Review: Agenda and Hallmarks Traditional Logic Manipulating Terms and Propositions 02—Traditional Logic I The Importance of Being Formal Martin Henz January 22, 2014 Generated on Wednesday 22 nd January, 2014, 09:51 The Importance of Being Formal 02—Traditional Logic I
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Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
02—Traditional Logic I
The Importance of Being Formal
Martin Henz
January 22, 2014
Generated on Wednesday 22nd January, 2014, 09:51
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
1 Review: Agenda and Hallmarks
2 Traditional Logic
3 Manipulating Terms and Propositions
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
1 Review: Agenda and Hallmarks
2 Traditional Logic
3 Manipulating Terms and Propositions
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
The Importance of Being Formal
First AgendaFind out in detail how formal systems work
GoalThorough understanding of formal logic as an example parexcellence for formal methods
ApproachStudy a series of logics: traditional, propositional, predicatelogic
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
The Importance of Being Formal
Second AgendaExplore fundamental boundaries of formal reasoning
GoalAppreciate Undecidability and Godel’s incompleteness results
ApproachStudy predicate logic deep enough to understand his formalarguments
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
The Importance of Being Formal
Third AgendaExplore formal methods across fields
ApproachStudents write essays and present their findings
GoalOverview of formal methods and their limitations in ourcivilization
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Hallmarks of Formal Methods
DiscretenessNamingAbstraction (classification)ReificationSelf-referenceForm vs contentSyntax vs semantics
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
1 Review: Agenda and Hallmarks
2 Traditional LogicOrigins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
3 Manipulating Terms and Propositions
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Traditional Logic
Origins
Greek philosopher Aristotle (384–322 BCE) wrote treatise PriorAnalytics; considered the earliest study in formal logic; widelyaccepted as the definite approach to deductive reasoning untilthe 19thcentury.
GoalFormalize relationships between sets; allow reasoning aboutset membership
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Example 1
All humans are mortal.All Greeks are humans.Therefore, all Greeks are mortal.
Makes “sense”, right?
Why?
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Example 2
All cats are predators.Some animals are cats.Therefore, all animals are predators.
Does not make sense!
Why not?
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Example 3
All slack track systems are caterpillar systems.All Christie suspension systems are slack tracksystems.Therefore, all Christie suspension systems arecaterpillar systems.
Makes sense, even if you do not know anything aboutsuspension systems.
Form, not contentIn 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
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Categorical Terms
Terms refer to setsTerm 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
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Models
MeaningA modelM fixes what elements we are interested in, and whatwe mean by each term
Fix universe
For a particularM, the universe UM contains all elements thatwe are interested in.
Meaning of termsFor a particularM and a particular term t , the meaning of t inM, denoted tM, is a particular subset of UM.
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Example 1A
For our examples, we haveTerm = {cats,humans,Greeks, . . .}.
First meaningMUM: the set of all living beings,catM the set of all cats,humansM the set of all humans,. . .
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Example 1B
Consider the same Term = {cats,humans,Greeks, . . .}.
Second meaningM′
UM′: A set of 100 playing cards, depicting living beings,
catM′: all cards that show cats,
humansM′: all cards that show humans,
. . .
The Importance of Being Formal 02—Traditional Logic I
Review: Agenda and HallmarksTraditional Logic
Manipulating Terms and Propositions
Origins and GoalsCategorical TermsCategorical Propositions and their MeaningAxioms, Lemmas and Proofs
Example 2A
Consider the following set of terms:Term = {even,odd,belowfour}
We allow ourselves to state definitions that may be convenient.Definitions are similar to axioms; they fix the properties of aparticular item for the purpose of a discussion.
Definition (ImmDef)The term immortal is considered equal to the term nonmortal.
The Importance of Being Formal 02—Traditional Logic I
For all terms t1 and t2, if the proposition Some non t1 are nont2 holds, then the proposition Some non t2 are not t1 alsoholds.
Proof.
1 Some non t1 are non t2 premise2 convert(Some non t2 are non t1) ConvDef 13 Some non t2 are non t1 ConvE1 24 obvert(Some non t2 are not t1) ObvDef 35 Some non t2 are not t1 ObvE 4
The Importance of Being Formal 02—Traditional Logic I