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PROPOSITIONAL LOGIC
FORMAL ANALYSIS Formal logic: evaluate the validity of argument based upon its form NOT the content of its premises and conclusion Much like math, variables take the place of statements and we deal solely with the variables Propositional logic: system of formal logic in which we can take simple atomic propositions and build more complex arguments
Propositional logic uses 2 main building blocks: propositions and propositional connectives Propositions: statement that is either true or false (has a truth value) “Atomic” without propositional connectives Propositional connectives: Used to connect smaller propositions into larger ones Very similar to mathematical connectives: */-+ Larger propositions that include connectives also have a truth value PROPOSITIONAL LOGIC
CONNECTIVES Conjunction, disjunction, negation, conditional & biconditional Each connective is governed by its own truth conditions (conditions under which propositions that include the connective are true) We can discover the truth conditions of non-atomic propositions that include many connectives through the use of truth tables Each connective has its own truth table
VARIABLES Replace propositions in English with variables that can stand in for any proposition Propositions, once replaced by variables, are put in propositional forms Propositional form: a pattern that can represent any number of actual propositions Example: p&q is a propositional form in which “p” and “q” can stand for any proposition Substitution instance: replace variables by actual propositions – many possible sub. instances for each prop. form
RULES Each proposition can be replaced by one or several variables in a series of propositional forms (argument) but each variable must represent the same proposition throughout P & Q can both represent the same proposition but P cannot represent two different propositions within the same series/argument Variables can represent atomic propositions or more complex ones that, themselves, include connectives
ARGUMENT FORMS Once we have propositional forms, we can combine them into argument forms Argument form: offers a pattern of argument that we is always valid pattern for any number of arguments Example: 1) p&q 	   2) p
ARGUMENT FORMS An argument is valid IF it is a valid argument form Note: not all valid arguments are so in virtue of their argument form – here we offer a sufficient, but not necessary, condition for validity An argument form is valid IF AND ONLY IF it has no substitution instances in which the premises are true and the conclusion false
CONJUNCTION Propositional conjunction: [while still in English] “and” expresses the conjunction of two or more propositions (called “conjuncts”) Non-propositional conjunction: “and” does not express the conjunction of two or more propositions Test: can you separate the proposition into two separate conjuncts without changing the meaning of the sentence?
DISJUNCTION Propositional connective: “or” Unless specified, meant as an inclusive “or” Exclusive “or” generally implied by context and not the actual proposition itself
NEGATION Propositional connective: “not” The negation of a proposition is true if and only if the proposition is false and vice versa “Not” is tricky so a good test for whether a proposition is an instance of negation is to reformulate the sentence so it starts with “It is not the case that x” … if it is possible without affecting the meaning of the sentence, it is probably an instance of negation

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Logic part1

  • 2. FORMAL ANALYSIS Formal logic: evaluate the validity of argument based upon its form NOT the content of its premises and conclusion Much like math, variables take the place of statements and we deal solely with the variables Propositional logic: system of formal logic in which we can take simple atomic propositions and build more complex arguments
  • 3. Propositional logic uses 2 main building blocks: propositions and propositional connectives Propositions: statement that is either true or false (has a truth value) “Atomic” without propositional connectives Propositional connectives: Used to connect smaller propositions into larger ones Very similar to mathematical connectives: */-+ Larger propositions that include connectives also have a truth value PROPOSITIONAL LOGIC
  • 4. CONNECTIVES Conjunction, disjunction, negation, conditional & biconditional Each connective is governed by its own truth conditions (conditions under which propositions that include the connective are true) We can discover the truth conditions of non-atomic propositions that include many connectives through the use of truth tables Each connective has its own truth table
  • 5. VARIABLES Replace propositions in English with variables that can stand in for any proposition Propositions, once replaced by variables, are put in propositional forms Propositional form: a pattern that can represent any number of actual propositions Example: p&q is a propositional form in which “p” and “q” can stand for any proposition Substitution instance: replace variables by actual propositions – many possible sub. instances for each prop. form
  • 6. RULES Each proposition can be replaced by one or several variables in a series of propositional forms (argument) but each variable must represent the same proposition throughout P & Q can both represent the same proposition but P cannot represent two different propositions within the same series/argument Variables can represent atomic propositions or more complex ones that, themselves, include connectives
  • 7. ARGUMENT FORMS Once we have propositional forms, we can combine them into argument forms Argument form: offers a pattern of argument that we is always valid pattern for any number of arguments Example: 1) p&q 2) p
  • 8. ARGUMENT FORMS An argument is valid IF it is a valid argument form Note: not all valid arguments are so in virtue of their argument form – here we offer a sufficient, but not necessary, condition for validity An argument form is valid IF AND ONLY IF it has no substitution instances in which the premises are true and the conclusion false
  • 9. CONJUNCTION Propositional conjunction: [while still in English] “and” expresses the conjunction of two or more propositions (called “conjuncts”) Non-propositional conjunction: “and” does not express the conjunction of two or more propositions Test: can you separate the proposition into two separate conjuncts without changing the meaning of the sentence?
  • 10. DISJUNCTION Propositional connective: “or” Unless specified, meant as an inclusive “or” Exclusive “or” generally implied by context and not the actual proposition itself
  • 11. NEGATION Propositional connective: “not” The negation of a proposition is true if and only if the proposition is false and vice versa “Not” is tricky so a good test for whether a proposition is an instance of negation is to reformulate the sentence so it starts with “It is not the case that x” … if it is possible without affecting the meaning of the sentence, it is probably an instance of negation