Prior Analytics 1.1 and 1.4 and 2.23

• Demonstration
• For Aristotle, 'demonstration' is a kind of argument to a conclusion (the other kind is 'dialectic').
• Demonstration has two types:
• Inductive
• Deductive
• 'Demonstrative knowledge' is knowledge that can be proven by deduction
• 'Demonstrative knowledge' can be thought of as 'science'
  
• With the caveat that Aristotle's demonstrative knowledge did not include the 'scientific method'  as understood in modern science
• A table of contents in 1.1 of  Prior Analytics :
• 24a12ff is a sort of table of contents: the order Aristotle will follow is:
• Premiss
• Term
• Deduction
• 'Complete' deduction and 'incomplete' deduction
• Part relationship
• 'Predicated of all' and 'predicated of none'
• Premiss
• A premiss is a kind of sentence, one which affirms or denies something.
• What is typically called a 'statement' is the sort of sentence that can be a premise.
  
• A sentence becomes a 'premiss' by being used in an argument.
  
• Thus it must be a declarative sentence: we might think of it as a sort of "proposition."
  
• A premiss that is affirmative says "F is G"
• A premiss that denies is "No F is G"
• Universal premisses:
• Have the universal quantifier "Every" or "No" in them
• The forms of universal premisses are:
  
• "Every B is A" (AaB in the Cambridge Companion, p. 34)
  
• "No B is A" (AeB)
• the reasons for 'a' and 'e' (and 'i' and 'o') will become clear later. For now, just try to work with them.
  
• Partial premisses:
• Have the quantifier "Some" in them
• The forms of partial premisses are:
• "Some B is A" (AiB)
• "Some B is not A" (AoB)
• 'Indeterminate' sentences are sentences in which it is not clear whether a partial or a universal is meant.
• For example, if given just the sentence "Pleasure is not a good," Aristotle says we cannot know "pleasure" whether it refers to all or only some pleasure
• Aristotle distinguishes between dialectical premisses and demonstrative premisses
• The distinction is not one of form:
• "No B is A," for example, could be a dialectical OR a demonstrative premiss
• It becomes one or the other by being used in a dialectical or a demonstrative argument
• In dialectic:
• There are 2 sides
• One poses questions
• The other answers 'yes' or 'no'
• Thus the argument relies on what one side will concede to the other
• The rules for proof are those of deductive argument (which Aristotle is about to tell us about)
  
• In demonstration
• There are no sides
• The demonstrative argument proceeds by making claims, which are posited as true
• The rules for proof are those of deductive argument (which Aristotle is about to tell us about)
  
• The form of a premiss
• A premiss has two terms that are connected by a verb of being (e.g."is," "are," "were") and sometimes a negative ("not")
• Typical premises look like
• A is not B.
• All A's are B.
• No A is B.
• Every C is A.
  
• The terms are labeled A, B, and C  purely for convenience here.
• They could have any label: we could call them Corinthafugue, Cleisenthene, and Elagabulus if we wanted to.
• They are variables, in other words.
• The first term of a premise can be qualified by the following UNIVERSAL qualifiers:
  
• "All/every"
• "No"
• Or by the following PARTIAL qualifier
  
• "Some"
• If there are no qualifiers on the first term, the premiss is called "indeterminate."
  
• So a premiss looks like the following:
• (Some/All/No)A is/isn't B.
• Fill in whatever you want for A and B, and chose from the qualifiers and whether or not to have "not."
• Every valid deduction, according to Aristotle, has two premises and a conclusion:
• An example of a deduction (also called a syllogism):
• Premise: All A's are B.
• Premise: No B's are C.
• Conclusion: Therefore, No A's are C.
• Filling in something for the variables, we might get:
• All mammals are warm-blooded animals.
• No warm-blooded animals are one-celled.
• Therefore no mammals are one-celled.
• Let's label the premises p and q and the conclusion r: again, any name would do.
• Remember we are talking about valid deduction, not sound deduction.
• "valid" means that the logic is fine: IF the premises are true, then the conclusion MUST be true
• "sound" means that the premisses are true
  
• Aristotle speaks of complete deductions and incomplete deductions
• complete deductions are ones in which the formula "p, q, and therefore r" needs no further premises to be evidently true.
• incomplete deductions need some further s premiss, and that s premiss must be one that follows from p and q.
• I'd rather not hazard an example.
• We can think of Aristotle's deductions as using classes: we can restate each premiss and conclusion in terms of members of classes and classes of members.
• Robin Smith's Chapter 2 of Cambridge Companion, P44, has an objection to this characterization which I don't fully understand, but for our introductory purposes, it needn't bother us. Take Philosophy 13 "Logic" if you want to understand it, or perhaps certain math classes.
  
• When talking about Aristotle's syllogism:
• p is called the "major premise"
• q is called the "minor premise"
• B is called the "middle term"
  

The First Figure

• Aristotle's First Figure
• First complete Deduction (25b32-40)
  
1. First form: Barbara
   
• Major Premise: A belongs to every B
• Alternate formulation: Every B is A.
• Schema: AaB
  
• Minor Premise: B belongs to every C
  
• Alternate formulation: Every C is B.
• Schema: BaC
• Conclusion: A belongs to every C
• Alternate formulation: Every C is A
• Schema: AaC
2. Second form: Celarent
   
• Major Premise: A belongs to no B
• Alternate formulation: No B is A.
  
• Schema: AeB
  
• Minor Premise: B belongs to every C
  
• Alternate formulation: Every C is B.
• Schema: BaC
• Conclusion: A belongs to no C.
• Alternate formulation: No C is B.
  
• Schema: AeC
• Examples
• First Form: Barbara
  
• Every Human is a primate.
• Every primate is an animal.
• Therefore, every human is an animal.
• Second Form: Celarent
  
• No human has gills.  OR  Being gilled belongs to no human.
  
• Being gilled belongs to every fish.  OR  Every fish has gills.
  
• Therefore no human is a fish.
• First case with no deduction possible (26a1-9)
3. Aristotle wants to claim that two premises of the form AaB and BeC lead to no conclusion. He shows this by showing that 2 different sets of content for A,B, and C lead to schematically incompatible conclusions.
   
• First set of content: Animal-Human-Horse
• Schema
• AaB
• BeC
• 
  
• Spelled out
  
• Being an animal belongs to every human.  OR  Every human is an animal.
• Being Human belongs to no horse.  OR No horse is human.
  
• 
  
• Note that both premises and the "conclusion" are true.
  
• Second set of content: Animal-Human-Stone
• Schema
• AaB
• BeC
• 
  
• Spelled out
• Being an animal belongs to every human. / Every human is an animal.
  
• Being a human belongs to no stone. No stone is a human.
  
• 
  
• Note that once again, both premises and the "conclusion" are true.
• WHY is it not a valid form?
  
• because IF there is a valid form of argument from the two premises AaB and BeC, the FORM of the conclusion must be AaC or AeC or AiC or AoC (there are no other forms for the conclusion to take)
  
• The FORM of the conclusion cannot be either of the following 2 forms:
  
• AeC (A belongs to no C/No C is A)
• AoC (A does not belong to some C / Some C is not A)
• BECAUSE animal-human-horse is compatible with AaC (which is incompatible with AeC and AoC)
• it cannot both be the case that AaC (Being an animal belongs to every horse) and that AeC (Being an animal does not belong to every horse)
  
• it also cannot both be the case that AaC (being an animal belongs to every horse) and that AoC (some horse is not an animal)
  
• Likewise,
  
• The FORM of the conclusion cannot be either of the following 2 forms:
• AaC
• AiC (A belongs to some C / Some C is A)
• BECAUSE animal-human stone is compatible with AeC (which is incompatible with AaC and AiC).
• Thus Aristotle has shown by means of two sets of terms that it is impossible for any form of conclusion to be validly proven by premises that take the forms AaB and BeC.
  
• Second case with no deduction possible (26a10-13)
4. Two premises of the form AeB, BeC:
• Fill them out with the following two sets of terms as we did with the previous case
  
• for a combination of three terms that has a positive relation between A and C, take science, geometrical line, and the discipline of medicine
• every geometric line (B) is not a science (A): or alternatively formulated it is not the case that being a science belongs to every geometric line
  
• every discipline of medicine (C) is not a geometric line (B): or alternatively formulated it is not the case that being a geometric line belongs to every discipline of medicine
• every discipline of medicine is a science
  
• for a combination of three terms that has a negative relation between A and C, take science, line, unit (as of distance)
• every geometric line is not a science
• every unit of distance is not a line
• every unit of distance is not a science
  
• Consider all the possible forms of conclusion (AaC, AeC, AiC, AoC) in the same manner as the previous case.
  
• There is no valid form of conclusion from AeC, BeC is the conclusion.
• Aristotle now pauses to say that he has considered all the "universal" options for premises: that is, ones that take the form "Every X is ..." and "No X is ...". There are four such options, two of which have valid conclusions, and two of which do not. (26a14-15)
  
• Then he says that he will continue to consider the "partial" options: that is, the ones that have "some" in their premises. (26a15-16)
• I will not go thru them all, but it is clear from reading them that he goes thru them in a systematic fashion and shows:
• which have valid conclusions
5. AaB and BiC, therefore AiC: Darii
   
6. AeB and BiC, therefore AoC: Ferio
   
• and which do not
7. AiB and BaC (use good, state, wisdom as terms and apply the schema above to prove that there is no valid form of conclusion to yourself)
   
8. AoB and BaC (use good, state, wisdom as terms and apply the schema above to prove that there is no valid form of conclusion to yourself)
9. AiB and BeC (use white, horse, and swan as terms and apply the schema above to prove that there is no valid form of conclusion to yourself)
10. AoB and BeC (use white, horse, raven) as terms and apply the schema above to prove that there is no valid form of conclusion to yourself)
11. AaB and BoC (26b1ff)
12. AeB and BoC (26b1ff.)
13. AiB and BiC (26b22ff.)
    
14. AiB and BoC (26b22ff.)
15. AoB and BiC (26b22ff.)
16. AoB and BoC (26b22ff.)

At this point, Aristotle has gone thru all 16 possible configurations that have the following pattern (called the first figure):
A is the predicate and B is the subject of the first premise.
B is the predicate and C is the subject of the second premise.

There are two more patterns of subject-predicate relations (See P. 35 of Cambridge Companion).