Sep 4-8 - Plato and Aristotle’s Logic

1. Plato’s Meno

• According to Plato, knowledge requires certainty, and certainty requires a derivation by an infallible method – a method that only ever results in the truth.

• What infallible method does he propose?

• Meno: “… what do you mean by saying that we do not learn, and that what we call learning is only a process of recollection? …”

• Does a side of twice the length give a square of twice the area?

• What side will give a square of twice the area?

• Socrates: “Do you observe, Meno, that I am not teaching the boy anything, but only asking him questions?”

• Is that a convincing answer to Meno’s question? Does it demonstrate the existence of an infallible method? Do we have an answer to Glymour’s third question?

2. Aristotle

• Subject-predicate structure of language mirrors the matter-form structure of reality:

Language	↔	Reality
		matter +
The student	↔	student-form = student +
The student is brilliant	↔	brilliance-form = brilliant student +
The brilliant student is late	↔	lateness = late brilliant student +
…	↔	…

• Remember Glymour’s third question for a theory of valid arguments – we have here the makings of an answer.

• TTT p42 Q1 – how does this example demonstrate that there is more to natural language than the subject-predicate form? What does that indicate about Aristotle’s project?

3. Syllogisms

• A form of argument for reasoning about types of things, given the subject-predicate picture of language and reality. Hence they are suitable for knowledge of generalisations – science.

• ‘Types’ of things, since syllogisms are not arguments about specific, named things, but involve quantifiers – all, some, none, not all.

• E.g., ‘Everything with property/form A has property/form B’ = ‘All As are Bs’

• E.g., ‘All cats are tabbies’

• Examples:

(i)	All As are Bs
	No Bs are Cs
	∴ No As are Cs

A valid form of argument – any concrete argument with A, B and C replaced by predicates is such that the premises necessitate the conclusion (infallibly).

Need such a concrete instance be sound?

(ii)	Some As are Bs
	Some Bs are Cs
	∴ Some As are Cs

Not a valid argument form because there are concrete arguments of this form with true premises and false conclusions – counter examples (e.g.?)

Hence having true premises and this form is not an infallible guarantee that the conclusion of an argument is true.

• TTT p49 Q1-2.

4. Aristotle’s Logic of Syllogisms

• A formal logic = a set of criteria for valid arguments, formulated in terms of general forms of language.

• Remember Glymour’s first question for a theory of valid arguments – we have here an answer.

• The system – TTT p50

• E.g., prove that Celarent and Ferio are valid syllogism forms.

	Syllogism
Second vowel – A, E, I, O	Quantifier As are Bs	Premise 1: Minor and middle terms
First vowel – A, E, I, O	Quantifier Bs are Cs	Premise 2: Major and middle terms
Third vowel – A, E, I, O	∴ Quantifier As are Cs	Conclusion: Major and minor terms

All	=	A
No/None	=	E
Some/At least one	=	I
Not all	=	O

• E.g., What would ‘Davidson’ of the 3rd figure (see TTT p50, Table 2.1) be?

• Is ‘Davidson’ a valid syllogism form? Give an abstract counter example in terms of set diagrams and a concrete counter example.

• TTT p52 Q1

• Proving the validity of syllogisms using given syllogisms: TTT p51

Rule 1	No As are Bs	⇒	No Bs are As
Rule 2	All As are Bs	⇒	Some As are Bs
Rule 3	Some As are Bs	⇒	Some Bs are As

• TTT p52 Q2

• How does Aristotle’s system do in addressing Glymour’s three questions?