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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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13941
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Are the truth-bearers sentences, utterances, ideas, beliefs, judgements, propositions or statements? [Cartwright,R]
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Full Idea:
What is it that is susceptible of truth or falsity? The answers suggested constitute a bewildering variety: sentences, utterances, ideas, beliefs, judgments, propositions, statements.
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From:
Richard Cartwright (Propositions [1962], 01)
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A reaction:
Carwright's answer is 'statements', which seem to be the same as propositions.
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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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7726
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Aristotelian logic dealt with inferences about concepts, and there were also proposition inferences [Weiner]
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Full Idea:
Till the nineteenth century, it was a common view that Aristotelian logic could evaluate inferences whose validity was based on relations between concepts, while propositional logic could evaluate inferences based on relations between propositions.
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From:
Joan Weiner (Frege [1999], Ch.3)
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A reaction:
Venn diagrams relate closely to Aristotelian syllogisms, as each concept is represented by a circle, and shows relations between sets. Arrows seem needed to represent how to go from one proposition to another. Is one static, the other dynamic?
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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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13944
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We can pull apart assertion from utterance, and the action, the event and the subject-matter for each [Cartwright,R]
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Full Idea:
We need to distinguish 1) what is asserted, 2) that assertion, 3) asserting something, 4) what is predicated, 5) what is uttered, 6) that utterance, 7) uttering something, 8) the utterance token, and 9) the meaning.
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From:
Richard Cartwright (Propositions [1962], 05-06)
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A reaction:
[summary of his overall analysis in the paper] It is amazingly hard to offer a critical assessment of this sort of analysis, but it gives you a foot in the door for thinking about the issues with increasing clarity.
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13947
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'It's raining' makes a different assertion on different occasions, but its meaning remains the same [Cartwright,R]
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Full Idea:
A person who utters 'It's raining' one day does not normally make the same statement as one who utters it the next. But these variations are not accompanied by corresponding changes of meaning. The words 'It's raining' retain the same meaning throughout.
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From:
Richard Cartwright (Propositions [1962], 10)
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A reaction:
This is important, because it shows that a proposition is not just the mental shadow behind a sentence, or a mental shadow awaiting a sentence. Unlike a sentence, a proposition can (and possibly must) include its own context. Very interesting!
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13946
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To assert that p, it is neither necessary nor sufficient to utter some particular words [Cartwright,R]
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Full Idea:
In order to assert that p it is not necessary to utter exactly those words. ...Clearly, also, in order to assert that p, it is not sufficient to utter the words that were actually uttered.
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From:
Richard Cartwright (Propositions [1962], 07)
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A reaction:
I take the first point to be completely obvious (you can assert one thing with various wordings), and the second seems right after a little thought (the words could be vague, ambiguous, inaccurate, contextual)
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13951
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Assertions, unlike sentence meanings, can be accurate, probable, exaggerated, false.... [Cartwright,R]
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Full Idea:
Whereas what is asserted can be said to be accurate, exaggerated, unfounded, overdrawn, probable, improbable, plausible, true, or false, none of these can be said of the meaning of a sentence.
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From:
Richard Cartwright (Propositions [1962], 12)
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A reaction:
That fairly firmly kicks into touch the idea that the assertion is the same as the meaning of the sentence.
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