince us
that it rains in Salalah every February. Who is to say that some year will not be
freakish year in which there is no rain in February in Salalah; even a single counter
example is enough to make the conclusion of the argument false.
Inductive arguments, even good arguments, are not deductively valid.
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LOGICAL TRUTH, CONSISTENCY &SOUND
logical notations:
In addition to deductive valid , we will be interested in some other logical concepts.
Truth Values:
True or false said to be truth values of sentence. We define sentences as things that
could be true or false. We could have said instead that sentences are thing that can
have truth values.
Logical truth:
In considering arguments formally, we care about what would be true if the premises
were true. Generally, we are not concerned with actual truth values of any particular
sentences- whether they are actually true or false. Yet there are some sentences that
must be true, just as a matter of logic.
Consider the sentences:
1. It is raining
2. Either it is raining or it is not
3. It is both raining or not raining
In order to know if sentence 1 is true, we would need to look outside or check the
weather channel. Logically speaking, it might be either true or false. Sentences like
this are called contingent sentences.
Sentence 2 is different. We do not need to look outside to know that it is true,
Regardless of what the weather is like, it is either raining or not. This sentence is
logically true; it is true only as a matter of logic, regardless of what the word is
actually like. A logically true sentence is called a TAUTOLOGY.
We do not need to check the weather to know about sentence 3, either. It must be
false, simply as a matter of logic. It might be raining here and not raining cross town, it
might be raining now but stop raining even as we need this, but it is impossible for it to
be both raining and not raining here at this moment.
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The third sentence is logically false; it is false regardless of what the world is like. A
logically false sentence is called a contradiction.
To be precise, we can define a contingent sentence as a sentence that neither a
tautology nor a contradiction.
Logical equivalence:
We can also ask bout the logical relation between two sentences. For example,
 Suhail went to store after the washed dishes
 Suhail washed the dishes before the went to the store
These sentences are both contingent, since Suhail might not have gone to store or
washed dishes at all .Yet they must have the same truth value. If either of the sentences
is true, then they both are; if either of the sentence is false, then they both are. When
two sentences necessarily have the same truth value, we say that they are logically
equivalent.
Consistency:
Consider these two sentences
 My only brother taller than I am
 My only brother shorter than I am
Logic alone can not tell us which, if either, of these sentences is true. Yet we can say
that if the first sentence is true, then second sentence must be false. And if second
sentence is true, then first sentence must be false. It can not be case that both of these
sentences are true.
If a set of sentences could not all be true at same time, like above two sentences, they
are said to be inconsistence. Otherwise they are consistent.
Soundness:
Validity is all about the logical connection between the premises and the
conclusion. When we are told that an argument is valid, this is not enough to tell us
anything about the actual truth or falsity of the premises or the conclusion. All we
know is that there is a logical connection between them, that the premises entail
the conclusion.
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If an argument is valid, and all the premises are true, then it is called a sound
argument. So given that a sound argument is valid and has true premises, its
conclusion must also be true. So if you have determined that an argument is indeed
sound, you can certainly accept the conclusion.
An argument that is not sound is an unsound argument. If an argument is unsound,
it might be that it is invalid, or maybe it has at least one false premise, or both.
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CHAPTER 3
SENTENTIAL LOGIC
Formal systems of logic:
Sentential logic is a formal system of logic. A system of logic is a set of rules that
tell us how to make use of special symbols to construct sentences and do proofs.
To define a particular system of logic, we need to specify:
1. The formal language of the system
2. The semantic rules for the formal language
3. The rules of proof for the language
A formal language in a system of logic is a language with precisely specified rules
that tell us how to construct grammatical sentences. Such rules are called syntactic
rules. They are equivalent to the rules of grammar you find in English or
Cantonese.
The semantic rules are rules for interpreting the sentences in the language. They
tell us what the sentences mean and the conditions under which the sentences are
true or false.
The rules of proof are rules that specify how logical proofs are to be constucted.
They tell us what conclusions can be derived given certain initial assumptions.
Advantages of formal systems of logic:
There are many advantages of formal systems of logic:

Systems of logic can be used to formalize arguments in natural languages.
By formalization we refer to the process of translating arguments or
sentences in natural languages into the notations of formal logic. The
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reason for carrying out formalization is that very often they can help us
understand the logical structure of arguments better, by identifying patterns
of valid arguments. By formalizing an argument we can use the rules of
proof to check whether the argument can indeed be proved to be valid.

Because the rules of formal systems of logic are defined very clearly, we can
program them into a computer and get a computer to construct and evaluate
proofs quickly and automatically. This is particularly important in areas
such as Artificial Intelligence, where many researchers teach computers to
use formal logic in reasoning.

Linguists are scientists who study natural languages. Many linguists also
study formal languages and use them to compare and contrast with natural
languages.

Many philosophers are also interested in formal systems of logic because
natural languages are sometimes not precise enough to express certain ideas
clearly. So sometimes they turn to formal systems of logic instead.

Formal systems of logic are also interesting in their own right. Logicians and
mathematicians are interested in finding out what they can or cannot prove,
and also their many other logical properties. Formal systems of logic also
play an important role in understanding the foundations of set theory and
mathematics.
The language of Sentential logic (SL):
To define the language of SL, we need to specify the symbols or the vocabulary of
SL. These are the basic building blocks out of which more complicated
expressions are to be constructed. There are three kinds of symbols inSL :
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1. Sentence letters:
A, B, C, etc. These capital letters are used to translate sentences. If
we run out of sentence letters we can always add subscripts to them
to make new ones, e.g. A1 , B274, etc..
2. Sentential connectives:
~ (tilde, or the negation sign)
& (ampersand, or the conjunction sign)
V (the wedge, or the disjunction sign)
→ (the arrow)
↔ (the double-arrow)
3. Open and close brackets: ( )
The set of sentence letters, connectives, and brackets constitutes the set of symbols
of SL. An expression of SL is simply any string of one or more symbols of SL:
1. ABCDF&&&&(())))→ABCB12356A
2. P
3. (P&Q)
4. ~~(P&Q))
Now we come to syntax, the rules that tell us which of the expressions of SL are
grammatical, and which are not. A grammatical expression is called a well-formed
formula (WFF). A WFF of SL is any expression of SL that can be constructed
according to these rules of formation :
4 Rules of formation for SL:
1. All sentence letters are WFFs.
2. If P is a WFF, then ~P is a WFF.
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3. If P and Q are WFFs, then (P&Q), (PVQ), (P→Q), (P↔Q) are
also WFFs.
4. Nothing else is a WFF.
Grammatical categories
Here are some useful terms for talking about WFFs and their parts. If P and Q are
WFFs, then :

(P&Q) is a conjunction where P and Q are the first and second
conjunct respectively.

(PVQ) is a disjunction where P and Q are the two disjuncts.

(P→Q) is a conditional sentence where P is the antecedent and
Q the consequent.

(P↔Q) is a biconditional sentence.

~P is the negation of P.
So for example:

"(P&Q)" is a conjunction.

"((P&Q) V (R↔Q))" is a disjunction.

"~ (P→ (QVS))" is the negation of "(P→ (QVS))".

"~ (P&Q)" is the antecedent of "(~ (P&Q) → ((P&S) ↔Q))".
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TRANSLATING INTO NATURAL LANGUAGE
Translating in to natural language connectives:
The sentential connectives can be translated into their natural language
counterparts as follows:
~
It is not the case that
→
if ... then ___
↔
if and only if
&
and
V
or
But in fact many other locutions can be translated using these five sentential
connectives:
1 Negation:
Suppose “P” translates the
sentence “Water is liquid”.
Then “~P” can be used to

Water is not liquid.

It is not the case that water is liquid.

It is false that water is liquid.
translate these sentences :
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2 Conjunction:
“(P&Q)” can be used to

P and Q.
translate the following :

P but Q.

Although P, Q.

P, also Q.

P as well as Q.
“(P Q)” can be used to

P or Q.
translate the following :

Either P or Q.

P unless Q.

Unless Q, P.
“(P→Q)” can be used to

If P then Q.
translate the following :

P only if Q.

Q if P.

Whenever P, Q.

Q provided that P.

P is sufficient for Q.

Q is necessary for P.
3. Disjunction:
4 Conditional:
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5 Biconditional:
“(P↔Q)” can be used to

P if and only if Q.
translate the following :

P iff (if and only if) Q.

P when and only when Q.

P is equivalent to Q.

P is both necessary and sufficient for Q.
FORMALIZATION IN SENTENTIAL LOGIC
Formalization in SL:
Consider this particular argument:
[Premise 1] The pollution index is high.
[Premise 2] If the pollution index is high, we should stay indoors.
[Conclusion] We should stay indoors.
This argument is of course valid. To use the methods of SL to show that it is
indeed valid, we need to translate it from English into the language of SL. This
process of translation is called formalization.
First of all we need to find sentence letters to translate the different parts of the
argument. Let us use the following translation scheme. A translation scheme in SL
is simply a pairing of sentence letters of SL with statements in natural language. In
carrying out formalization you should always write down the translation scheme
first.
Translation scheme :
P : The pollution index is high.
Q : We should stay indoors.
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Remember that “→” is used to translate “if ... then ___”. So using the above
translation scheme we can formalize the argument as follows :
Premise #1 : P
Premise #2 : (P→Q)
Conclusion : Q
In SL we can rewrite this argument as a one line sequent, with the premises
separated by comma :
P, (P→Q)
Q
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CHAPTER 4
TRUTH TABLES
Introdcution: Truth tables provide the systematic method for determining the
validity of sequents or arguments in SL.
What a truth-table does is to show how the truth-value of a complex argument
depends on the truth-values of its component statements. So what are truth-values?
In SL there are only two truth-values: T and F, which stands for truth and falsity.
To say that a statement has truth-value T is just to say that it is true. To say that its
truth-value is F is to say that it is false. Notice that in SL we assume the principle
of bivalence: a WFF either has truth-value T or F. Some systems of logic (e.g.
fuzzy logic) reject the principle of bivalence. Notice also that some logic or
engineering textbooks use "1" and "0" in place of "T" and "F".
Let us now look at the truth-table of each of the sentential connectives.
Negation:
P ~P
Consider the statement "Water is liquid." Let us use the sentenceletter "P" to translate this statement. This just means we are now
T F
F T
taking the symbol "P" to have the same meaning as "Water is
liquid." But suppose you disagree with the statement. Then you
might express your disagreement by saying things like :

Water is not liquid.

It is not true that water is liquid.

It is not the case that water is liquid.
In sentential logic, these three different sentences are all translated
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as "~P". Obviously, "P" and "~P" have opposite truth-values - if
one is true then the other one must be false, and vice versa.
Conjunction:
P Q (P&Q)
Now consider these two statements : "water is liquid" and
"water quenches thirst". Each of them can be either true or false
T T
T
T F
F
F T
F
1. Water is liquid. Water quenches thirst.
F F
F
2. Water is liquid. But water does not quench thirst.
independently of the other, so in terms of their truth-values
there are four different possibilities :
3. Water is not liquid. But water quenches thirst.
4. Water is not liquid. Water also does not quench thirst.
The first row of truth-values tells us that when the first and
second conjuncts are true, the whole WFF is true. The other
three rows tell us that the conjunction is false in all other
situations.
Disjunction:
P Q (PVQ)
The disjunction symbol "V" is usually used to translate "or".
The truth-table on the left tells us that a disjunction is false
T T
T
T F
T
when both disjuncts are false. Otherwise it is always true.
Notice that the first row tells us that the disjunction is true when
F T
T
both disjuncts are true or when any one of the disjunts are true.
In other words, if "(PVQ)" is used to translate "Either I want
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F F
F
tea, or I want coffee.", and if I get both , then the whole
complex statement is still true.
Conditional:
P Q (P→Q)
The arrow sign is often translated as "If ... then Consider the
conditional statement such as "If I have lots of money then I
T T
T
T F
F
F T
T
F F
T
shall be happy." This conditional statement will be false only
when you have a lot of money but still you are not happy. In
other words, when the antecedent is true and the consequent is
false, the whole conditional statement is false. This is exactly
what the second row of the truth-table says. Just remember that
the conditional is true in all other situations.
Biconditional:
P Q (P↔Q)
The symbol "↔" is used to translate "if and only if". Consider
the biconditional statement "This is water if and only if this is
T T
T
T F
F
F T
F
F F
T
H2O." The biconditional statement will be true if either both the
statements in it are true or both are false. If any one of the
statement is false then the whole statement will be false. This
sense of "if and only if" (or "iff") is captured in the truth-table
on the left.
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LOGICAL PROPERTIES AND RELATIONS
Logical status:
Armed with truth-tables we can now use them to classify WFFs in SL according to
their different logical status :
Tautology:
A tautology is any WFF in SL that is true under all assignments of truth-values to
its sentence letters, e.g.
P Q ((P&Q)→(PVQ))
P (PV~P) P (P→P) P (P↔P) T T
T
T
T
T
T
T
T
T F
T
F
T
F
T
F
T
F T
T
F F
T
Inconsistency:
An inconsistent WFF is any WFF in SL that is false under all assignments of truthvalues to its sentences letters, e.g.
P Q ((P&~P)V(Q&~Q))
P (P&~P) P (P↔~P) T T
F
T
F
T
F
T F
F
F
F
F
F
F T
F
F F
F
If a WFF is not inconsistent, then it is consistent.
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Contingency:
A contingent WFF is any WFF in SL that is not inconsistent and not a tautology. In
other words, there is at least one assignment under which it is true, and at least
another assignment under which it is false. All single sentence letters are of course
contingent. These WFFs are also contingent : (P&Q), (PVQ), ~(P→~Q).
The full truth table method:
In this tutorial we study how to make use of full truth-table method to check the
validity of a sequent in SL. Consider this valid sequent:
P, (P→Q)
Q
To prove that it is valid, we draw a table where the top row contains all the
different sentence letters in the argument, followed by the premises, and then the
conclusion :
Then using the same method as in drawing complex truth-tables, we list all the
possible assignments of truth-values to the sentence letters on the left. In our
particular example, since there are only two sentence letters, there should be 4
assignments:
P Q P (P→Q) Q
T T
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T F
F T
F F
The next step is to draw the truth-table for each of the premise and also the
conclusion. First, the first premise :
P Q P (P→Q) Q
T T T
T F T
F T F
F F F
Then the second premise:
P Q P (P→Q) Q
T T T
T
T F T
F
F T F
T
F F F
T
Finally for the conclusion:
P Q P (P→Q) Q
T T T
T
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T F T
F
F
F T F
T
T
F F F
T
F
In the completed truth-table, the first two cells in each row give us the assignment
of truth-values, and the next three cells tell us the truth-values of the premises and
the conclusion under each of the assignment. If an argument is valid, then every
assignment where the premises are all true is also an assignment where the
conclusion is true. It so happens that there is only one assignment (the first row)
where both premises are true. We can see from the last cell of the row that the
conclusion is also true under such an assignment. So this argument has been shown
to be valid.
More examples:
Remember that “(P→Q), ~P, therefore ~Q” is invalid. Verify for yourself that this
is the truth-table for the sequent :
P Q (P→Q) ~P ~Q
T T
T
F
F
T F
F
F
T
F T
T
T
F
F F
T
T
T
Check for yourself that the truth-table is done correctly. A sequent is invalid if
there is at least one assignment where the premises are true and the conclusion is
false. The third row of the table shows that this is indeed the case. When “P” is
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false and “Q” is true, the two premises are true but the conclusion is false. So the
argument is not valid. We call such an assignment an invalidating assignment (a
counterexample) for the sequent.
So far we have only looked at sequents where each WFF contains at most one
sentential connective. Now let us look at sequents with more complex WFFs, such
as this one here :
(~P∨ Q), ~(Q→P)
(Q↔~P)
Again we draw a truth-table for the premises and the conclusion :
P Q (~P∨ Q) ~(Q→P) (Q↔~P)
T T FT T T
FTTT
T F FT
T F
FT F F
FFTT
F T FT
F T TF T T
TTFF
T T TF
F F
FFTF
F F TF
TF T F
To help us calculate the truth-values of the WFFs under each assignment, we use
the full truth-table method to write down the truth-values of the sentence letters
first, and then work out the truth-values of the whole WFFs step by step. The truthvalues of the complete WFFs under each assignment is written beneath the main
operator of the WFFs. As you can see, there is no assignment where the premises
are true and the conclusion is false. So the sequent is valid.
Some informal comments on decidability:
One thing you might notice about the full truth-table method is that it can help us
determine the validity of any sequent in SL. A program can be written that, given
any finite sequent in SL as input, after a finite number of processing steps,
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produces an output "Yes" if the sequent is valid, or an output "No" if it is not. Of
course, the computer would need to have a lot of memory if the sequent is a long
one, but in principle it can be done. This is roughly what logicians mean when they
say that validity in SL is decidable. Intuitively, it is a matter of whether there is an
algorithm or computer program that can come up with a proof of either the validity
or the invalidity of a sequent.
The indirect method in SL:
The full-truth-table method can be used to determine whether any given sequent in
SL is valid or not. But as the number of sentence letters in the sequent increases,
the number of rows we have to fill in increases exponentially. Here we introduce a
more efficient method for determining validity.
Here is the reasoning behind the so-called indirect method (or the reduction
method): the full truth-table method shows that an argument is valid by examining
all possible assignments of truth-values. However, to show that an argument is not
valid, all we need to do is to find one assignment where all the premises are true
and the conclusion is false. So in this method we first assume that the argument is
invalid, and try to find one invalidating assignment. If we succeed, then the
argument has been shown to be invalid. Otherwise it will lead to inconsistency,
and we can conclude that the argument is valid after all.
The limitations of Sentential Logic:
An example:
Consider this argument :
All students are intelligent.
Salim is a student.
So Salim is intelligent.
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This argument is obviously valid, but its validity cannot be demonstrated in SL.
This is because the best we can do in SL is to translate the premises and the
conclusion using distinct sentence letters, as in :
P, Q
R.
Such a sequent is of course not valid in SL. To demonstrate the validity of the first
argument, we need to analyse the internal structure of the premises and the
conclusion in more details. A more powerful formal system, predicate logic,
allows us to do that. Like SL, PL is also a formal system of logic. It is at least as
powerful as SL in that PL includes all the WFFs of SL, and any logical truth of SL
can also be proved in PL. But in addition, PL can be used to express certain logical
connections between statements that SL cannot, and carry out more complicated
proofs.
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