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Propositional Logic - Connectives and Truth Tables

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Roberto Ruiz

on 2 December 2012

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Transcript of Propositional Logic - Connectives and Truth Tables

Propositional Logic Connectives and
Truth Tables Basic Truth Tables Want more? Conjunction Compound (or complex) Propositions Variables
and Connectives In propositional logic we use symbols to stand for statements and for the relationships between statements. 1) A compound statement has entire statements as its components. To learn how to apply these concepts to
complex arguments, click on the following link: The symbols that stand for statements are called variables. The symbols that stand for the relationships between statements are called connectives. And because the validity of a deductive argument depends exclusively on the form of the argument, propositional logic helps us assess the validity of an argument without having to get distracted by non-formal elements of the argument, such as its content or the language used to express that content. Because these connectives specify the relationships between statements, they determine the form of the argument. 2) The component statements must be coherent in themselves. 3) The compound statement must remain sensible after one (or both) of its components are replaced. 4) The truth of the compound proposition is dependent on the truth of its logical components. Simple: Compound: The sun is shining. The sun is shining and
the birds are singing. Simple: Compound: The people who supported Mary
and Jessica were their families. The sun is shining and
the birds are singing. Simple: Compound: The child who ate the cookie
was sweet. The child who ate ice is
frozen water. "He envisions a world
where humans are immortal." "The sun is shining and
the birds are singing." The truth of the statement does not depend on
whether its "component" is true or false. The truth of the compound statement depends on whether both statements are true. Symbols for Connectives Conjunction Disjunction Negation Conditional Connective Meaning Symbol Common Indicators and or not & v ~ if it is not the case that, it is false that but, while, also, yet, moreover unless, either or, neither nor if ... then, only if, provided, unless, whenever p q p & q

T T

T F

F T

F F T F F F The only way in which a conjunction can be true is when both conjunctive terms are true If just one statement in a conjunction is false, the whole conjunction is false. Disjunction p q p v q

T T

T F

F T

F F T T T F If just one statement in a disjunction is true, the whole disjunction is true. The only way in which a disjunction can be false is when both disjunctive terms are false. Negation p p

T

F T F A negation reverses the truth value of a statement. Conditional p q p > q

T T

T F

F T

F F T F T T The only way in which a conditional can be false is when the antecedent is true and the consequent is false. For instance, suppose that
p = it's raining, and
q = it's wet ~ If a statement is true, its negation is false. If a statement is false, its negation is true. A double negation is the same thing as no negation. The only time in which the conditional p > q would have to be false is when we claim that
it's raining, but it's not wet. In a conditional statement,
the following are some common indicators to introduce the antecedent: In a conditional statement,
the following are some common indicators to introduce the consequent: if,

provided,

unless,

whenever then,

only if A conjunction is formed by putting "&" between two conjunctive terms. Not a Conjunction: Conjunction: Mary and Jessica are sisters. Mary is funny, while Jessica is witty. A note on the grammar of conjunctions If the conjuncts share a common subject or common predicate, we may omit stating the common term twice in the conjunction. Alvin is a musician and Alvin is an athlete. Alvin is a musician and James is a musician. Alvin is a musician and an athlete. Alvin and James are musicians. A disjunction is formed by putting "v" between two disjunctive terms. Inclusive Disjunction: Exclusive Disjunction: You may buy bread or (you may buy) potatoes [or both]. Either one is good or one is evil [but not both]. A note on the grammar of disjunctions Like conjunctions, when disjuncts share a common subject or predicate, we may omit stating the common term twice in the disjunction. Alvin is a musician or Alvin is an alien. Alvin is the musician or James is the musician. Alvin is either a musician or an alien. Either Alvin or James is the musician. Antecedent Consequent http://berto-meister.blogspot.com/2012/11/propositional-logic-using-truth-tables.html Check out a quick introduction to conditional statements
from Kevin deLaplante's great website
Critical Thinking Academy:

http://www.criticalthinkeracademy.com/academy/basic-concepts-in-propositional-logic/conditionals/ Keep in mind that,
in a conditional statement,
the indicator "unless"
introduces the antecedent,
and negates it. For instance, the statement:
"I will be late to class unless I drive"
would get translated to the if-then statement
"If I do NOT drive, I will be late to class." As we have seen before, the truth value of a compound statement depends on the truth value of its component parts. Truth tables are logical representations
of every possible combination of the truth value
of different propositions in order to help us
determine the truth value
of the compound statements
of which they are a part. So what are truth tables?
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