Combining Texts

All the ideas for 'On Interpretation', 'Set Theory and Its Philosophy' and 'The Will to Believe'

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34 ideas

2. Reason / B. Laws of Thought / 4. Contraries
In "Callias is just/not just/unjust", which of these are contraries? [Aristotle]
     Full Idea: Take, for example, "Callias is just", "Callias is not just", and "Callias is unjust"; which of these are contraries?
     From: Aristotle (On Interpretation [c.330 BCE], 23a31)
3. Truth / B. Truthmakers / 10. Making Future Truths
It is necessary that either a sea-fight occurs tomorrow or it doesn't, though neither option is in itself necessary [Aristotle]
     Full Idea: It is not necessary for a sea-battle to take place tomorrow, nor for one not to take place tomorrow - though it is necessary for one to take place OR not take place tomorrow.
     From: Aristotle (On Interpretation [c.330 BCE], 19a30)
3. Truth / C. Correspondence Truth / 1. Correspondence Truth
Statements are true according to how things actually are [Aristotle]
     Full Idea: Statements are true according to how things actually are.
     From: Aristotle (On Interpretation [c.330 BCE], 19a33)
4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
Aristotle's later logic had to treat 'Socrates' as 'everything that is Socrates' [Potter on Aristotle]
     Full Idea: When Aristotle moved from basic name+verb (in 'De Interpretatione') to noun+noun logic...names had to be treated as special cases, so that 'Socrates' is treated as short for 'everything that is Socrates'.
     From: comment on Aristotle (On Interpretation [c.330 BCE]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 02 'Supp'
     A reaction: Just the sort of rewriting that Russell introduced for definite descriptions. 'Twas ever the logicians' fate to shoehorn ordinary speech into awkward containers.
Square of Opposition: not both true, or not both false; one-way implication; opposite truth-values [Aristotle]
     Full Idea: Square of Opposition: horizontals - 'contraries' can't both be true, and 'subcontraries' can't both be false; verticals - 'subalternatives' have downwards-only implication; diagonals - 'contradictories' have opposite truth values.
     From: Aristotle (On Interpretation [c.330 BCE], Ch.12-13)
     A reaction: This is still used in modern discussion (e.g. by Stalnaker against Kripke), and there is a modal version of it (Fitting and Mendelsohn p.7). Corners read: 'All F are G', 'No F are G', 'Some F are G' and 'Some F are not G'.
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
Modal Square 1: □P and ¬◊¬P are 'contraries' of □¬P and ¬◊P [Aristotle, by Fitting/Mendelsohn]
     Full Idea: Modal Square of Opposition 1: 'It is necessary that P' and 'It is not possible that not P' are the contraries (not both true) of 'It is necessary that not P' and 'It is not possible that P'.
     From: report of Aristotle (On Interpretation [c.330 BCE], Ch.12a) by M Fitting/R Mendelsohn - First-Order Modal Logic 1.4
Modal Square 2: ¬□¬P and ◊P are 'subcontraries' of ¬□P and ◊¬P [Aristotle, by Fitting/Mendelsohn]
     Full Idea: Modal Square of Opposition 2: 'It is not necessary that not P' and 'It is possible that P' are the subcontraries (not both false) of 'It is not necessary that P' and 'It is possible that not P'.
     From: report of Aristotle (On Interpretation [c.330 BCE], Ch.12b) by M Fitting/R Mendelsohn - First-Order Modal Logic 1.4
Modal Square 3: □P and ¬◊¬P are 'contradictories' of ¬□P and ◊¬P [Aristotle, by Fitting/Mendelsohn]
     Full Idea: Modal Square of Opposition 3: 'It is necessary that P' and 'It is not possible that not P' are the contradictories (different truth values) of 'It is not necessary that P' and 'It is possible that not P'.
     From: report of Aristotle (On Interpretation [c.330 BCE], Ch.12c) by M Fitting/R Mendelsohn - First-Order Modal Logic 1.4
Modal Square 4: □¬P and ¬◊P are 'contradictories' of ¬□¬P and ◊P [Aristotle, by Fitting/Mendelsohn]
     Full Idea: Modal Square of Opposition 4: 'It is necessary that not P' and 'It is not possible that P' are the contradictories (different truth values) of 'It is not necessary that not P' and 'It is possible that P'.
     From: report of Aristotle (On Interpretation [c.330 BCE], Ch.12d) by M Fitting/R Mendelsohn - First-Order Modal Logic 1.4
Modal Square 5: □P and ¬◊¬P are 'subalternatives' of ¬□¬P and ◊P [Aristotle, by Fitting/Mendelsohn]
     Full Idea: Modal Square of Opposition 5: 'It is necessary that P' and 'It is not possible that not P' are the subalternatives (first implies second) of 'It is not necessary that not P' and 'It is possible that P'.
     From: report of Aristotle (On Interpretation [c.330 BCE], Ch.12e) by M Fitting/R Mendelsohn - First-Order Modal Logic 1.4
Modal Square 6: □¬P and ¬◊P are 'subalternatives' of ¬□P and ◊¬P [Aristotle, by Fitting/Mendelsohn]
     Full Idea: Modal Square of Opposition 6: 'It is necessary that not P' and 'It is not possible that P' are the subalternatives (first implies second) of 'It is not necessary that P' and 'It is possible that not P'.
     From: report of Aristotle (On Interpretation [c.330 BCE], Ch.12f) by M Fitting/R Mendelsohn - First-Order Modal Logic 1.4
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter]
     Full Idea: Set theory has three roles: as a means of taming the infinite, as a supplier of the subject-matter of mathematics, and as a source of its modes of reasoning.
     From: Michael Potter (Set Theory and Its Philosophy [2004], Intro 1)
     A reaction: These all seem to be connected with mathematics, but there is also ontological interest in set theory. Potter emphasises that his second role does not entail a commitment to sets 'being' numbers.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Usually the only reason given for accepting the empty set is convenience [Potter]
     Full Idea: It is rare to find any direct reason given for believing that the empty set exists, except for variants of Dedekind's argument from convenience.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.3)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Infinity: There is at least one limit level [Potter]
     Full Idea: Axiom of Infinity: There is at least one limit level.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.9)
     A reaction: A 'limit ordinal' is one which has successors, but no predecessors. The axiom just says there is at least one infinity.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
Nowadays we derive our conception of collections from the dependence between them [Potter]
     Full Idea: It is only quite recently that the idea has emerged of deriving our conception of collections from a relation of dependence between them.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.2)
     A reaction: This is the 'iterative' view of sets, which he traces back to Gödel's 'What is Cantor's Continuum Problem?'
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter]
     Full Idea: We group under the heading 'limitation of size' those principles which classify properties as collectivizing or not according to how many objects there are with the property.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 13.5)
     A reaction: The idea was floated by Cantor, toyed with by Russell (1906), and advocated by von Neumann. The thought is simply that paradoxes start to appear when sets become enormous.
4. Formal Logic / G. Formal Mereology / 1. Mereology
Mereology elides the distinction between the cards in a pack and the suits [Potter]
     Full Idea: Mereology tends to elide the distinction between the cards in a pack and the suits.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1)
     A reaction: The example is a favourite of Frege's. Potter is giving a reason why mathematicians opted for set theory. I'm not clear, though, why a pack cannot have either 4 parts or 52 parts. Parts can 'fall under a concept' (such as 'legs'). I'm puzzled.
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
We can formalize second-order formation rules, but not inference rules [Potter]
     Full Idea: In second-order logic only the formation rules are completely formalizable, not the inference rules.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 01.2)
     A reaction: He cites Gödel's First Incompleteness theorem for this.
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
In talking of future sea-fights, Aristotle rejects bivalence [Aristotle, by Williamson]
     Full Idea: Unlike Aristotle, Stoics did not reject Bivalence for future contingencies; it is true or false that there will be a sea-fight tomorrow.
     From: report of Aristotle (On Interpretation [c.330 BCE], 19a31) by Timothy Williamson - Vagueness 1.2
     A reaction: I'd never quite registered this simple account of the sea-fight. As Williamson emphasises, one should not lightly reject the principle of bivalence. Has Aristotle entered a slippery slope? Stoics disagreed with Aristotle.
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
A prayer is a sentence which is neither true nor false [Aristotle]
     Full Idea: A prayer is a sentence which is neither true nor false.
     From: Aristotle (On Interpretation [c.330 BCE], 17a01)
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter]
     Full Idea: A 'supposition' axiomatic theory is as concerned with truth as a 'realist' one (with undefined terms), but the truths are conditional. Satisfying the axioms is satisfying the theorem. This is if-thenism, or implicationism, or eliminative structuralism.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 01.1)
     A reaction: Aha! I had failed to make the connection between if-thenism and eliminative structuralism (of which I am rather fond). I think I am an if-thenist (not about all truth, but about provable truth).
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
If set theory didn't found mathematics, it is still needed to count infinite sets [Potter]
     Full Idea: Even if set theory's role as a foundation for mathematics turned out to be wholly illusory, it would earn its keep through the calculus it provides for counting infinite sets.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.8)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter]
     Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms).
     From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2)
     A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying.
7. Existence / A. Nature of Existence / 3. Being / e. Being and nothing
Non-existent things aren't made to exist by thought, because their non-existence is part of the thought [Aristotle]
     Full Idea: It is not true to say that what is not, since it is thought about, is something that is; for what is thought about it is not that it is, but that it is not.
     From: Aristotle (On Interpretation [c.330 BCE], 21a31)
     A reaction: At least there has been one philosopher who was quite clear about the distinction between a thought and what the thought is about (its content). Often forgotten!
7. Existence / A. Nature of Existence / 5. Reason for Existence
Maybe necessity and non-necessity are the first principles of ontology [Aristotle]
     Full Idea: Perhaps the necessary and non-necessary are first principles of everything's either being or not being.
     From: Aristotle (On Interpretation [c.330 BCE], 23a18)
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
A relation is a set consisting entirely of ordered pairs [Potter]
     Full Idea: A set is called a 'relation' if every element of it is an ordered pair.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 04.7)
     A reaction: This is the modern extensional view of relations. For 'to the left of', you just list all the things that are to the left, with the things they are to the left of. But just listing the ordered pairs won't necessarily reveal how they are related.
9. Objects / B. Unity of Objects / 2. Substance / b. Need for substance
If dependence is well-founded, with no infinite backward chains, this implies substances [Potter]
     Full Idea: The argument that the relation of dependence is well-founded ...is a version of the classical arguments for substance. ..Any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3)
     A reaction: Thus the iterative conception of set may imply a notion of substance, and Barwise's radical attempt to ditch the Axiom of Foundation (Idea 13039) was a radical attempt to get rid of 'substances'. Potter cites Wittgenstein as a fan of substances here.
9. Objects / C. Structure of Objects / 8. Parts of Objects / b. Sums of parts
Collections have fixed members, but fusions can be carved in innumerable ways [Potter]
     Full Idea: A collection has a determinate number of members, whereas a fusion may be carved up into parts in various equally valid (although perhaps not equally interesting) ways.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1)
     A reaction: This seems to sum up both the attraction and the weakness of mereology. If you doubt the natural identity of so-called 'objects', then maybe classical mereology is the way to go.
10. Modality / A. Necessity / 1. Types of Modality
Priority is a modality, arising from collections and members [Potter]
     Full Idea: We must conclude that priority is a modality distinct from that of time or necessity, a modality arising in some way out of the manner in which a collection is constituted from its members.
     From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3)
     A reaction: He is referring to the 'iterative' view of sets, and cites Aristotle 'Metaphysics' 1019a1-4 as background.
19. Language / A. Nature of Meaning / 2. Meaning as Mental
For Aristotle meaning and reference are linked to concepts [Aristotle, by Putnam]
     Full Idea: In 'De Interpretatione' Aristotle laid out an enduring theory of reference and meaning, in which we understand a word or any other sign by associating that word with a concept. This concept determines what the word refers to.
     From: report of Aristotle (On Interpretation [c.330 BCE]) by Hilary Putnam - Representation and Reality 2 p.19
     A reaction: Sounds right to me, despite all this Wittgensteinian stuff about beetles in boxes. When you meet a new technical term in philosophy, you must struggle to fully grasp the concept it proposes.
19. Language / D. Propositions / 4. Mental Propositions
Spoken sounds vary between people, but are signs of affections of soul, which are the same for all [Aristotle]
     Full Idea: Spoken sounds are symbols of affections in the soul, ...and just as written marks are not the same for all men, neither are spoken sounds. But what these are in the first place signs of - affections of the soul - are the same for all.
     From: Aristotle (On Interpretation [c.330 BCE], 16a03-08)
     A reaction: Loux identifies this passage as the source of the 'conceptualist' view of propositions, which I immediately identify with. The view that these propositions are 'the same for all' is plausible for normal objects, but dubious for complex abstractions.
19. Language / F. Communication / 3. Denial
It doesn't have to be the case that in opposed views one is true and the other false [Aristotle]
     Full Idea: It is not necessary that of every affirmation and opposite negation one should be true and the other false. For what holds for things that are does not hold for things that are not but may possibly be or not be.
     From: Aristotle (On Interpretation [c.330 BCE], 19a39)
     A reaction: Thus even if Bivalence holds, and the only truth-values are T and F, it doesn't follow that Excluded Middle holds, which says that every proposition must have one of those two values.
23. Ethics / E. Utilitarianism / 4. Unfairness
Imagine millions made happy on condition that one person suffers endless lonely torture [James]
     Full Idea: Consider a case in which millions could be made permanently happy on the one simple condition that a certain lost soul on the far-off edge of things should lead a life of lonely torture.
     From: William James (The Will to Believe [1896], p.188), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.2
     A reaction: This seems to be one of the earliest pinpointings of a key problem with utilitiarianism, which is that other values than happiness (in this case, fairness) seem to be utterly overruled. If we ignore fairness, why shouldn't we ignore happiness?
27. Natural Reality / D. Time / 1. Nature of Time / g. Growing block
Things may be necessary once they occur, but not be unconditionally necessary [Aristotle]
     Full Idea: To say that everything that is, is of necessity, when it is, is not the same as saying unconditionally that it is of necessity.
     From: Aristotle (On Interpretation [c.330 BCE], 19a25)