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All the ideas for 'fragments/reports', 'Mathematics and Indispensibility' and 'Intro to Non-Classical Logic (1st ed)'

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29 ideas

1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined
Even pointing a finger should only be done for a reason [Epictetus]
     Full Idea: Philosophy says it is not right even to stretch out a finger without some reason.
     From: Epictetus (fragments/reports [c.57], 15)
     A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!).
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
Free logic is one of the few first-order non-classical logics [Priest,G]
     Full Idea: Free logic is an unusual example of a non-classical logic which is first-order.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], Pref)
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets [Priest,G]
     Full Idea: X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets, the set of all the n-tuples with its first member in X1, its second in X2, and so on.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.0)
<a,b&62; is a set whose members occur in the order shown [Priest,G]
     Full Idea: <a,b> is a set whose members occur in the order shown; <x1,x2,x3, ..xn> is an 'n-tuple' ordered set.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.10)
a ∈ X says a is an object in set X; a ∉ X says a is not in X [Priest,G]
     Full Idea: a ∈ X means that a is a member of the set X, that is, a is one of the objects in X. a ∉ X indicates that a is not in X.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2)
{x; A(x)} is a set of objects satisfying the condition A(x) [Priest,G]
     Full Idea: {x; A(x)} indicates a set of objects which satisfy the condition A(x).
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2)
{a1, a2, ...an} indicates that a set comprising just those objects [Priest,G]
     Full Idea: {a1, a2, ...an} indicates that the set comprises of just those objects.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2)
Φ indicates the empty set, which has no members [Priest,G]
     Full Idea: Φ indicates the empty set, which has no members
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.4)
{a} is the 'singleton' set of a (not the object a itself) [Priest,G]
     Full Idea: {a} is the 'singleton' set of a, not to be confused with the object a itself.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.4)
X⊂Y means set X is a 'proper subset' of set Y [Priest,G]
     Full Idea: X⊂Y means set X is a 'proper subset' of set Y (if and only if all of its members are members of Y, but some things in Y are not in X)
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6)
X⊆Y means set X is a 'subset' of set Y [Priest,G]
     Full Idea: X⊆Y means set X is a 'subset' of set Y (if and only if all of its members are members of Y).
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6)
X = Y means the set X equals the set Y [Priest,G]
     Full Idea: X = Y means the set X equals the set Y, which means they have the same members (i.e. X⊆Y and Y⊆X).
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6)
X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets [Priest,G]
     Full Idea: X ∩ Y indicates the 'intersection' of sets X and Y, which is a set containing just those things that are in both X and Y.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8)
X∪Y indicates the 'union' of all the things in sets X and Y [Priest,G]
     Full Idea: X ∪ Y indicates the 'union' of sets X and Y, which is a set containing just those things that are in X or Y (or both).
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8)
Y - X is the 'relative complement' of X with respect to Y; the things in Y that are not in X [Priest,G]
     Full Idea: Y - X indicates the 'relative complement' of X with respect to Y, that is, all the things in Y that are not in X.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8)
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
The 'relative complement' is things in the second set not in the first [Priest,G]
     Full Idea: The 'relative complement' of one set with respect to another is the things in the second set that aren't in the first.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8)
The 'intersection' of two sets is a set of the things that are in both sets [Priest,G]
     Full Idea: The 'intersection' of two sets is a set containing the things that are in both sets.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8)
The 'union' of two sets is a set containing all the things in either of the sets [Priest,G]
     Full Idea: The 'union' of two sets is a set containing all the things in either of the sets
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.8)
The 'induction clause' says complex formulas retain the properties of their basic formulas [Priest,G]
     Full Idea: The 'induction clause' says that whenever one constructs more complex formulas out of formulas that have the property P, the resulting formulas will also have that property.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.2)
A 'singleton' is a set with only one member [Priest,G]
     Full Idea: A 'singleton' is a set with only one member.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.4)
A 'member' of a set is one of the objects in the set [Priest,G]
     Full Idea: A 'member' of a set is one of the objects in the set.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2)
An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order [Priest,G]
     Full Idea: An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.10)
A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets [Priest,G]
     Full Idea: A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.10)
A 'set' is a collection of objects [Priest,G]
     Full Idea: A 'set' is a collection of objects.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.2)
The 'empty set' or 'null set' has no members [Priest,G]
     Full Idea: The 'empty set' or 'null set' is a set with no members.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.4)
A set is a 'subset' of another set if all of its members are in that set [Priest,G]
     Full Idea: A set is a 'subset' of another set if all of its members are in that set.
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6)
A 'proper subset' is smaller than the containing set [Priest,G]
     Full Idea: A set is a 'proper subset' of another set if some things in the large set are not in the smaller set
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6)
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
The empty set Φ is a subset of every set (including itself) [Priest,G]
     Full Idea: The empty set Φ is a subset of every set (including itself).
     From: Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.1.6)
7. Existence / D. Theories of Reality / 11. Ontological Commitment / e. Ontological commitment problems
All scientific tests will verify mathematics, so it is a background, not something being tested [Sober]
     Full Idea: If mathematical statements are part of every competing hypothesis, then no matter which hypothesis comes out best in the light of observations, they will be part of the best hypothesis. They are not tested, but are a background assumption.
     From: Elliott Sober (Mathematics and Indispensibility [1993], 45), quoted by Charles Chihara - A Structural Account of Mathematics
     A reaction: This is a very nice objection to the Quine-Putnam thesis that mathematics is confirmed by the ongoing successes of science.