Combining Texts

Ideas for 'Are there propositions?', 'On Formally Undecidable Propositions' and 'Questions on Aristotle's Posterior Analytics'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts


6 ideas

5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Logic studies consequence, compatibility, contradiction, corroboration, necessitation, grounding.... [Ryle]
     Full Idea: Logic studies the way in which one thing follows from another, in which one thing is compatible with another, contradicts, corroborates or necessitates another, is a special case of another or the nerve of another. And so on.
     From: Gilbert Ryle (Are there propositions? [1930], IV)
     A reaction: I presume that 'and so on' would include how one thing proves another. This is quite a nice list, which makes me think a little more widely about the nature of logic (rather than just about inference). Incompatibility isn't a process.
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
     Full Idea: The inherent limitations of the axiomatic method were first brought to light by the incompleteness theorems.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Koellner - On the Question of Absolute Undecidability 1.1
5. Theory of Logic / K. Features of Logics / 2. Consistency
Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P]
     Full Idea: Second Incompleteness Theorem: roughly, nice theories that include enough basic arithmetic can't prove their own consistency.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.5
     A reaction: On the face of it, this sounds less surprising than the First Theorem. Philosophers have often noticed that it seems unlikely that you could use reason to prove reason, as when Descartes just relies on 'clear and distinct ideas'.
5. Theory of Logic / K. Features of Logics / 3. Soundness
If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh]
     Full Idea: Gödel showed PA cannot be proved consistent from with PA. But 'reflection principles' can be added, which are axioms partially expressing the soundness of PA, by asserting what is provable. A Global Reflection Principle asserts full soundness.
     From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Halbach,V/Leigh,G.E. - Axiomatic Theories of Truth (2013 ver) 1.2
     A reaction: The authors point out that this needs a truth predicate within the language, so disquotational truth won't do, and there is a motivation for an axiomatic theory of truth.
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel]
     Full Idea: Where Gödel's First Theorem sabotages logicist ambitions, the Second Theorem sabotages Hilbert's Programme.
     From: comment on Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 36
     A reaction: Neo-logicism (Crispin Wright etc.) has a strategy for evading the First Theorem.
The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
     Full Idea: My undecidable arithmetical sentence ...is not at all absolutely undecidable; rather, one can always pass to 'higher' systems in which the sentence in question is decidable.
     From: Kurt Gödel (On Formally Undecidable Propositions [1931]), quoted by Peter Koellner - On the Question of Absolute Undecidability 1.1
     A reaction: [a 1931 MS] He says the reals are 'higher' than the naturals, and the axioms of set theory are higher still. The addition of a truth predicate is part of what makes the sentence become decidable.