Combining Texts

All the ideas for 'The Philosopher's Toolkit', 'On Minerals' and 'On Formally Undecidable Propositions'

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37 ideas

2. Reason / B. Laws of Thought / 2. Sufficient Reason
4643 The Principle of Sufficient Reason does not presuppose that all explanations will be causal explanations [Baggini /Fosl]
2. Reason / B. Laws of Thought / 3. Non-Contradiction
4633 You cannot rationally deny the principle of non-contradiction, because all reasoning requires it [Baggini /Fosl]
2. Reason / C. Styles of Reason / 1. Dialectic
4635 Dialectic aims at unified truth, unlike analysis, which divides into parts [Baggini /Fosl]
3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
21752 Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
4632 'Natural' systems of deduction are based on normal rational practice, rather than on axioms [Baggini /Fosl]
4631 In ideal circumstances, an axiom should be such that no rational agent could possibly object to its use [Baggini /Fosl]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17835 Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M]
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
4638 The principle of bivalence distorts reality, as when claiming that a person is or is not 'thin' [Baggini /Fosl]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17886 The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
5. Theory of Logic / K. Features of Logics / 2. Consistency
10071 Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P]
5. Theory of Logic / K. Features of Logics / 3. Soundness
19123 If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
10621 Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel]
17888 The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10132 There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10072 First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P]
9590 Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman]
3198 Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
11069 Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna]
10118 First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman]
10122 Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman]
10611 There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P]
10867 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
8747 Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro]
9. Objects / C. Structure of Objects / 2. Hylomorphism / a. Hylomorphism
16756 Substantial forms must exist, to explain the stability of metals like silver and tin [Albertus Magnus]
9. Objects / F. Identity among Objects / 3. Relative Identity
4640 If identity is based on 'true of X' instead of 'property of X' we get the Masked Man fallacy ('I know X but not Y') [Baggini /Fosl, by PG]
9. Objects / F. Identity among Objects / 4. Type Identity
4647 'I have the same car as you' is fine; 'I have the same fiancée as you' is not so good [Baggini /Fosl]
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
4639 Leibniz's Law is about the properties of objects; the Identity of Indiscernibles is about perception of objects [Baggini /Fosl]
10. Modality / A. Necessity / 3. Types of Necessity
4646 Is 'events have causes' analytic a priori, synthetic a posteriori, or synthetic a priori? [Baggini /Fosl]
12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori
4645 'A priori' does not concern how you learn a proposition, but how you show whether it is true or false [Baggini /Fosl]
13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / b. Basic beliefs
4582 Basic beliefs are self-evident, or sensual, or intuitive, or revealed, or guaranteed [Baggini /Fosl]
14. Science / A. Basis of Science / 6. Falsification
4644 A proposition such as 'some swans are purple' cannot be falsified, only verified [Baggini /Fosl]
14. Science / C. Induction / 1. Induction
4584 The problem of induction is how to justify our belief in the uniformity of nature [Baggini /Fosl]
14. Science / C. Induction / 4. Reason in Induction
4583 How can an argument be good induction, but poor deduction? [Baggini /Fosl]
14. Science / D. Explanation / 3. Best Explanation / a. Best explanation
4634 Abduction aims at simplicity, testability, coherence and comprehensiveness [Baggini /Fosl]
4637 To see if an explanation is the best, it is necessary to investigate the alternative explanations [Baggini /Fosl]
17. Mind and Body / C. Functionalism / 2. Machine Functionalism
3192 Basic logic can be done by syntax, with no semantics [Gödel, by Rey]
18. Thought / A. Modes of Thought / 5. Rationality / a. Rationality
4629 Consistency is the cornerstone of rationality [Baggini /Fosl]