display all the ideas for this combination of texts
13 ideas
| 12376 | Demonstrations by reductio assume excluded middle [Aristotle] |
| Full Idea: Demonstrations by reduction to the impossible assume that everything is asserted or denied. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 77a23) | |
| A reaction: This sounds like the lynchpin of classical logic. |
| 15326 | Doubt is thrown on classical logic by the way it so easily produces the liar paradox [Horsten] |
| Full Idea: Aside from logic, so little is needed to generate the liar paradox that one wonders whether the laws of classical logic are unrestrictedly valid after all. (Many theories of truth have therefore been formulated in nonclassical logic.) | |
| From: Leon Horsten (The Tarskian Turn [2011], 01.2) | |
| A reaction: Kripke uses Strong Kleene logic for his theory. The implication is that debates discussed by Horsten actually have the status of classical logic at stake, as well as the nature of truth. |
| 12373 | Something holds universally when it is proved of an arbitrary and primitive case [Aristotle] |
| Full Idea: Something holds universally when it is proved of an arbitrary and primitive case. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 73b33) | |
| A reaction: A key idea in mathematical logic, but it always puzzles me. If you snatch a random person in London, and they are extremely tall, does that prove that people of London are extremely tall? How do we know the arbitrary is representative? |
| 8078 | Modus ponens is one of five inference rules identified by the Stoics [Chrysippus, by Devlin] |
| Full Idea: Modus ponens is just one of the five different inference rules identified by the Stoics. | |
| From: report of Chrysippus (fragments/reports [c.240 BCE]) by Keith Devlin - Goodbye Descartes Ch.2 | |
| A reaction: Modus ponens strikes me as being more like a definition of implication than a 'rule'. Implication is what gets you from one truth to another. All the implications of a truth must also be true. |
| 15341 | Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms [Horsten] |
| Full Idea: The Deduction Theorem says ψ is derivable in classical predicate logic from ψ iff the sentence φ→ψ is a theorem of classical logic. Hence inferring φ to ψ is truth-preserving iff the axiom scheme φ→ψ is provable. | |
| From: Leon Horsten (The Tarskian Turn [2011], 02.2) | |
| A reaction: Horsten offers this to show that the Tarski bi-conditionals can themselves be justified, and not just the rule of inference involved. Apparently you can only derive something if you first announce that you have the ability to derive it. Odd. |
| 12363 | Everything is either asserted or denied truly [Aristotle] |
| Full Idea: Of the fact that everything is either asserted or denied truly, we must believe that it is the case. | |
| From: Aristotle (Posterior Analytics [c.327 BCE], 71a14) | |
| A reaction: Presumably this means that every assertion which could possibly be asserted must come out as either true or false. This will have to include any assertions with vague objects or predicates, and any universal assertions, and negative assertions. |
| 6023 | Every proposition is either true or false [Chrysippus, by Cicero] |
| Full Idea: We hold fast to the position, defended by Chrysippus, that every proposition is either true or false. | |
| From: report of Chrysippus (fragments/reports [c.240 BCE]) by M. Tullius Cicero - On Fate ('De fato') 38 | |
| A reaction: I am intrigued to know exactly how you defend this claim. It may depend what you mean by a proposition. A badly expressed proposition may have indeterminate truth, quite apart from the vague, the undecidable etc. |
| 15328 | A theory is 'non-conservative' if it facilitates new mathematical proofs [Horsten] |
| Full Idea: A theory is 'non-conservative' if it allows us to prove mathematical facts that go beyond what the background mathematical theory can prove on its own. | |
| From: Leon Horsten (The Tarskian Turn [2011], 01.4) | |
| A reaction: This is an instance of the relationship with mathematics being used as the test case for explorations of logic. It is a standard research method, because it is so precise, but should not be mistaken for the last word about a theory. |
| 15349 | It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F) [Horsten] |
| Full Idea: It is easier to imagine what it is like for a sentence to lack a truth value than what it is like for a sentence to be both truth and false. So I am grudgingly willing to entertain the possibility that certain sentences (like the Liar) lack a truth value. | |
| From: Leon Horsten (The Tarskian Turn [2011], 02.5) | |
| A reaction: Fans of truth value gluts are dialethists like Graham Priest. I'm with Horsten on this one. But in what way can a sentence be meaningful if it lacks a truth-value? He mentions unfulfilled presuppositions and indicative conditionals as gappy. |
| 15366 | Satisfaction is a primitive notion, and very liable to semantical paradoxes [Horsten] |
| Full Idea: Satisfaction is a more primitive notion than truth, and it is even more susceptible to semantical paradoxes than the truth predicate. | |
| From: Leon Horsten (The Tarskian Turn [2011], 06.3) | |
| A reaction: The Liar is the best known paradox here. Tarski bases his account of truth on this primitive notion, so Horsten is pointing out the difficulties. |
| 13004 | Aristotle's axioms (unlike Euclid's) are assumptions awaiting proof [Aristotle, by Leibniz] |
| Full Idea: Aristotle's way with axioms, rather than Euclid's, is as assumptions which we are willing to agree on while awaiting an opportunity to prove them | |
| From: report of Aristotle (Posterior Analytics [c.327 BCE], 76b23-) by Gottfried Leibniz - New Essays on Human Understanding 4.07 | |
| A reaction: Euclid's are understood as basic self-evident truths which will be accepted by everyone, though the famous parallel line postulate undermined that. The modern view of axioms is a set of minimum theorems that imply the others. I like Aristotle. |
| 15353 | The first incompleteness theorem means that consistency does not entail soundness [Horsten] |
| Full Idea: It is a lesson of the first incompleteness theorem that consistency does not entail soundness. If we add the negation of the gödel sentence for PA as an extra axiom to PA, the result is consistent. This negation is false, so the theory is unsound. | |
| From: Leon Horsten (The Tarskian Turn [2011], 04.3) |
| 15355 | Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated [Horsten] |
| Full Idea: The Strengthened Liar sentence says 'this sentence is not true in any context'. It is not hard to figure out that there is no context in which the sentence can be coherently evaluated. | |
| From: Leon Horsten (The Tarskian Turn [2011], 04.6) |