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19125
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If we define truth, we can eliminate it [Halbach/Leigh]
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Full Idea:
If truth can be explicitly defined, it can be eliminated.
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From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.3)
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A reaction:
That we could just say p corresponds to the facts, or p coheres with our accepted beliefs, or p is the aim of our enquiries, and never mention the word 'true'. Definition is a strategy for reduction or elimination.
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19127
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The T-sentences are deductively weak, and also not deductively conservative [Halbach/Leigh]
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Full Idea:
Although the theory is materially adequate, Tarski thought that the T-sentences are deductively too weak. …Also it seems that the T-sentences are not conservative, because they prove in PA that 0=0 and ¬0=0 are different, so at least two objects exist.
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From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 3.2)
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A reaction:
They are weak because they can't prove completeness. This idea give two reasons for looking for a better theory of truth.
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19124
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A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness [Halbach/Leigh]
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Full Idea:
If a natural theory of truth is added to Peano Arithmetic, it is not necessary to add explicity global reflection principles to assert soundness, as the truth theory proves them. Truth theories thus prove soundess, and allows its expression.
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From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.2)
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A reaction:
This seems like a big attraction of axiomatic theories of truth for students of metamathematics.
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19126
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If deflationary truth is not explanatory, truth axioms should be 'conservative', proving nothing new [Halbach/Leigh]
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Full Idea:
If truth does not have any explanatory force, as some deflationists claim, the axioms of truth should not allow us to prove any new theorems that do not involve the truth predicate. That is, a deflationary axiomatisation of truth should be 'conservative'.
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From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.3)
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A reaction:
So does truth have 'explanatory force'? These guys are interested in explaining theorems of arithmetic, but I'm more interested in real life. People do daft things because they have daft beliefs. Logic should be neutral, but truth has values?
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19129
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The FS axioms use classical logical, but are not fully consistent [Halbach/Leigh]
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Full Idea:
It is a virtue of the Friedman-Sheard axiomatisation that it is thoroughly classical in its logic. Its drawback is that it is ω-inconsistent. That is, it proves &exists;x¬φ(x), but proves also φ(0), φ(1), φ(2), …
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From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 4.3)
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A reaction:
It seems the theory is complete (and presumably sound), yet not fully consistent. FS also proves the finite levels of Tarski's hierarchy, but not the transfinite levels.
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19130
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KF is formulated in classical logic, but describes non-classical truth, which allows truth-value gluts [Halbach/Leigh]
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Full Idea:
KF is formulated in classical logic, but describes a non-classical notion of truth. It allow truth-value gluts, making some sentences (such as the Liar) both true and not-true. Some authors add an axiom ruling out such gluts.
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From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 4.4)
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A reaction:
[summary, which I hope is correct! Stanford is not wholly clear]
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19121
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We can reduce properties to true formulas [Halbach/Leigh]
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Full Idea:
One might say that 'x is a poor philosopher' is true of Tom instead of saying that Tom has the property of being a poor philosopher. We quantify over formulas instead of over definable properties, and thus reduce properties to truth.
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From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.1)
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A reaction:
[compressed] This stuff is difficult (because the axioms are complex and hard to compare), but I am excited (yes!) about this idea. Their point is that you need a truth predicate within the object language for this, which disquotational truth forbids.
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11214
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We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt]
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Full Idea:
The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A.
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From:
Ian Rumfitt ("Yes" and "No" [2000], IV)
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A reaction:
[compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role).
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18202
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The concept of a field gradually replaced the substances in explaining relations between charges [Einstein/Infeld]
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Full Idea:
In the beginning the field concept was no more than a means of facilitating the understanding of phenomena. ...In the new field language it is the field and not the charges themselves which is essential. The substance was overshadowed by the field.
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From:
Einstein,A/Infeld,L (The Evolution of Physics [1938], p.151), quoted by Penelope Maddy - Naturalism in Mathematics II.4
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A reaction:
This is very important for philosophical metaphysicians, especially those like me who want to explain the universe by the nature of the stuff that composes it. The 'stuff' had better not be simplistic individual 'substances'.
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