Categorical Terms and their Meaning Propositions, Axioms, Lemmas, Proofs Manipulating Terms and Propositions Arguments and Syllogisms 02—Traditional Logic CS 3234: Logic and Formal Systems Martin Henz and Aquinas Hobor August 19, 2010 Generated on Wednesday 18 th August, 2010, 23:10 CS 3234: Logic and Formal Systems 02—Traditional Logic
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Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
02—Traditional Logic
CS 3234: Logic and Formal Systems
Martin Henz and Aquinas Hobor
August 19, 2010
Generated on Wednesday 18th August, 2010, 23:10
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
1 Categorical Terms and their Meaning
2 Propositions, Axioms, Lemmas, Proofs
3 Manipulating Terms and Propositions
4 Arguments and Syllogisms
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
1 Categorical Terms and their MeaningOrigins and GoalsForm, not ContentCategorical TermsMeaning through models
2 Propositions, Axioms, Lemmas, Proofs
3 Manipulating Terms and Propositions
4 Arguments and Syllogisms
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
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.
GoalExpress relationships between sets; allow reasoning about setmembership
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 1
All humans are mortal.All Greeks are humans.Therefore, all Greeks are mortal.
Makes “sense”, right?
Why?
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Example 2
All cats are predators.Some animals are cats.Therefore, all animals are predators.
Does not make sense!
Why not?
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
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.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
Categorical Terms
Terms refer to setsTerm animals refers to the set of animals,term brave refers to the set of brave persons, etc
Terms
The set Terms contains all terms under consideration
Examples
animals ∈ Terms
brave ∈ Terms
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
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.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
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,. . .
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
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,. . .
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Origins and GoalsForm, not ContentCategorical TermsMeaning through models
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.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
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
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
Lemma (AllNonNon)For any terms t1 and t2, the proposition All non t1 are nont2 holds iff the proposition All t2 are t1 holds.
All non t1 are non t2
All t2 are t1
All t2 are t1
All non t1 are non t2
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
1 Categorical Terms and their Meaning
2 Propositions, Axioms, Lemmas, Proofs
3 Manipulating Terms and Propositions
4 Arguments and SyllogismsArgumentsSyllogismsBarbaraFun With Barbara
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Argument
An argument has the form
If premises then conclusion
Sometimes also
premises therefore conclusion
Example:
Lemma (SomeNon)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.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Syllogisms
A syllogism is an argument with two premises, in which threedifferent terms occur, and in which every term occurs twice, butnever twice in the same proposition.
ExampleAll cats are predators.Some animals are cats.Therefore, all animals are predators.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Barbara
Axiom (B)For all terms minor, middle, and major, if All middle are majorholds, and All minor are middle holds, then All minor aremajor also holds.
All middle are major All minor are middle
All minor are major[B]
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Why is Barbara valid?
If the first premise holds, then areas 1 and 4 are empty, and ifthe second premise holds, then areas 2 and 3 are empty. Theconclusion simply states that areas 1 and 2 are empty, whichclearly follows from the premises, regardless what otherproperties the model under consideration has.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Example
LemmaThe proposition All Greeks are mortal holds.
Proof.
1 All Greeks are humans GH2 All humans are mortal HM3 All Greeks are mortal B 1,2
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Officers as Poultry?
PremisesNo ducks waltz.No officers ever decline to waltz.All my poultry are ducks.
ConclusionNo officers are my poultry.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Formulation in Term Logic
Lemma (No-Officers-Are-My-Poutry)If
No ducks are things-that-waltz holds,No officers are non things-that-waltz holds,andAll my-poutry are ducks holds,
then No officers are my-poultry also holds.
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Proof
1 No officers are nonthings-that-waltz
premise
2 obvert(All officers arethings-that-waltz)
ObvDef 1
3 All officers arethings-that-waltz)
ObvE 2
4 No ducks arethings-that-waltz)
premise
5 convert(No things-that-waltzare ducks)
ConvDef 4
6 No things-that-waltz areducks
ConvE2 5
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Proof (continued)
7 No things-that-waltz are nonnon ducks
NNI 6
8 obvert(All things-that-waltzare non ducks)
ObvDef 7
9 All things-that-waltz arenon ducks
ObvE 8
10 All my-poultry are ducks premise11 All my-poultry are non non
ducksNNI 10
12 All non non my-poultry arenon non ducks
NNI 11
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Proof (continued)
13 contrapose(All non ducks arenon my-poultry)
ContrDef 12
14 All non ducks are nonmy-poultry
ContrE1 13
15 All things-that-waltz arenon my-poultry
B 9,14
16 All officers are nonmy-poultry
B 3,15
17 obvert(No officers aremy-poultry)
ObvDef 16
18 No officers are my-poultry ObvE 17
CS 3234: Logic and Formal Systems 02—Traditional Logic
Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs
Manipulating Terms and PropositionsArguments and Syllogisms
ArgumentsSyllogismsBarbaraFun With Barbara
Admin
Assignment 1: out on module homepage; due 26/8,11:00amCoq Homework 1: out on module homepage; due 27/8,9:30pmMonday, Wednesday: Office hoursTuesday: Tutorials (clarification of assignment)Wednesday: Labs (Coq Homework 1; start earlier!)
CS 3234: Logic and Formal Systems 02—Traditional Logic