|
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]
|
|
7485
|
For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson]
|
|
Full Idea:
For Pythagoreans, one, 1, is not a true number but the 'essence' of number, out of which the number system emerges.
|
|
From:
report of Pythagoras (reports [c.530 BCE], Ch.8) by Peter Watson - Ideas Ch.8
|
|
A reaction:
I think this is right! Counting and numbers only arise once the concept of individuality and identity have arisen. Counting to one is no more than observing the law of identity. 'Two' is the big adventure.
|
|
3425
|
Reduction has been defined as deriving one theory from another by logic and maths [Nagel,E, by Kim]
|
|
Full Idea:
Ernest Nagel defines reduction as the possibility of deriving all laws of one theory by logic and mathematics to another theory, with appropriate 'bridging principles' (either definitions, or empirical laws) connecting the expressions of the two theories.
|
|
From:
report of Ernest Nagel (The Structure of Science [1961]) by Jaegwon Kim - Philosophy of Mind p.213
|
|
A reaction:
This has been labelled as 'weak' reduction, where 'strong' reduction would be identity, as when lightning is reduced to electrical discharge. You reduce x by showing that it is y in disguise.
|
|
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.
|
|
3053
|
Pythagoras taught that virtue is harmony, and health, and universal good, and God [Pythagoras, by Diog. Laertius]
|
|
Full Idea:
Pythagoras taught that virtue is harmony, and health, and universal good, and God.
|
|
From:
report of Pythagoras (reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.1.19
|
|
A reaction:
I like the link with health, because I consider that a bridge over the supposed fact-value gap. Very Pythagorean to think that virtue is harmony. Plato liked that thought.
|
|
5244
|
For Pythagoreans, justice is simply treating all people the same [Pythagoras, by Aristotle]
|
|
Full Idea:
Some even think that what is just is simple reciprocity, as the Pythagoreans maintained, because they defined justice simply as having done to one what one has done to another.
|
|
From:
report of Pythagoras (reports [c.530 BCE], 28) by Aristotle - Nicomachean Ethics 1132b22
|
|
A reaction:
One wonders what Pythagoreans made of slavery. Aristotle argues that officials, for example, have superior rights. The Pythagorean idea makes fairness the central aspect of justice, and that must at least be partly right.
|