16 ideas
| 6021 | It is only when we say a proposition that we speak truly or falsely [Sext.Empiricus] |
| Full Idea: It is only when we say a proposition that we speak truly or falsely. | |
| From: Sextus Empiricus (Against the Professors (six books) [c.180], 8.74) | |
| A reaction: This makes assertions truth-bearers, rather than propositions. But a proposition can be true or false if it is stamped with a date and/or place. "Shakespeare was born in Stratford on 23rd April 1664". No one needs to assert that. |
| 9540 | A 'value-assignment' (V) is when to each variable in the set V assigns either the value 1 or the value 0 [Hughes/Cresswell] |
| Full Idea: A 'value-assignment' (V) is when to each variable in the set V assigns either the value 1 or the value 0. | |
| From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1) | |
| A reaction: In the interpreted version of the logic, 1 and 0 would become T (true) and F (false). The procedure seems to be called nowadays a 'valuation'. |
| 9541 | The Law of Transposition says (P→Q) → (¬Q→¬P) [Hughes/Cresswell] |
| Full Idea: The Law of Transposition says that (P→Q) → (¬Q→¬P). | |
| From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1) | |
| A reaction: That is, if the consequent (Q) of a conditional is false, then the antecedent (P) must have been false. |
| 9543 | The rules preserve validity from the axioms, so no thesis negates any other thesis [Hughes/Cresswell] |
| Full Idea: An axiomatic system is most naturally consistent iff no thesis is the negation of another thesis. It can be shown that every axiom is valid, that the transformation rules are validity-preserving, and if a wff α is valid, then ¬α is not valid. | |
| From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1) | |
| A reaction: [The labels 'soundness' and 'consistency' seem interchangeable here, with the former nowadays preferred] |
| 15375 | If terms change their designations in different states, they are functions from states to objects [Fitting] |
| Full Idea: The common feature of every designating term is that designation may change from state to state - thus it can be formalized by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3) | |
| A reaction: Specifying the objects sounds OK, but specifying states sounds rather tough. |
| 15376 | Intensional logic adds a second type of quantification, over intensional objects, or individual concepts [Fitting] |
| Full Idea: To first order modal logic (with quantification over objects) we can add a second kind of quantification, over intensions. An intensional object, or individual concept, will be modelled by a function from states to objects. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.3) |
| 15378 | Awareness logic adds the restriction of an awareness function to epistemic logic [Fitting] |
| Full Idea: Awareness logic enriched Hintikka's epistemic models with an awareness function, mapping each state to the set of formulas we are aware of at that state. This reflects some bound on the resources we can bring to bear. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) | |
| A reaction: [He cites Fagin and Halpern 1988 for this] |
| 15379 | Justication logics make explicit the reasons for mathematical truth in proofs [Fitting] |
| Full Idea: In justification logics, the logics of knowledge are extended by making reasons explicit. A logic of proof terms was created, with a semantics. In this, mathematical truths are known for explicit reasons, and these provide a measure of complexity. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.6.1) |
| 11026 | Classical logic is deliberately extensional, in order to model mathematics [Fitting] |
| Full Idea: Mathematics is typically extensional throughout (we write 3+2=2+3 despite the two terms having different meanings). ..Classical first-order logic is extensional by design since it primarily evolved to model the reasoning of mathematics. | |
| From: Melvin Fitting (Intensional Logic [2007], §1) |
| 6020 | 'Man is a rational mortal animal' is equivalent to 'if something is a man, that thing is a rational mortal animal' [Sext.Empiricus] |
| Full Idea: Definitions are identical to universal propositions in meaning, and only differ in syntax, for whoever says 'Man is a rational mortal animal' says the same thing in meaning as whoever says 'If something is a man, that thing is a rational mortal animal'. | |
| From: Sextus Empiricus (Against the Professors (six books) [c.180], 11.8) | |
| A reaction: How strikingly like Bertrand Russell's interest and solutions. Sextus shows a straightforward interest in logical form, of a kind we associate with the twentieth century. Did Sextus Empiricus invent quantification? |
| 11028 | λ-abstraction disambiguates the scope of modal operators [Fitting] |
| Full Idea: λ-abstraction can be used to abstract and disambiguate a predicate. De re is [λx◊P(x)](f) - f has the possible-P property - and de dicto is ◊[λxP(x)](f) - possibly f has the P-property. Also applies to □. | |
| From: Melvin Fitting (Intensional Logic [2007], §3.3) | |
| A reaction: Compare the Barcan formula. Originated with Church in the 1930s, and Carnap 1947, but revived by Stalnaker and Thomason 1968. Because it refers to the predicate, it has a role in intensional versions of logic, especially modal logic. |
| 9544 | A system is 'weakly' complete if all wffs are derivable, and 'strongly' if theses are maximised [Hughes/Cresswell] |
| Full Idea: To say that an axiom system is 'weakly complete' is to say that every valid wff of the system is derivable as a thesis. ..The system is 'strongly complete' if it cannot have any more theses than it has without falling into inconsistency. | |
| From: GE Hughes/M Cresswell (An Introduction to Modal Logic [1968], Ch.1) | |
| A reaction: [They go on to say that Propositional Logic is strongly complete, but Modal Logic is not] |
| 15377 | Definite descriptions pick out different objects in different possible worlds [Fitting] |
| Full Idea: Definite descriptions pick out different objects in different possible worlds quite naturally. | |
| From: Melvin Fitting (Intensional Logic [2007], 3.4) | |
| A reaction: A definite description can pick out the same object in another possible world, or a very similar one, or an object which has almost nothing in common with the others. |
| 6026 | How can you investigate without some preconception of your object? [Sext.Empiricus] |
| Full Idea: A preconception and conception must precede every object of investigation, for how can anyone even investigate without some conception of the object of investigation? | |
| From: Sextus Empiricus (Against the Professors (six books) [c.180], 8.331a) | |
| A reaction: The Duhem-Quine thesis about the 'theory-ladenness of observation' is just a revival of some routine ancient scepticism. As well as a conceptual scheme to accommodate the observation, there must also be some motivation for the investigation. |
| 6032 | Right actions, once done, are those with a reasonable justification [Sext.Empiricus] |
| Full Idea: Right action is whatever, once it has been done, has a reasonable justification. | |
| From: Sextus Empiricus (Against the Professors (six books) [c.180], 7.158) | |
| A reaction: Why does he add 'once it has been done'? Wouldn't a proposed action be right if it had a reasonable justification? This grows out of the classical and Stoic emphasis on reason in ethics, and leads towards Scanlon's Contractualism. |
| 1517 | The tektraktys (1+2+3+4=10) is the 'fount of ever-flowing nature' [Sext.Empiricus] |
| Full Idea: The tektraktys (1+2+3+4=10) is the 'fount of ever-flowing nature', because nature is a harmony of three concords (4th,5th and octave), and these ratios (4:3, 3:2, and 2:1) are found in the tektraktys. | |
| From: Sextus Empiricus (Against the Professors (six books) [c.180], 7.95) |