19125
|
If we define truth, we can eliminate it [Halbach/Leigh]
|
|
Full Idea:
If truth can be explicitly defined, it can be eliminated.
|
|
From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.3)
|
|
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.
|
19127
|
The T-sentences are deductively weak, and also not deductively conservative [Halbach/Leigh]
|
|
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.
|
|
From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 3.2)
|
|
A reaction:
They are weak because they can't prove completeness. This idea give two reasons for looking for a better theory of truth.
|
19124
|
A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness [Halbach/Leigh]
|
|
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.
|
|
From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.2)
|
|
A reaction:
This seems like a big attraction of axiomatic theories of truth for students of metamathematics.
|
19126
|
If deflationary truth is not explanatory, truth axioms should be 'conservative', proving nothing new [Halbach/Leigh]
|
|
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'.
|
|
From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.3)
|
|
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?
|
19129
|
The FS axioms use classical logical, but are not fully consistent [Halbach/Leigh]
|
|
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), …
|
|
From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 4.3)
|
|
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.
|
19130
|
KF is formulated in classical logic, but describes non-classical truth, which allows truth-value gluts [Halbach/Leigh]
|
|
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.
|
|
From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 4.4)
|
|
A reaction:
[summary, which I hope is correct! Stanford is not wholly clear]
|
21222
|
Logicians presuppose a world, and ignore logic/world connections, so their logic is impure [Husserl, by Velarde-Mayol]
|
|
Full Idea:
Husserl maintained that because most logicians have not studied the connection between logic and the world, logic did not achieve its status of purity. Even more, their logic implicitly presupposed a world.
|
|
From:
report of Edmund Husserl (Formal and Transcendental Logic [1929]) by Victor Velarde-Mayol - On Husserl 4.5.1
|
|
A reaction:
The point here is that the bracketing of phenomenology, to reach an understanding with no presuppositions, is impossible if you don't realise what your are presupposing. I think the logic/world relationship is badly neglected, thanks to Frege.
|
21224
|
Pure mathematics is the relations between all possible objects, and is thus formal ontology [Husserl, by Velarde-Mayol]
|
|
Full Idea:
Pure mathematics is the science of the relations between any object whatever (relation of whole to part, relation of equality, property, unity etc.). In this sense, pure mathematics is seen by Husserl as formal ontology.
|
|
From:
report of Edmund Husserl (Formal and Transcendental Logic [1929]) by Victor Velarde-Mayol - On Husserl 4.5.2
|
|
A reaction:
I would expect most modern analytic philosophers to agree with this. Modern mathematics (e.g. category theory) seems to have moved beyond this stage, but I still like this idea.
|
19121
|
We can reduce properties to true formulas [Halbach/Leigh]
|
|
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.
|
|
From:
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.1)
|
|
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.
|