02Traditional Logic I The Importance of Being Formal Martin Henz - PowerPoint PPT Presentation

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 3 All slack track systems are caterpillar systems. All Christie suspension systems are slack track systems. Therefore, all Christie suspension systems are caterpillar systems. Makes sense, even if you do not know anything about suspension systems. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 3 All slack track systems are caterpillar systems. All Christie suspension systems are slack track systems. Therefore, all Christie suspension systems are caterpillar systems. Makes sense, even if you do not know anything about suspension systems. Form, not content In logic, we are interested in the form of valid arguments, irrespective of any particular domain of discourse. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Categorical Terms Terms refer to sets Term animals refers to the set of animals, term brave refers to the set of brave persons, etc The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Categorical Terms Terms refer to sets Term animals refers to the set of animals, term brave refers to the set of brave persons, etc Term The set Term contains all terms under consideration The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Categorical Terms Terms refer to sets Term animals refers to the set of animals, term brave refers to the set of brave persons, etc Term The set Term contains all terms under consideration Examples animals ∈ Term brave ∈ Term The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Models Meaning A model M fixes what elements we are interested in, and what we mean by each term The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Models Meaning A model M fixes what elements we are interested in, and what we mean by each term Fix universe For a particular M , the universe U M contains all elements that we are interested in. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Models Meaning A model M fixes what elements we are interested in, and what we mean by each term Fix universe For a particular M , the universe U M contains all elements that we are interested in. Meaning of terms For a particular M and a particular term t , the meaning of t in M , denoted t M , is a particular subset of U M . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 1A For our examples, we have Term = { cats , humans , Greeks , . . . } . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 1A For our examples, we have Term = { cats , humans , Greeks , . . . } . First meaning M U M : the set of all living beings, The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 1A For our examples, we have Term = { cats , humans , Greeks , . . . } . First meaning M U M : the set of all living beings, cat M the set of all cats, The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 1A For our examples, we have Term = { cats , humans , Greeks , . . . } . First meaning M U M : the set of all living beings, cat M the set of all cats, humans M the set of all humans, . . . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 1B Consider the same Term = { cats , humans , Greeks , . . . } . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 1B Consider the same Term = { cats , humans , Greeks , . . . } . Second meaning M ′ U M ′ : A set of 100 playing cards, depicting living beings, The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 1B Consider the same Term = { cats , humans , Greeks , . . . } . Second meaning M ′ U M ′ : A set of 100 playing cards, depicting living beings, cat M ′ : all cards that show cats, The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 1B Consider the same Term = { cats , humans , Greeks , . . . } . Second meaning M ′ U M ′ : A set of 100 playing cards, depicting living beings, cat M ′ : all cards that show cats, humans M ′ : all cards that show humans, . . . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2A Consider the following set of terms: Term = { even , odd , belowfour } The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2A Consider the following set of terms: Term = { even , odd , belowfour } First meaning M 1 U M 1 = { 0 , 1 , 2 , 3 , . . . } , The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2A Consider the following set of terms: Term = { even , odd , belowfour } First meaning M 1 U M 1 = { 0 , 1 , 2 , 3 , . . . } , even M 1 = { 0 , 2 , 4 , . . . } , The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2A Consider the following set of terms: Term = { even , odd , belowfour } First meaning M 1 U M 1 = { 0 , 1 , 2 , 3 , . . . } , even M 1 = { 0 , 2 , 4 , . . . } , odd M 1 = { 1 , 3 , 5 , . . . } , and The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2A Consider the following set of terms: Term = { even , odd , belowfour } First meaning M 1 U M 1 = { 0 , 1 , 2 , 3 , . . . } , even M 1 = { 0 , 2 , 4 , . . . } , odd M 1 = { 1 , 3 , 5 , . . . } , and belowfour M 1 = { 0 , 1 , 2 , 3 } . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2B Consider the same Term = { even , odd , belowfour } The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2B Consider the same Term = { even , odd , belowfour } Second meaning M 2 U M 2 = { a , b , c , . . . , z } , The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2B Consider the same Term = { even , odd , belowfour } Second meaning M 2 U M 2 = { a , b , c , . . . , z } , even M 2 = { a , e , i , o , u } , The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2B Consider the same Term = { even , odd , belowfour } Second meaning M 2 U M 2 = { a , b , c , . . . , z } , even M 2 = { a , e , i , o , u } , odd M 2 = { b , c , d , . . . } , and The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Example 2B Consider the same Term = { even , odd , belowfour } Second meaning M 2 U M 2 = { a , b , c , . . . , z } , even M 2 = { a , e , i , o , u } , odd M 2 = { b , c , d , . . . } , and belowfour M 2 = ∅ . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Categorical Propositions All cats are predators expresses a relationship between the terms cats (subject) and predators (object). The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Categorical Propositions All cats are predators expresses a relationship between the terms cats (subject) and predators (object). Intended meaning Every thing that is included in the class represented by cats is also included in the class represented by predators . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Four Kinds of Categorical Propositions Quantity universal particular affirmative All t 1 are t 2 Some t 1 are t 2 Quality negative No t 1 are t 2 Some t 1 are not t 2 The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Four Kinds of Categorical Propositions Quantity universal particular affirmative All t 1 are t 2 Some t 1 are t 2 Quality negative No t 1 are t 2 Some t 1 are not t 2 Example Some cats are not brave is a particular , negative proposition. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Meaning of Universal Affirmative Propositions In a particular model M , All Greeks are mortal means that Greeks M is a subset of mortal M The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Meaning of Universal Affirmative Propositions In a particular model M , All Greeks are mortal means that Greeks M is a subset of mortal M The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Meaning of Universal Negative Propositions In a particular model M , No Greeks are cats means that the intersection of Greeks M and cats M is empty. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Meaning of Universal Negative Propositions In a particular model M , No Greeks are cats means that the intersection of Greeks M and cats M is empty. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Meaning of Particular Affirmative Propositions In a particular model M , Some humans are Greeks means that the intersection of humans M and Greeks M is not empty. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Meaning of Particular Affirmative Propositions In a particular model M , Some humans are Greeks means that the intersection of humans M and Greeks M is not empty. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Meaning of Particular Negative Propositions In model M , Some Greeks are not vegetarians means the difference of Greeks M and vegetarians M is not empty. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Meaning of Particular Negative Propositions In model M , Some Greeks are not vegetarians means the difference of Greeks M and vegetarians M is not empty. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Axioms Axioms are propositions that are assumed to hold. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Axioms Axioms are propositions that are assumed to hold. Axiom (HM) The proposition All humans are mortal holds. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Axioms Axioms are propositions that are assumed to hold. Axiom (HM) The proposition All humans are mortal holds. Axiom (GH) The proposition All Greeks are humans holds. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Graphical Notation [ HumansMortality ] All humans are mortal The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Lemmas Lemmas are affirmations that follow from all known facts. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Lemmas Lemmas are affirmations that follow from all known facts. Proof obligation A lemma must be followed by a proof that demonstrates how it follows from known facts. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Trivial Example of Proof Lemma The proposition All humans are mortal holds. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Trivial Example of Proof Lemma The proposition All humans are mortal holds. Proof. [ HM ] All humans are mortal The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Unusual Models We can choose any model for our terms, also “unusual” ones. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Unusual Models We can choose any model for our terms, also “unusual” ones. Example U M = { 0 , 1 } , humans M = { 0 } , mortal M = { 1 } The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Unusual Models We can choose any model for our terms, also “unusual” ones. Example U M = { 0 , 1 } , humans M = { 0 } , mortal M = { 1 } Here All humans are mortal does not hold. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Asserting Axioms Purpose of axioms By asserting an axiom A , we are focusing our attention to only those models M for which A M = T . The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Asserting Axioms Purpose of axioms By asserting an axiom A , we are focusing our attention to only those models M for which A M = T . Consequence The lemmas that we prove while utilizing an axiom only hold in the models in which the axiom holds. The Importance of Being Formal 02—Traditional Logic I

Origins and Goals Review: Agenda and Hallmarks Categorical Terms Traditional Logic Categorical Propositions and their Meaning Manipulating Terms and Propositions Axioms, Lemmas and Proofs Asserting Axioms Purpose of axioms By asserting an axiom A , we are focusing our attention to only those models M for which A M = T . Consequence The lemmas that we prove while utilizing an axiom only hold in the models in which the axiom holds. Validity A proposition is called valid , if it holds in all models. The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Review: Agenda and Hallmarks 1 Traditional Logic 2 Manipulating Terms and Propositions 3 Complement Conversion Contraposition Obversion Combinations The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Complement We allow ourselves to put non in front of a term. The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Complement We allow ourselves to put non in front of a term. Meaning of complement In a model M , the meaning of non t is the complement of the meaning of t The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Complement We allow ourselves to put non in front of a term. Meaning of complement In a model M , the meaning of non t is the complement of the meaning of t More formally In a model M , ( non t ) M = U M / t M The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Double Complement Axiom (NonNon) For any term t, the term non non t is considered equal to t. The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Double Complement Axiom (NonNon) For any term t, the term non non t is considered equal to t. · · · t · · · [ NNI ] · · · non non t · · · · · · non non t · · · [ NNE ] · · · t · · · The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Rule Schema · · · t · · · [ NNI ] · · · non non t · · · is a rule schema. An instance is: Some t 1 are t 2 Some non non t 1 are t 2 The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Definitions We allow ourselves to state definitions that may be convenient. Definitions are similar to axioms; they fix the properties of a particular item for the purpose of a discussion. The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Definitions We allow ourselves to state definitions that may be convenient. Definitions are similar to axioms; they fix the properties of a particular item for the purpose of a discussion. Definition (ImmDef) The term immortal is considered equal to the term non mortal . The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Writing a Proof Graphically Lemma The proposition All humans are non immortal holds. The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Writing a Proof Graphically Lemma The proposition All humans are non immortal holds. Proof. [ HM ] All humans are mortal [ NNI ] All humans are non non mortal [ ImmDef ] All humans are non immortal The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Writing a Text-based Proof Lemma The proposition All humans are non immortal holds. The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Writing a Text-based Proof Lemma The proposition All humans are non immortal holds. Proof. 1 HM All humans are mortal 2 NNI 1 All humans are non non mortal 3 ImmDef 2 All humans are non immortal The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Conversion switches subject and object Definition (ConvDef) For all terms t 1 and t 2 , we define convert ( All t 1 are t 2 ) = All t 2 are t 1 convert ( Some t 1 are t 2 ) = Some t 2 are t 1 convert ( No t 1 are t 2 ) = No t 2 are t 1 convert ( Some t 1 are not t 2 ) = Some t 2 are not t 1 The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Which Conversions Hold? If All Greeks are humans holds in a model, then does All humans are Greeks hold? The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Valid Conversions Axiom (ConvE1) If, for some terms t 1 and t 2 , the proposition convert ( Some t 1 are t 2 ) holds, then the proposition Some t 1 are t 2 also holds. The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations Valid Conversions Axiom (ConvE2) If, for some terms t 1 and t 2 , the proposition convert ( No t 1 are t 2 ) holds, then the proposition No t 1 are t 2 also holds. The Importance of Being Formal 02—Traditional Logic I

Complement Review: Agenda and Hallmarks Conversion Traditional Logic Contraposition Manipulating Terms and Propositions Obversion Combinations In Graphical Notation In graphical notation, two rules correspond to the two cases. convert ( Some t 1 are t 2 ) [ ConvE 1 ] Some t 1 are t 2 convert ( No t 1 are t 2 ) [ ConvE 2 ] No t 1 are t 2 The Importance of Being Formal 02—Traditional Logic I