syllogism and truth tables

8/11/2019 Syllogism and Truth Tables

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A syllogism (Greek : syllogismos "conclusion,"

"inference") is a kind of logical argument that applies deductive

reasoning to arrive at a conclusion based on two or

more propositions that are asserted or assumed to be true.

In its earliest form, defined by Aristotle, from the combination of a

general statement (the major premise) and a specific statement (the

minor premise), a conclusion is deduced. For example, knowing that all

men are mortal (major premise) and that Socrates is a man (minor

premise), we may validly conclude that Socrates is mortal. Syllogistic

arguments are usually represented in a three-line form (withoutsentence-terminating periods):

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

The word "therefore" is usually either omitted or replaced by the symbol

" "

Aristotle defines the syllogism as, "...a discourse in which certain(specific) things having been supposed, something different from thethings supposed results of necessity because these things are so." [1]

1. Aristotle , "Prior Analytics", 24b18 20

In logic , a categorical proposition , or categorical statement , isa proposition that asserts or denies that all or some of the members ofone category (the subject term ) are included in another (the predicateterm ). The study of arguments using categorical statements(i.e., syllogisms ) forms an important branch of deductive reasoning thatbegan with the Ancient Greeks .
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The Ancient Greeks such as Aristotle identified four primary distinct

types of categorical proposition and gave them standard forms (now

often called A, E , I , and O). If, abstractly, the subject category is

named S and the predicate category is named P , the four standardforms are:

All S are P . ( A form)

All S are not P . (E form)

Some S are P . ( I form)

Some S are not P . (O form)

Name Statement Quantity Quality

A All S are P. universal affirmative

E No S are P. universal negative

I Some S are P. particular affirmative

O Some S are not P. particular negative

In predicate logic , an existential quantification is a type of quantifier , a logical constant which is interpreted as "there exists," "there is at leastone," or "for some." existential quantifier (" x" or " (x)")It expresses that a propositional function can be satisfied by at leastone member of a domain of discourse . In other terms, it isthe predication of a property or relation to at least one member of the
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domain. It asserts that a predicate within the scope of an existentialquantifier is true of at least one value of a predicate variable .

Universal quantification is a type of quantifier , a logical constant whichis interpreted as "given any" or "for all". It expresses that a propositionalfunction can be satisfied by every member of a domain of discourse . In other terms, it is the predication of a property or relation to everymember of the domain. It asserts that a predicate within the scope of auniversal quantifier is true of every value of a predicate variable . universal quantifier (" x", " (x)")

A propositional function in logic , is a statement expressed in a way

that would assume the value of true or false , except that within thestatement is a variable ( x ) that is not defined or specified, which leavesthe statement undetermined.

A truth-bearer is an entity that is said to be either true or false andnothing else. Truth-bearer candidatesinclude propositions , sentences , statements , concepts , beliefs , thoughts , intuitions , utterances , and judgements but different authors exclude oneor more of these, deny their existence, argue that they are true only in aderivative sense, assert or assume that the terms are synonymous , [1] orseek to avoid addressing their distinction or do not clarify it.

the type token distinction is a distinction that separatesa concept from the objects which are particular instances of the concept.For example, the particular bicycle in your garage is a token of the typeof thing known as "The bicycle". Whereas the bicycle in your garage is ina particular place at a particular time, that is not true of "the bicycle" asused in the sentence: " The bicycle has become more popular recently".

In mathematical logic , a propositional calculus or logic (alsocalled sentential calculus or sentential logic ) is a formal system inwhich formulas of a formal language may be interpreted torepresent propositions .
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An interpretation is an assignment of meaning to the symbols ofa formal language . Many formal languages used in mathematics , logic , and theoretical computer science are defined in solely syntactic terms,and as such do not have any meaning until they are given some

interpretation. The general study of interpretations of formal languages iscalled formal semantics .

In propositional logic , m o d u s p o n e n d o p o n en s (Latin for "the way thataffirms by affirming"; and also affirming the antecedent , oftenabbreviated to MP or m o d u s p o n e n s i s a valid , simple argumentform and rule of inference

It can be summarized as " P implies Q; P is asserted to be true, sotherefore Q must be true."

Modus ponens can be stated formally as:

In propositional logic , mod u s t o l l en s [1][2][3][4] (or m od us t o l l end oto l lens and also denying the consequent )[5] (Latin f or "the way that

denies by denying" )[6]

is a valid argument form and a rule of inference .

The inference rule modus tollens , also known as the law of

contrapositive , validates the inference from implies and the

contradictory of , to the contradictory of .

The modus tollens rule can be stated formally as:

The logical form of a sentence(or proposition or statement or truthbearer ) or set of sentences is theform obtained by abstracting from the subject matter of its content termsor by regarding the content terms as mere placeholders or blanks on a
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form. In an ideal logical language, the logical form can be determinedfrom syntax alone; formal languages used in formal sciences areexamples of such languages.

Logical conjunction

Logical conjunction is an operation on two logical values , typically the

values of two propositions , that produces a value of true if both of its

operands are true.

The truth table for p AND q (also written as p q , Kpq , p & q , or p q )

is as follows:

Logical Conjunction

p q p q

T T T

T F F

F T F

F F F

In ordinary language terms, if both p and q are true, then the

conjunction p q is true. For all other assignments of logical values

to p and to q the conjunction p q is false.
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Logical disjunction

Logical disjunction is an operation on two logical values , typically the

values of two propositions , that produces a value of true if at least one of

its operands is true.

The truth table for p OR q (also written as p q , Apq , p || q , or p + q ) is

as follows:

Logical Disjunction

p q p q

T T T

T F T

F T T

F F F
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Logical implication Logical implication or the material conditional are both associated withan operation on two logical values , typically the values oftwo propositions , that produces a value of false just in the singular casethe first operand is true and the second operand is false.

The truth table associated with the material conditional if p then

q (symbolized as p q ) and the logical implication p impliesq (symbolized as p q , or Cpq ) is as follows:

Logical Implication

p q p q

T T T

T F F

F T T

F F T
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Logical equality

Logical equality (also known as biconditional) is an operation on

two logical values , typically the values of two propositions , that produces

a value of true if both operands are false or both operands are true.

The truth table for p XNOR q (also written as p q , Epq , p = q , or p

q ) is as follows:

Logical Equality

p q p q

T T T

T F F

F T F

F F T

So p EQ q is true if p and q have the same truth value (both true or both

false), and false if they have different truth values.
http://en.wikipedia.org/wiki/Logical_equalityhttp://en.wikipedia.org/wiki/Logical_equalityhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_valuehttp://en.wikipedia.org/wiki/Logical_valuehttp://en.wikipedia.org/wiki/Logical_valuehttp://en.wikipedia.org/wiki/Propositionhttp://en.wikipedia.org/wiki/Propositionhttp://en.wikipedia.org/wiki/Propositionhttp://en.wikipedia.org/wiki/Truth_valuehttp://en.wikipedia.org/wiki/Truth_valuehttp://en.wikipedia.org/wiki/Truth_valuehttp://en.wikipedia.org/wiki/Truth_valuehttp://en.wikipedia.org/wiki/Propositionhttp://en.wikipedia.org/wiki/Logical_valuehttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_equality


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Exclusive disjunction

Exclusive disjunction is an operation on two logical values , typically the

values of two propositions , that produces a value of true if one but not

both of its operands is true.

The truth table for p XOR q (also written as p q , Jpq , or p q ) is as

follows:

Exclusive Disjunction

p q p q

T T F

T F T

F T T

F F F

Source: Wikipedia
http://en.wikipedia.org/wiki/Exclusive_disjunctionhttp://en.wikipedia.org/wiki/Exclusive_disjunctionhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Logical_valuehttp://en.wikipedia.org/wiki/Logical_valuehttp://en.wikipedia.org/wiki/Logical_valuehttp://en.wikipedia.org/wiki/Propositionhttp://en.wikipedia.org/wiki/Propositionhttp://en.wikipedia.org/wiki/Propositionhttp://en.wikipedia.org/wiki/Propositionhttp://en.wikipedia.org/wiki/Logical_valuehttp://en.wikipedia.org/wiki/Logical_operationhttp://en.wikipedia.org/wiki/Exclusive_disjunction

syllogism and truth tables

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