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Propositional Logic
In common sense, Logic is the Science of Methods of Thinking. But precisely, it isregarded as a systematic study of the Forms of Valid Inferences within a givenLogical System. An Inference is a process in Reasoning which turns someStatements called Premises into another Statement called Conclusion. This means,a Valid Inference is an Inference in which Premises, to certain extent, “support”the Conclusion. On the other hand, a Logical System or simply a Logic is aFormalized Inference System or Formal Axiomatic System called Formal Systemtogether with a Semantics consists of a Mathematical Structure calledInterpretation Model, and an Interpretation Mapping. Different branches ofModern Logic study different Logical Systems together with the Forms of ValidInference within such Systems. In simplest terms, once we defined a LogicalSystem, we are able to talk about Logic.
Definition 1 (Metalanguage and Object Language)A Metalanguage is a Language, which can be Natural or Formal, used forstudying other Language. The Language being studied by a Metalanguage is anObject Language.
Example 1In this article, the Metalanguage is English plus Mathematics, while the ObjectLanguage is the Language of Propositional Logic.
Definition 2 (Metalogic or Metamathematics and Logic)Logic can be defined as the systematic study of the Forms of Valid Inferenceswithin a Logical System.Metalogic or Metamathematics can be defined as the systematic study ofproperties of Formal Systems. Metalogic or Metamathematics emphasizes ondistinguishing Proofs outside any Formal System from Proofs inside a FormalSystem. Intuitively, Metalogic or Metamathematics can be regarded as theMathematics of Logic and Mathematics.
Definition 3 (Syntax and Semantics)A Syntax is either one of the following or the both:1. A Grammatical or Prescribed or Conventional or Defined Form or Pattern or
Structure of an Abstract Object (Concept), without regard to any of itsMeanings or Truth Value;
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2. A Grammar or a set of Rules or Conventions or Definitions for Constructingor Formulating or Transforming a Class of Abstract Objects (Concept),without specifying any of their Meanings or Truth Values.
Therefore, a Syntactic Object is either one of the following or the both:1. A Pure Grammatical or Prescribed or Conventional or Defined Form or
Pattern or Structure of an Abstract Object (Concept);
2. A Grammar or a set of Rules or Conventions or Definitions for Constructingor Formulating or Constructing a Class of Abstract Objects (Concept);
A Semantics or an Interpretation, in contrast, is a Meaning or a Truth Value of anAbstract Object (Concept). And a Semantic Object is merely a Meaning or a TruthValue of an Abstract Object (Concept).Note that a Syntax of an Abstract Object (Concept) is different from an Identifieror a Signifier of that Abstract Object (Concept) in that the former must beconsistent with a Semantics of that Abstract Object (Concept) and must beLogically possible while the later can be any arbitrary symbol to refer to thatAbstract Object (Concept).
Example 2The Syntax of a Sentence is its Grammatical Form or Grammatical Structure or aGrammar for forming a Sentence;
The Syntax of the Statement “All Functions are Relations” is the form of theStatement represented by Statement Variables, for example, it is represented by
QP , where P is the Statement “x is a Function” and Q is the Statement “x is
a Relation”, in Propositional Logic, while it is represented by
n(x)Relatio)Function(xx , where )Function(x “x is a Function” and
(x)Relation “x is a Relation” are Unary Predicates defined on the Universal
Class of all Sets, in First Order Predicate Logic;
The Syntax of a Unary Function f is f(x)Y : yXx, yf (as a Relation)
or Yf : X defined by f(x)yx (as a Correspondence or its Intension or a
Rule) or fX, Y, Gf (as a Mathematical Structure);
The Syntax of a Group G is X, G , where X is a Class of Objects and is
a Binary Operation defined on X ;
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The Syntax of a Formal System F is m , r, , r r, A, GS, G, LF 21 , m .
All these Syntax are Syntactical Objects while Sentence, Statement, Functionand Group are Abstract Objects which possess such Syntactical Objects.
The Semantics of a Sentence is any of its Meaning or its Truth Value;
The Semantics of the Sentence “All Functions are Relations” is its usual Meaningor its Truth Value: True;
The Interpretation of a Unary Function f is its Graph
f(x)Y : yXx, yfG ;
The Semantics of a Group G is its Domain;
The Semantics of a Formal System F consists of: An Interpretation Model which isa Mathematical Structure and an Interpretation Mapping which assigns
Meanings and Truth Values to all Well Formed Formulas in GL .
Definition 4 (Correspondence Theory of Truth)
According to Correspondence Theory of Truth, something is defined to be True,denoted by T or 1, if and only if it is consistent with the Reality; It is defined to beFalse, denoted by F or 0, if and only if it is NOT True. T or 1 and F or 0 arecalled Truth Values.
Definition 5 (Concept)Something named by “C” is defined as a Concept or Term or Sign if and only if it
is a Mathematical Structure , , PP, I, , EEC 2121 , where:
1. , , , , i, iE i 321 are Domains, each of which is a Set or Class of
Objects referred by C and is called Extension or Referent of C. In particular,
1E is called the Entity Class which contains all the Instances of C, and one of
the Extension is the Boolean Domain F, T ;
2. I is the Structure Set called Comprehension of C which is a set of Predicates
, , , j, j, nF, TEE : P j, nj, j 211 , where each jP is called
an Intension or Signified and it represents either:
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a. a Property F, T : EPj 1 possessed by elements of 1E , or
b. an Attribute , nEE : EP j, nj, j 11 possessed by elements
of 1E which corresponds to , nF, TEEE : P j, nj, j 11 , or
c. a Relation , nEEP j, nj, j 1 between elements of
j, nj, , E, E 1 which corresponds to the Predicate
, nF, TEE : P j, nj, j 1 ;
3. “C ” is called the Name or Identifier or Signifier of C and it generally takesthe form of a String of symbols or a Sequence of sounds.
Definition 6 (Statement and Sentence)A Proposition or Statement is defined as a Declarative Sentence which is eitherTrue or False.A Sentence is a formulation that represents a Statement.A Statement can be represented by many different Sentences.
Definition 7 (Negation of a Statement)
In Propositional Logic, the Negation of a Statement p is defined as a Statement
with Syntax: “Not p” or simply “ p ”. p has the opposite Truth Value as p.
Definition 8 (Conjunction of Statements)In Propositional Logic, a Conjunction is defined as a Statement with Syntax: “p
and q” or simply “ qp ”, where p and q are any two Statements.
The only case qp is True is when both p and q are True.
Definition 9 (Disjunction of Statements)In Propositional Logic, a Disjunction is defined as a Statement with Syntax: “p or
q” or simply “ qp ”, where p and q are any two Statements.
The only case qp is False is when both p and q are False.
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Definition 10 (Conditional Statement and Logical Consequence)In Propositional Logic, a Conditional Statement or Material Conditional Statement
is defined as a Statement with Syntax: “If p then q” or simply “ qp ”, where p
and q are any two Statements such that p is called Antecedent or Hypothesis orPremise and q is called Consequent or Conclusion. q is also called the MaterialConsequence of p.
The only case qp is False is when p is True and q is False. In any Natural
Language, we cannot refute the Truth of a Conditional Statement qp simply
when p is False since in this case the False Premise p has nothing works with the
whole Conditional Statement qp and what qp merely asserts is: if the
Premise p is True, then the Conclusion q will also be True.
For any qp , p is called the Sufficient Condition of q and q is called the
Necessary Condition of p.For any two Statements p and q, we define: q is the Logical Consequence of p, or
p Implies or Infers q, or q is Entailed by p, denoted by qp , to mean:
1. There is NO Interpretation in which p is True and q is False;
2. qp is a True Statement under any Interpretation;
3. The Class of Interpretations under which p is True is a SubClass of the Classof Interpretations under which q is True.
In general, for any Statements , q, p, , pp n21 , n , q is the Logical
Consequence of n, p, , pp 21 , or n, p, , pp 21 together Implies or Infers q, or q is
Entailed by n, p, , pp 21 , denoted by qppp n 21 , is defined as:
1. There is NO Interpretation in which n, p, , pp 21 are all True and q is False;
2. qppp n 21 is a True Statement under any Interpretation (The
Proof of this is provided in Metalogical and Semantic Theorem 1);
3. The Class of Interpretations under which n, p, , pp 21 are all True is a
SubClass of the Class of Interpretations under which q is True.Logical Consequence can be regarded as an Order Relation in Logic but it is aSemantic concept.
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Definition 11 (Types of Conditional Statement)
For any Conditional Statement qp , where p and q are any two Statements:
1. q p is called the Inverse of qp ;
2. pq is called the Converse of qp ;
3. p q is called the Contrapositive of qp .
Definition 12 (Biconditional Statement and Logical Equivalence)In Propositional Logic, a Biconditional Statement or Material BiconditionalStatement or Material Equivalence is defined as a Statement with Syntax: “p if
and only if q” or “p is Materially Equivalent to q” or “p iff q” simply “ qp ”,
where p and q are any two Statements.
The notation of “ qp ” suggests that qp is defined as qp and pq ,
so qp is True when both qp and pq are True, or equivalently, either
when both p and q are True or when both p and q are False.
For any qp , p and q is called the Necessary and Sufficient Condition of each
other.For any two Statements p and q, we define: p is Logically Equivalent to q, or p
and q are Logical Equivalence, denoted by qp , to mean:
1. There is NO Interpretation in which p and q have different Truth Values;
2. qp is a True Statement under any Interpretation (The Proof of this is
provided in Metalogical and Semantic Theorem 2);3. The Class of Interpretations under which p is True is equal to the Class of
Interpretations under which q is True.Logical Equivalence can be regarded as an Equality Relation in Logic but it is aSemantic concept.
Metalogical and Semantic Theorem 1 (Logical Consequence and Conditional)
Let , q, p, , pp n21 , n are any Statements. Then q is the Logical Consequence
of n, p, , pp 21 if and only if qppp n 21 is a True Statement under
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any Interpretation. Briefly, Logical Consequence is Logically Equivalent to TrueConditional.
Proof
Suppose qppp n 21 is a True Statement under any Interpretation.
Then by the Definition of Conditional Statement, this means when the Premise
nppp 21 is True, q is also True and CANNOT be False. Now by the
Definition of Conjunction, nppp 21 is True only when n, p, , pp 21 are
all True. Therefore, q is True when n, p, , pp 21 are all True, or equivalently,
there is NO Interpretation in which n, p, , pp 21 are all True and q is False. So q
is the Logical Consequence of n, p, , pp 21 .
Conversely, suppose q is the Logical Consequence of n, p, , pp 21 and consider
the Conditional Statement qppp n 21 . Since when n, p, , pp 21 are
all True, q is also True, so qppp n 21 is True in this case. On the other
hand, by the Definition of Conjunction, when at least one of , n, , ipi 1 is
False, then nppp 21 is False, but qppp n 21 is still True.
Therefore, qppp n 21 is always True in any case.
Metalogical and Semantic Theorem 2 (Logical Equivalence and Biconditional)
Let p, q are any Statements. Then p and q is Logical Equivalence or p is Logically
Equivalent to q if and only if qp is a True Statement under any Interpretation.
Briefly, Logical Equivalence is Logically Equivalent to True Biconditional.
Proof
Suppose qp is a True Statement under any Interpretation. Then by the
Definition of Biconditional Statement, this means p and q have the same TruthValue under any Interpretation. Therefore, there is NO Interpretation in which pand q have different Truth Values. So p is Logically Equivalent to q.
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Conversely, suppose p is Logically Equivalent to q and consider the Biconditional
Statement qp . Since under any Interpretation, either p and q are both True
or p and q are both False, so qp is True under any Interpretation.
Definition 13 (Logical Connectives, Atomic and Compound Statements) , , , , are called Logical Connectives. A Statement which contains no
Logical Connective is called an Atomic Statement. Logical Connectives can beapplied to Atomic Statements to construct more complex Statements, suchStatements are called Compound Statements. Similarly, Logical Connectives canbe applied to Compound Statements to construct even more complex CompoundStatements.
Example 3
Any p, q represent Atomic Statements and qq, pq, pq, p p, p are all
Compound Statements, while srq p is a Complex Compound
Statement.
Definition 14 (Classification of Statements)A Statement is defined as:1. A Logical Truth or Tautology if and only if it is True in all possible
Interpretations.a) In general Logical Truth can be simply denoted by the Truth Value T;
b) A Conditional Statement , nqppp n21 where q is True
when n, p, , pp 21 are all True is always True. In this case, q is a Logical
Consequence of n, p, , pp 21 . So , nqppp n21 is a
Logical Truth and is denoted by , nqppp n21 ;
c) A Biconditional Statement qp where p and q always have the same
Truth Value is always True. So it is a Logical Truth and is denoted by
qp ;
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2. A Fact or Contingency or Consistent Statement if and only if it is True in atleast one possible Interpretation;
3. A Contradiction or an Inconsistent Statement if and only if it is False in allpossible Interpretations. In general, Contradiction can be simply denoted byTruth Value F.
Concept and Definition are closely related to and are essential to each other. AnyConcept requires a Definition to provide its Meaning. Conversely, any Definitionmust attach to the Concept of some Class of Objects. Without Concept, aDefinition would have nothing refer to. Conversely, without Definition, it wouldbe difficult to master a Concept.
Definition 15 (Structure of a Definition)A Definition is, generally a set of Statements, which gives a general Meaning to aConcept. This means, a Definition either lists the Entity Class of a Concept orspecifies an Intension of a Concept. In a Definition:1. The Concept being defined is called the Definiendum.2. The part of a Definition except the Definiendum, which conveys the Meaning
of the Definiendum, is called the Definiens.The process of formulating a Concept’s Definition or giving a Meaning to aConcept is called Defining the Concept. Given a Definition of a Concept:1. The Meaning of the Concept Defines the Concept (the Definiendum);2. The Concept (the Definiendum) is Defined by its Meaning.
Definition 16 (Types of Definition)There are two main Types of Definition:1. (Intensional Definition): An Intensional Definition of a Concept gives a
Meaning to the Concept by specifying a Property (an Intension) of theConcept that any Object must possess in order to be an Instance of theConcept.In other words, an Intensional Definition gives a Meaning to a Concept byspecifying a Property of the Concept, which is a Necessary and SufficientCondition possessed by all Instances of the Concept. So any Object is anInstance of the Concept if and only if it possesses the specified Property ofthe Concept. A Mathematical Definition is almost of one of the following twoTypes of Intensional Definition:a. Genus-Differentia Definition;b. Recursive Definition.
2. (Extensional Definition): An Extensional Definition of a Concept gives aMeaning to the Concept by listing all Instances of the Concept. In otherwords, any Object is an Instance of the Concept if and only if it is one of thelisted Instance of the Concept.
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Definition 17 (Genus-Differentia Definition)
Let , , PP, I, , EEC 2121 and , , PP, J, , FFG 2121 are two
Concepts, where , , , iE i 21 are Extensions and 1E is the Entity Class of
C, , , , iFi 21 are Extensions and 1F is the Entity Class of G, while I, J are
Comprehensions of C and G respectively. Then:
1. G is a Genus of C if and only if 11 FE and JI .
2. A n-ary Predicate 1 , jI, jPj is a Differentia of C if and only if
nj , x, x, , x : PFxE 111 .
3. A Genus-Differentia Definition of a Concept gives a Meaning to the Conceptby specifying a Genus of the Concept and a distinguishing Intension of theConcept. It generally takes one of the following forms:
(1) (Common): A C is defined as a G (Genus) such that nj , x, x, , xP 1
(Differentia);(2) (Detailed): For any Object x, x is an Instance of C if and only if or is
defined as or is defined by or means that x is an Instance of G (Genus)
and nj , x, x, , xP 1 (Differentia);
(3) (Formal):
njj
def
, x, x, , x, P, jI, jPFx Exx, 111 1 .
Example 4Consider an Intensional Definition of Natural Number :(Common): A Natural Number n is an Integer such that 0n .(Detailed): For any Object n, n is an Instance of Natural Number if and only if n
is an Instance of Integer Ζ and nP , where ""xxΖ, Px 0 .
(Formal): n", P"xxΖ, PxΖnnn, 0 .
In this Definition: The Definiendum, which is a Concept, is the NaturalNumber ; The Genus of Natural Number , which is another Concept, is the
Integer Ζ ; The Differentia, which is a Unary Predicate F, TP : Ζ such
that ""xxP 0 .
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Definition 18 (Recursive Definition)A Recursive Definition or Inductive Definition of a Concept gives a Meaning tothe Concept by specifying a Relation between Instances of the Concept, so thatall Instances of the Concept can be theoretically generated one by one, and hencethe whole Entity Class of the Concept can also be theoretically constructed. ARecursive Definition generally contains:1. At least one Object is specified to be an Instance of the Concept;2. Any Object stands to a specified Relation with other Instances of the
Concept is specified to be an Instance of the Concept;3. No other Object is an Instance of the Concept.
Definition 19 (Formal Definition of Natural Number by Peano Axioms)Let be the Concept of Natural Number or the Entity Class of the Concept ofNatural Number (Set of all Natural Numbers). Define the Successor Function
s : such that nsn , thencan be defined Recursively by:
1. 0 ;
2. nsn ;
3. nsmsnm, m, n (s is an Injection);
4. 0 n, sn (0 is NOT the Successor of any Natural Number);
5. n, PnSSnsSnS, xx : PS 0
(Principle of Mathematical Induction, a Second Order Logical Form).Each of the , n, , , , 210 defined above is called a Natural Number.This Recursive or Inductive Definition of the Natural Number Concept wasproposed by the Italian Mathematician Giuseppe Peano, so it is called PeanoAxioms.
Based on the ZF Axiomatic Set Theory, Peano Axioms and the Theory of NaturalNumbers can be derived.
Definition 20 (Formal Definition of Natural Numbers as Ordinals)Let be the Concept of Natural Number or the Entity Class of the Concept ofNatural Number (Set of all Natural Numbers). Define the Successor
Function s : such that nnnsn , then can be defined
Recursively or Inductively by:
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1. 0 ;
2. , n, n, , nnnsn 110 ;
3. is the Intersection of all Sets satisfying Axioms 1 and 2.Each of the Sets , n, , , , 210 defined above is called a Natural Number or anOrdinal.Therefore, the Natural Numbers defined in this way are:
0 ,
00001 s ,
1000001112 , , s ,
210101010102223 , , , , , , , s ,
and so on...This Recursive or Inductive Definition of the Natural Number Concept wasproposed by the American Mathematician and Computer Scientist John VonNeumann.
Based on the Naive Set Theory, Peano Axioms and the Theory of Natural Numberscan be derived.
Definition 21 (Formal Definition of Natural Numbers as Cardinals)Let be the Concept of Natural Number or the Entity Class of the Concept ofNatural Number (Set of all Natural Numbers). Define:
1. The Equivalence Relation of Equinumerosity and let AX : XA be the
Equivalence Class of set A,
2. The Successor Function s : such that , n, , , nsn 210
, n, , , X : X 210 , that is, ns is a Class of all sets, each of which has
exactly as many elements as , n, , , 210 ,
thencan be defined Recursively or Inductively by:
1. X : X0 ;
2. , n, , , X : X, n, , , nsn 210210 ;
3. is the Intersection of all Sets satisfying Axioms 1 and 2.
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Each of the Classes , n, , , , 210 defined above is called a Natural Number ora Cardinal.Therefore, the Natural Numbers defined in this way are:
X : X0 ;
001 X : X ;
10102 , X : X, ;
2102103 , , X : X, , ;
and so on...This Recursive or Inductive Definition of the Natural Number Concept wasproposed independently by German Mathematician, Logician and PhilosopherGottlob Frege and British Mathematician, Logician and Philosopher BertrandRussell.
Definition 22 (Formal Definition of Natural Numbers as Categories)Let be the Concept of Natural Number or the Entity Class of the Concept ofNatural Number (Set of all Natural Numbers). Define the Successor Function
s : such that , , Dom, Cod, , jj : ii, , n, nsn 0 ,
thencan be defined Recursively or Inductively by:
1. , , Dom, Cod, , , 0 ;
2. , , Dom, Cod, , jj : ii, , n, , , , nsn 3210 ,
where:
a. , n, , , , nsOb 3210 is a Class of Categories;
b. jj : iinsAr is a Class of Arrows where nsObi, j ;
c. , are two Binary Relations defined on nsOb such that
jObiObji and jObiObji ;
d. is a Ternary Relation called Arrow Composition defined on nsAr :
nsArns : Ar 2 , kijikjkj, ji ,
nsObi, j, k and kji .
3. is the Intersection of all Sets satisfying Axioms 1 and 2.
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Each of the Categories , n, , , , 210 defined above is called a Natural Number.Therefore, the Natural Numbers defined in this way are:
, , Dom, Cod, , , 0 ,
, , Dom, Cod, , , 0001 ,
, , Dom, Cod, , , , , , 111000102 ,
and so on ... .
Note that in the Definition of Category ns , the number of Arrows in nsAr is
given by
221
1
nnins
iand the Arrow Composition is clearly Associative and
has Identity Arrows ii , nsObi .
Definition 23 (Logical Form)A Logical Form or Formalized Form of a set of Statements is defined as a Syntaxof all Statements represented in a Formal Language which is the result ofAbstraction from the Semantic contents other than Truth Values of allStatements.A Logical Form can be represented by many different Syntax within a FormalLanguage.
Example 5The Logical Form of three Statements: “If something is a Function, then it is aRelation” or “All Functions are Relations”, “f is a Function” and “f is a Relation”
are given by: Q(x)P(x)x , where P(x)= “x is a Function” andQ(x)= “x is a
Relation” are Unary Predicates defined on the Universal Class of all Sets, P(f)
andQ(f)respectively, in First Order Predicate Logic.
Definition 24 (Logical Argument and Logical Inference)
A Logical Argument A is a Sequence of Statements , c, p, , pA : p n21 , n ,
such that n, p, , pp 21 are called Premises and c is called Conclusion.
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Therefore, for any Logical Argument , c, p, , pA : p n21 , n , it corresponds to
a Conditional Statement cppp n 21 .
For any Logical Argument , c, p, , pA : p n21 , n , the Reasoning Process
which turns all n, p, , pp 21 into c is called a Logical Inference.
Definition 25 (Valid Logical Argument and Valid Logical Inference)
A Logical Argument , c, p, , pA : p n21 , n , is Valid if and only if there is NO
Interpretation for all Statements in A in which n, p, , pp 21 are all True and c is
False, or equivalently, c is a Logical Consequence of n, p, , pp 21 .
Therefore, for any Valid Logical Argument , c, p, , pA : p n21 , n , it
corresponds to a Logical Truth or Tautology cppp n 21 .
For any Valid Logical Argument , c, p, , pA : p n21 , n , the Reasoning
Process which turns all n, p, , pp 21 into c is called a Valid Logical Inference.
Definition 26 (Sound Logical Argument and Sound Logical Inference)
A Logical Argument , c, p, , pA : p n21 , n , is Sound if and only if A is Valid
and n, p, , pp 21 are all True.
For any Sound Logical Argument , c, p, , pA : p n21 , n , the Reasoning
Process which turns all True Premises n, p, , pp 21 into the True Conclusion c is
called a Sound Logical Inference.
Metalogical Axiom 1 (Fundamental Axiom in Logic: Law of Identity)
For any Statement p, we have: pp .
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Metalogical Axiom 2 (Fundamental Axiom in Logic: Law of Non-contradiction)
For any Statement p, we have: F pp or pp .
Metalogical Axiom 3 (Fundamental Axiom in Logic: Law of Excluded Middle)
For any Statement p, we have: pp .
Axioms 1 to 3 are fundamental and essential to almost all disciplines especiallyMathematics, Logic and Computer Science and they are called The Law ofThought or The Fundamental Principles of Logic. Even though some LogicalSystems were developed without founding on Axioms 3, such as IntuitionisticLogic which does not admit Axiom 3 or Fuzzy Logic which asserts UncountablyMany Truth Values, Axioms 1 to 3 can never be violated. For a simple example,
let p be the Statement πp : x . Then it is totally unfounded or ridiculous to
claim that πxπx pp is at least a Contingency or a Fact
because it is nonsense and we cannot even imagine it. So it is a Contradiction.For another example, an Intuitionist may claim that the Truth Value of aStatement p is neither True nor False but is a third value: Unknown, because the
proof for p only shows that p is False and p has not been shown to be True.
Although F p means Fp , p has not been shown to be True does not mean p
is not True. p is still possible to be True simply we are not able to prove p directly.
So it does not make sense to claim that Fp and Tp and the Truth Value of p
is a third value: unknown. And even if any Intuitionist does not admit Axiom 3, itis still Valid since it exhausts all definite Truth possibilities of any Statement. Forone more example, in Fuzzy Logic, the fundamental concept of Truth Value isreplaced by the concept of Degree of Truth and the Domain of Truth Value,
namely the Boolean Domain F, T or 10, , is replaced by 10, . This means
in Fuzzy Logic, the Degree of Truth of any Statement or Proposition can take any
10, x . However, this is just a matter of different viewpoints. Because if
we consider, let say, the Partition F, T , ., ., 150500 of 10, , then the
Degree of Truth of any Statement in Fuzzy Logic will be either F or T and cannotbe both. Also, if the Degree of Truth of any Statement in Fuzzy Logic is not F (or
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T), then it must be T (or F) and no other possibility. Therefore, Axioms 2 and 3are also Valid in Fuzzy Logic.
Definition 27 (Formal Grammar or Phase Structure Grammar)
AMathematical Structure n, P, f, P, s, S, N, T, SG 10 , n , where:
1. S is an Alphabet or a Vocabulary;2. SN is a set of Nonterminal Symbols;3. ST is a set of Terminal Symbols or Allowed Symbols such that NST ;
4. S is the Kleen Star of S which is the set of all Countable Strings over S;
5. SNs 0 is a set of Start Symbol or Sentence Symbol;
6. SSf : S such that xyx, yfx, y is a Binary Operation
on S called String Concatenation;
7. SSy, zx, ff, zx, yff, xyzSN, yS : x, zxyz, sPi
, i , , n, i 1 , is a Binary Relation defined on S called Production
Rule,is an Abstract Mathematical Structure called Formal Grammar or PhaseStructure Grammar proposed by American MIT Linguist and Philosopher NoamChomsky.
Definition 28 (Derive Relation on S )
Let n, P, f, P, s, S, N, T, SG 10 , n , be a Formal Grammar.
1. S, z, yx, y 21 , if Szxy, wSzxyw 2211 and 21 yyPi , for some
ni 1 , then Sw2 is called to be Directly Derivable from Sw1 ,
denoted by 21 ww ;
2. , n, , , iSwi 21 , if nn wwww 121 , then Swn is called
to be Derivable from Sw1 , denoted by nww 1 . The Sequence of steps
used to derive nw from 1w is called a Derivation;
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3. is a Binary Relation defined on S by G, called the Derive Relation.
Definition 29 (Formal Language and L-Structure)
Let n, P, f, P, s, S, N, T, SG 10 , n , be a Formal Grammar. Then:
1. A SL is called a Formal Language over S ;
2. The Sw : sTwGL 0 , is called the Language of G, which is a
Formal Language over S and is also a set of all Well Formed Formulas
(WFFs) generated by G. In other words, SGL is the set of all Properly
Produced Strings or Words or Sentences or Formulas generated by G.
A Mathematical Structure , , f, f, , RR, , , DDΜ 212121 which is
formulated by the Language L is called a L-Structure.
Example 6
Let c, w, a, b,sS 0 , , wsN 0 and a, b, cT .
Consider the Formal Grammar 3210 , P, Pf, P, s, S, N, T, SG , where
w, c, Pw, bbw, P, awsP 3201 .
Start with 0s , then we can Derive:
c, nabwabwababbbbwwababbwaws nn 420 , and we
can stop at cabn , since it contains only Terminal Symbols and nothing more can
be Derived by applying the Production Rule P from this point. So
c : nabGL n .
Definition 30 (Formal System or Formal Axiomatic System)
AMathematical Structure m , r, , r r, A, GS, G, LF 21 , m , where:
1. S is an Alphabet or a Vocabulary;
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2. n, P, f, P, s, S, N, T, SG 10 , n is a Formal Grammar which can
be used for producing Well-Formed Formulas (WFF, which are properlyproduced Strings);
3. SGL is the Language of G;
4. , nGL, a, , a aA n21 , is a finite set of Axioms and each Axiom
must be a WFF;
5. GLG : Lr ki such that kik , w, , wwrw, w, , ww 2121 , k ,
i , , m, i 1 , is a (k+1)-ary Relation defined on GL called k-ary Rule of
Inference,is an Abstract Mathematical Structure called Formal System or FormalAxiomatic System. In a Formal System F, it is common to represent each
, m, , iri 1 as GLG : Lr ki defined by w , w, , ww Fk 21 . All
k, w, , ww 21 are called Premises and w is called Conclusion or Theorem.
WFFs k, w, , ww 21 are called together Syntactically Implies or Infers or Proves
or Concludes WFF w inside F.In simplest terms, a Formal System F is a generative system which has two mainfunctions:1. Constructing its own Language by using its own set of symbols andGrammatical Rules;2. Deriving Theorems by applying its own Rules of Inference to some of its ownAxioms or derived Theorems.So the product of a Formal System F is a Sequence of WFFs written in theLanguage of G, is called a Formal Theory.
Metalogical Axiom 4 (Conditional Proof or Direct Proof)
Let qp be a Conditional Statement, where p and q are any two Statements.
To show that: qp is True or qp . ASSUME the Premise p is True and
derive q is necessarily True, this is called a Conditional Proof or Direct Proof
for qp . If this is the case, it is denoted by qp .
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In general, let , nqppp n21 be a Conditional Statement, where
, q, p, , pp n21 are any Statements.
To show that qppp n 21 is True or qppp n 21 , ASSUME
all Premises n, p, , pp 21 are True and derive q is necessarily True, this is called
a Conditional Proof or Direct Proof for qppp n 21 . If this is the case,
it is denoted by q , p, , pp n 21 .
Definition 31 (Formal Proof, Formal Theorem and Metatheorem)
Let m , r, , r r, A, GS, G, LF 21 , m be a Formal System.
For any WFFs , pGL, w, w, , ww p21 , a Formal Proof or a Formal
Derivation for a Conditional Statement wwww p 21 or for GLw in F
is defined as a Valid Logical Argument , w, ω, , ωP : ω n21 , n which is a
Sequence of WFFs in SGL such that:
1. Each Premise , n, , iωi 1 in P is either one of the following:
a) An Axiom in A;
b) Any one of the , p, , jw j 1 ;
c) A WFF in SGL obtained by applying a , m, , krk 1 to some
11 , i, , iωi ,
2. The Conclusion in P is SGLw , which is called a Formal Theorem and
is denoted by wF .
The Conditional Statement wwww p 21 is called a Metatheorem and is
denoted by w , w, , ww Fp 21 .
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Definition 32 (Syntactic and Semantic Consequences in a Formal System)
Let m , r, , r r, A, GS, G, LF 21 , m be a Formal System. Then:
1. GLw is defined as the Syntactic Consequence or the Material
Consequence of GL, w, , ww p 21 in F if and only if there exists a
Formal Proof in F for the Statement wwww p 21 .
In this case, wwww p 21 is a Metatheorem. p, w, , ww 21 are called
together Syntactically Implies or Infers or Proves or Concludes w, and this is
denoted by w , w, , ww Fp 21 .
2. GLw is defined as the Semantic Consequence or the Logical Consequence
of GL, w, , ww p 21 in F if and only if:
a) There is NO Interpretation in which p, w, , ww 21 are all True and w is
False;
b) wwww p 21 is a True Statement under any Interpretation;
c) The Class of Interpretations under which p, w, , ww 21 are all True is a
SubClass of the Class of Interpretations under which w is True.
In this case, p, w, , ww 21 are called together Semantically Implies or Infers
or Proves or Concludes w, and this is denoted by w , w, , ww Fp 21 or
wwww p 21 .
Semantic Consequence can be regarded as an Order Relation in Logic but itis a Semantic concept.
Definition 33 (Syntactic and Semantic Equivalences in a Formal System)
Let m , r, , r r, A, GS, G, LF 21 , m be a Formal System. Then:
1. GLw 1 is defined as Syntactically Equivalent or Materially Equivalent to
GLw 2 in F or GLw 1 and GLw 2 is defined as Syntactic Equivalence
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or Material Equivalence in F if and only if there exist Formal Proofs in F for
Statements 21 ww and 12 ww .
In this case, 122121 wwwwww is a Metatheorem.
2. GLw 1 is defined as Semantically Equivalent or Logically Equivalent to
GLw 2 in F or GLw 1 and GLw 2 is defined as Semantic Equivalence
or Logical Equivalence in F if and only if:
a) There is NO Interpretation in which 1w and 2w have different Truth
Values;
b) 21 ww is a True Statement under any Interpretation;
c) The Class of Interpretations under which 1w is True is equal to the Class
of Interpretations under which 2w is True.
In this case, 21 ww is a Logical Truth or Tautology.
Semantic Equivalence can be regarded as an Equality Relation in Logic but itis a Semantic concept.
Example 7 (Example of Formal Direct Proof in Propositional Logic)Given the Premises “If I study in HKU, then I will major in Mathematics.”, “If Ido not study in HKU, then I will go to study in Tokyo University.” and “If I go tostudy in Tokyo University, then I will major in Computer Science” give a FormalProof for the Conclusion “If I do not major in Mathematics, then I will major inComputer Science.”Let p, q, r, and s are Statements “I study in HKU”, “I major in Mathematics”, “Igo to study in Tokyo University” and “I major in Computer Science”
respectively. Then the Premises are qp , r p and sr . The required
Formal Theorem is s q .
The following Logical Argument is a Formal Proof for s q .
1. qp (Premise 1);
2. p q (Apply 20R Proof by Contrapositive in Definition 37 to 1);
3. r p (Premise 2);
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4. r q (Apply 23R Hypothetical Syllogism in Definition 37 to 2 and 3);
5. sr (Premise 3);
6. s q (Apply 23R Hypothetical Syllogism in Definition 37 to 4 and 5).
Definition 34 (Formal Theory, Model, Mathematical Theory and Truth)
Let m , r, , r r, A, GS, G, LF 21 , m , be a Formal System. Then:
1. A set GLT which contains WFFs of GL is called a Formal Theory
of GL or of F. In general, a Formal Theory is defined as a subset of a
Formal Language,2. A Mathematical Theory is theoretically possible to be Formalized to a Formal
Theory, so a Mathematical Theory is defined as a Formal Theory whichcontains all Axioms and derived Theorems of a corresponding FormalAxiomatic System,
3. A proved WFF in F is a Formal Theorem. An Axiom in F is also regarded asa Formal Theorem,
4. A Formal Theorem or a Metatheorem in a Mathematical Theory is called aMathematical Truth,
5. A Mathematical Structure which satisfies all WFFs of a Formal Theory iscalled a Model of that Formal Theory.
Example 8 (Arithmetic)
Let , , , , , , , , , , , n, n, s, , , SA 210 be the
Alphabet of Arithmetical Symbols and AG be the Formal Grammar for
formulating Well-Formed Arithmetical Formulas (WFAFs) such as
basbsa . Then:
1. The Formal Language AA SGL is called the Arithmetical Language which
contains all WFAFs. The Peano Axioms of Natural Number (PA) stated in
Definition 19 is just a set of WFAFs in AGL , that is, AGLPA is a Formal
Theory. This means that all Axioms in PA are WFAFs formulated by
Arithmetical Symbols in AS ;
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2. The Formal System , , rr, PA, G, L, GSF AAAA 21 , where
, , , iri 21 are usual Logical Rules of Inference, is called the Peano
Arithmetical System. By the Peano Arithmetical System, many FormalTheories of Arithmetic can be developed;
3. A Model of PA is a Mathematical Structure ns, , 0 which satisfies
all Axioms in PA. Examples of Standard Model of PA are:
a. The Von Neumann Ordinal Model nnns, , O 0 , which
is formulated by Well-Formed Set-theoretical Formulas and Symbols andArithmetical Symbols , n, , , 210 ,
b. The Frege-Russell Cardinal Model , n, , ns, , C 100 ,
which is formulated by Well-Formed Set-theoretical Formulas and SymbolsandArithmetical Symbols , n, , , 210 ,
c. The Categorical Model ns, , , Dom, Cod, , , , C 0 ,
where , , Dom, Cod, , jj : ii, , n, , , , ns 3210 ,
which is formulated by Well-Formed Categorical Formulas and SymbolsandArithmetical Symbols , n, , , 210 .
Example 9
If S is the set of all “Allowable Mathematical and Logical Symbols” and G is the“Formal Grammar” for formulating Well-Formed Mathematical and LogicalFormulas (WFMLFs), then:
1. SGL will be the Formal Language which is called the Mathematical and
Logical Language and it contains all WFMLFs;
2. , , rr, A, GS, G, LF 21 , where A is a set of some fundamental
Mathematical and Logical Axioms and Definitions such that all arenecessary for this article and all can be completely Formalized into
WFMLFs, , , , iri 21 are the usual Logical Rules of Inference, will be a
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Formal System, for deriving Theorems in this articles which are all
WFMLFs in GL ;
3. GLTA which contains all Axioms and Definitions in A and all
Theorems derived within F, is a Formal Theory or more precisely aMathematical Theory. Hence this article is a representation of MathematicalTheory T.
Similarly, all my other articles each of which is a representation of some otherMathematical Theory.
Example 10Mathematics, as a typical Formal Science, contains thousands of examples ofMathematical Theory. Each Mathematical Theory is theoretically possible to beFormalized to a Formal Theory which is the product of some Formal AxiomaticSystem. Here are some examples:1. Axiomatic Set Theory is a Formal Theory which has Set Structures as its
Models;2. Number Theory is a Mathematical (and hence Formal) Theory which has
Integers Structure as its Model;3. Arithmetic is a Mathematical (and hence Formal) Theory which has Real
Numbers Structure as its Model;4. Euclidean Geometry is a Formal Theory which has Points Structure, Lines
Structure and Planes Structure as its Models (proposed by GermanMathematician David Hilbert);
5. Calculus is a Mathematical (and hence Formal) Theory which hasReal-valued Functions Space as its Model;
6. Abstract Algebra is a Mathematical (and hence Formal) Theory which hasvarious Algebraic Structures such as Groups, Rings and Fields, etc, as itsModels;
7. Universal Algebra is a Mathematical (and hence Formal) Theory which hasvarious Classes of Algebraic Structures as its Models;
8. Model Theory is a Mathematical (and hence Formal) Theory which hasvarious Mathematical Structures as its Models, each of which is a Model ofsome Formal Theory of some Formal Language;
9. Proof Theory is a Mathematical (and hence Formal) Theory which hasvarious Classes of Formal Systems as its Models;
Mathematical Logic, which contains Metamathematics or Metalogic, is aMathematical (and hence Formal) Theory studies various MathematicalTheories together with their respective Formal Systems and Models. It hasvarious Logical Systems as its Models.
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Definition 35 (Logical System or Formal Logic)
Let , m , r, , r r, A, GS, G, LF m21 , be a Formal System. Then a
Formal System , mIF, M, , r, , rI, r, A, M, GS, G, LL m21 ,
where:
1. S is an Alphabet or a Vocabulary;
2. n, P, f, P, s, S, N, T, SG 10 , n is a Formal Grammar;
3. SGL is the Language of G;
4. , nGL, a, , a aA n21 is a finite set of Axioms;
5. M is a Mathematical Structure called Interpretation Model which assigns
meanings and assigns Truth Values to all WFFs of SGL by an
Interpretation Mapping I. M together with I forms a Semantics of L;
6. GLG : Lr ki such that kik , w, , wwrw, w, , ww 2121 , k ,
i , , m, i 1 , is a k-ary Rule of Inference defined on SGL ,
is called a Logical System or Formal Logic or simply a Logic.So a Logic is simply a Formal System plus a Semantics which consists of anInterpretation Model and an Interpretation Mapping.
Example 11 (Logic of Sets)
Let , , , , , , , , , , , , , , , , x, y, z, SS
be the Alphabet of Set-theoretic Symbols and SG be the Formal Grammar for
formulating Well-Formed Set-theoretic Formulas (WFSFs) such as
yxyzxzz, y, x, . Then:
1. The Formal Language SS SGL is called the Set-theoretic Language which
contains all WFSFs. The ZF Axiomatic Set Theory (ZF) is a set of WFSFs
in SGL , that is, SGLZF is a Formal Theory. This means that all
formulas in ZF are WFSFs formulated by the Set-theoretic Symbols in SS ;
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2. The Formal System , , rr, ZFA, G, L, GSF SSSZF 21 , where ZFA is the
set of Axioms in ZF and , , , iri 21 are usual Logical Rules of Inference, is
called the ZF Axiomatic Set System. The product of ZFF is ZF;
3. A Model of ZF is a Mathematical Structure ZFM which satisfies all WFSFs in
ZF;
4. The Mathematical Structure , , r, rI, , ZFA, MG, L, GSL ZFZFSSSS 21
= ZFZFZF I, , MF , where ZFM is an Interpretation Model which assigns
Meanings and assigns Truth Values to all WFSFs in SGL by an
Interpretation Mapping ZFI , is called a Logical System of Sets or Logic of Sets
or simply Sets Logic. Therefore, Sets Logic is a main branch in MathematicalLogic.
A Formal System or Formal Axiomatic System or Logical System is not simplyany Syntax represented in the Formal Language of Mathematics and Logic, anypractical Formal System or Formal Axiomatic System or Logical System mustsatisfy the properties given in the next Definition.
Definition 36 (Important Properties of Formal System or Logical System)
Let m , r, , r r, A, GS, G, LF 21 , m be a Formal System or Formal
Axiomatic System. Then F is expected to possess as many the followingproperties (listed by the order of importance) as possible.
1. (Consistency): F is Consistent if and only if GLT , GLTw such
that wT wT FF .
2. (Validity): F is Valid if and only if GLG : Lr ki defined by
w , w, , ww Fk 21 , , m, i 1 , it always have w , w, , w w Fk 21 .
3. (Completeness):
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a) F is Semantically Complete if and only if any Logical Truth or Tautology
in GL is a Formal Theorem. That is, ww , GLw FF ;
b) F is Strongly Complete if and only if GL, w, , w w n 21 and
GLw , w , w, , w ww , w, , w w FnFn 2121 ;
c) F is Syntactically Complete or Deductively Complete if and only if
GLw , either w or w is a Formal Theorem in F. That is, GLw ,
ww FF .
4. (Soundness, the Converse of Completeness):
a) F is Weak Soundness if and only if GLw , ww FF ;
b) F is Strong Soundness if and only if GL, w, , w w n 21 and
GLw , w , w, , w ww , w, , w w FnFn 2121 .
Definition 37 (Propositional Logic or Propositional Calculus)
A Logical System m , r, , rI, r, A, M, G, C, G, LF, TS, V, L 21 , m , is
called a Propositional Logical System or Propositional Logic or PropositionalCalculus or Sentential Logical System or Sentential Logic or Sentential Calculusor Statement Logic or 0th Order Logic (In contrast to 1st Order Logic and HigherOrder Logic), where:
1. , , , , , , , F, T, , , R, , Q, PS kji , i, j, k is an
Alphabet or a Vocabulary, where:
a) , i, j, k, R, QP kji are Atomic Formulas or Propositional Variables, each
of which contains no Logical Constant;b) , , , , are Logical Connectives;
c) , , , , , F, T, are Logical Constants or Logical Symbols.
Each Logical Constant has the same meaning under every Interpretation,
2. S, , R, , Q, PV kji , i, j, k is the set of all Atomic Formula or
Propositional Variables,
3. S, , , , C is the set of all Logical Connectives,
4. 543210 , P, P, P, PP, , SS, SG is a Formal Grammar, where:
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a) SF, TV, F, T, , R, , Q, PS iii 0 ;
b) The Production Rules are given by:P1. Logical Constants or Logical Symbols T and F are WFFs;P2. All Propositional Variables are WFFs;P3. If w is a WFF, then w is also a WFF;
P4. If 1w and 2w are WFFs, then 212121 ww, ww, ww and
21 ww are also WFFs;
P5. Nothing else is a WFF,
5. SGL is the Formal Language of G which is a set of all WFFs built up
from symbols of S according to the Production Rules P1 to P5,
6. GL, a, aaA 321 is a set of Axioms which contains the following Axioms:
For any Atomic Formula or Propositional Variable VP , we have:
PP : a 1 or PP (Law of Identity);
F PP : a 2 or PP (Law of Non-Contradiction);
P : Pa 3 (Law of Excluded Middle),
7. , , , , , T, FM is the Interpretation Model of L and is a
Mathematical Structure called Boolean Structure which assigns Truth Valuesto Logical Constants: F, T and all Propositional Variables and assignsBoolean Operations to all Logical Connectives, by the Interpretation MappingI, where:
a) T, F is the Domain called Boolean Domain which is a set of Truth
Values;
b) , , , , is a set of Boolean Operations defined on T, F :
i. T, FT, F : , called Negation or NOT, is an Unary Operation
or Function defined by:
x (x)
F TT F
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ii. T, FT, F : 2 , called Conjunction or AND, is a Binary
Operation defined by:
x y yx
T T TT F FF T FF F F
iii. T, FT, F : 2 , called Disjunction or OR, is a Binary Operation
defined by:
x y yx
T T TT F TF T TF F F
iv. T, FT, F : 2 , called Conditional or Material Conditional or
Material Consequence or IF...THEN..., is a Binary Operation definedby:
x y yx
T T TT F FF T TF F T
v. T, FT, F : 2 , called Biconditional or Material Biconditional
or Material Equivalence or IF AND ONLY IF, is a Binary Operationdefined by:
x y yx
T T TT F F
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F T FF F T
8. I is a Surjective Mapping called Interpretation Mapping:
, , , , F, T, CVF, TI : defined by:
FI(F) ; TI(T) ; F or TV, I(P)P but not both;
F, TF, T : )I( ;
F, TF, T : )I( 2 ;
F, TF, T : )I( 2 ;
F, TF, T : )I( 2 ;
F, TF, T : )I( 2 .
So given an Interpretation Mapping I:a) The Logical Constants F and T correspond to the Truth Values False (or
0 or a False Statement) and True (or 1 or a True Statement) respectively;b) Any Propositional Variable is assigned a Truth Value and so it represents
a Statement which contains no Logical Constant, that is, an AtomicStatement. Notice that different Interpretation Mappings assigndifferent Truth Values and hence different Atomic Statement to the same
Propositional Variable. For example, SGLVP and any two
Interpretation Mappings 21 , II , it may happen that 01 (P)I (so VP
represents a False Statement and this is denoted by FP under the
Interpretation Mapping 1I ) and 12 (P)I (so VP represents a True
Statement and this is denoted by TP under the Interpretation
Mapping 2I );
c) Any WFF in SGL which is neither a Propositional Variable nor a
Logical Constant F or T is a Formula represents a Compound Statement.The Truth Value of any Compound Statement depends on and can becalculated from the Truth Values of all its Atomic Statements. Therefore,although I only assigns Truth Values to all Propositional Variables, I in
fact assigns Truth Values to all WFF in SGL .
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The Interpretation Model M together with an Interpretation Mapping Iforms the Semantics of Propositional Logic L,
9. 2121 , r, , rr is a set of Rules of Inference defined on SGL which
contains the following Inference Rules for deriving Theorems. Formally, ir ,
i , , m, i 1 can be defined as a (k+1)-ary Relation GLG : Lr ki
such that kk , w, , wwRw, w, , ww 2121 , where k .
1. (Negation Introduction)
GLG : Lr 21 such that 12121 w ww, ww ;
2. (Negation Elimination)
GLG : Lr 2 such that 211 ww w ;
3. (Double Negative Elimination)
GLG : Lr 3 such that 11 w w ;
4. (Conjunction Introduction)
GLG : Lr 24 such that 2121 ww, ww ;
5. (Tautology 1)
GLG : Lr 5 such that 111 www ;
6. (Conjunction Elimination 1)
GLG : Lr 6 such that 121 www ;
7. (Conjunction Elimination 2)
GLG : Lr 7 such that 221 www ;
8. (Disjunction Introduction 1)
GLG : Lr 8 such that 211 www ;
9. (Disjunction Introduction 2)
GLG : Lr 9 such that 212 www ;
10. (Tautology 2)
GLG : Lr 10 such that 111 www ;
11. (Disjunction Elimination or Proof by Cases)
GLG : Lr 311 such that 2312321 www, ww, ww ;
12. (De Morgan’s Law 1)
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GLG : Lr 12 such that 2121 w www ;
13. (De Morgan’s Law 2)
GLG : Lr 13 such that 2121 w www ;
14. (Conditional Introduction)
GLG : Lr 14 such that 2121 w www ;
15. (Biconditional Introduction)
GLG : Lr 215 such that 211221 wwww, ww ;
16. (Biconditional Elimination 1)
GLG : Lr 16 such that 2121 wwww ;
17. (Biconditional Elimination 2)
GLG : Lr 17 such that 1221 wwww ;
18. (Modus Ponens)
GLG : Lr 218 such that 2211 www, w ;
19. (Modus Tollens)
GLG : Lr 219 such that 1212 www, w ;
20. (Proof by Contrapositive)
GLG : Lr 20 such that 1221 w www ;
21. (Proof by Contradiction 1 or Reductio Ad Absurdum 1)
GLG : Lr 21 such that 11 wF w ;
22. (Proof by Contradiction 2 or Reductio Ad Absurdum 2)
GLG : Lr 22 such that 2121 wwF ww ;
23. (Hypothetical Syllogism)
GLG : Lr 223 such that 313221 wwww, ww .
The Propositional Logic is the simplest type of Formal Logic among all.
It can be proved that each Inference Rule in Propositional Logic indeedcorresponds to a Logical Truth, and hence the Propositional Logic is Valid. Infact, all Inference Rules in Propositional Logic are True-preserving, this meansgiven any set of True Axioms and any set of True Premises, any Inference Rulecan turn all True Statements into another True Statement. But notice that: First,Validity is a Semantic fact, so it cannot be proved Syntactically inside thePropositional Logical System. Second, this is also a property regarding
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Propositional Logic, so it is Metalogical. Therefore, the following Theorems areall Metalogical and Semantic.The Proof of Validity of Propositional Logic is given by: First, define eachInference Rule as a Biconditional or a Conditional Statement. And second, UseTruth Table to show that the defined Biconditional or Conditional Statement is aLogical Truth. We shall only prove some of the above Inference Rules.Although Propositional Variables will be used in all the following Metalogicaland Semantic Theorems, same results also hold if they are replaced by anyWFFs.
Metalogical and Semantic Theorem 3 (Negation Introduction)
p qpqp : r 1 .
Proof
p q qp qp qpqp p qpqp
T T T F F TT F F T F TF T T T T TF F T T T T
Metalogical and Semantic Theorem 4 (Double Negative Elimination)
p p : r 3 .
Proofp p p p p
T F T TF T F T
Metalogical and Semantic Theorem 5 (Proof by Cases)
qrpqrqp : r 11 .
Proof
p q r qp qr rp qrpqrqp
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T T T T T T TT T F T T T TT F T F F T TF T T T T T TT F F F T T TF T F T T F TF F T T F T TF F F T T F T
Metalogical and Semantic Theorem 6 (De Morgan’s Law 1)
q pqp : r 12
Proof
p q p q qp q p q pqp
T T F F F F TT F F T T T TF T T F T T TF F T T T T T
Metalogical and Semantic Theorem 7 (De Morgan’s Law 2)
q pqp : r 13
Proof
p q p q qp q p q pqp
T T F F F F TT F F T F F TF T T F F F TF F T T T T T
Metalogical and Semantic Theorem 8 (Conditional Introduction)
q pqp : r 14 .
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Proof
p q qp q p q pqp
T T T T TT F F F TF T T T TF F T T T
Metalogical and Semantic Theorem 9 (Biconditional Introduction)
qppqqp : r 15
Proof
p q qp pq pqqp qp qppqqp
T T T T T T TT F F T F F TF T T F F F TF F T T T T T
Metalogical and Semantic Theorem 10 (Modus Ponens)
qqpp : r 18
Proof
p q qp qpp qqpp
T T T T TT F F F TF T T F TF F T F T
Metalogical and Semantic Theorem 11 (Proof by Contrapositive)
p qqp : r 20
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Proof
p q qp p q p qqp
T T T T TT F F F TF T T T TF F T T T
Since p qqp is True under all possible Interpretations, so
p qqp is a Logical Truth or Tautology and qp always have
the same Truth Value as p q .
So to prove qp , we first assume q is True and try to derive p . If this can
be done, then we actually proved p q . Since qp and p q are True
at the same time, this means we also proved qp . This Method of Indirect Proof
is called Proof by Contrapositive.
Example 12 (Example of Proof by Contrapositive)
Prove that: Ζn , If 2n is Even, then n is also Even.
ProofNotice that the Logical Form of the Statement we want to prove is:
kΖ, nkmΖ, nmΖ, n 222 .
Let p “ mΖ, nm 22 ”and q “ kΖ, nk 2 ”. Then we want to prove
qp .
Suppose q which is the Statement “ kΖ, nk 2 ” = “ kΖ, nk 2 ”.
This means n is Odd and Ζt such that 12 tn . This implies
122214412 2222 tttttn . So 2n is also Odd and this implies
“ mΖ, nm 22 ” = “ mΖ, nm 22 ” p is True. Hence we have
proved p q .
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By the Method of Proof by Contrapositive since p qqp , so
qpΖ, n is also True and therefore qpΖ, n .
Metalogical and Semantic Theorem 12 (Proof by Contradiction 1)
F p : pr 21
Proof
p p F p F pp
T F T TF T F T
Since F pp is True under all possible Interpretations, so
F pp is a Logical Truth or Tautology and p always have the same
Truth Value as F p .
So to prove p, we first assume p is True and try to derive a Contradiction F. If
this can be done, then we actually proved F p . Since p and F p are
True at the same time, this means we also proved p. This Method of IndirectProof is called Proof by Contradiction or Reductio Ad Absurdum.
Metalogical and Semantic Theorem 13 (Proof by Contradiction 2)
F qpqp : r 22
Proof
p q qp qp F qp F qpqp
T T T F T TT F F T F TF T T F T TF F T F T T
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Since F qpqp is True under all possible Interpretations, so
F qpqp is a Logical Truth or Tautology and qp always
have the same Truth Value as F qp .
So to prove qp , we first assume qp qpq p is True and
try to derive a Contradiction F. If this can be done, then we actually proved
F qp . Since qp and F qp are True at the same time, this
means we also proved qp . This Method of Indirect Proof is called Proof by
Contradiction or Reductio Ad Absurdum.
Example 13 (Example of Proof by Contradiction)
Prove that 2 is an Irrational Number.
Proof
Notice that the Statement we want to prove can be rephrased as: “If 2 ,
then 2 is an Irrational Number.”
Let p “ 2 ”and q “ 2 is an Irrational Number”. Then we want to prove
qp .
Suppose on the Contrary that q “ 2 is NOT an Irrational Number”. This
means that Q2 and 0 Ζ, ba, b where a, b have no Common Divisor other
than 1 such thatba
2 . This implies 222
222
222 baba
ba
and 2a is
Even. By Example 12, this means a is also Even and mΖ, am 2 . So we have
222222 22422 mbbmbm and 2b is Even. By Example 12 again, this
means b is also Even and nΖ, bn 2 . But this impliesnm
nm
ba
222 . So
we have proved: a, b have no Common Divisor other than 1 and a, b have a
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Common Divisor 2. Obviously, this is a Contradiction. Therefore, we have proved
F qp .
By the Method of Proof by Contradiction since F qpqp , so
qp is also True and therefore qp .