Ideas of Ian Rumfitt, by Theme

[British, fl. 2014, Pupil of Dummett. At University College, Oxford. Then Professor at Birkbeck, then Birmingham.]

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1. Philosophy / E. Nature of Metaphysics / 6. Metaphysics as Conceptual
Logic doesn't have a metaphysical basis, but nor can logic give rise to the metaphysics
     Full Idea: There is surely no metaphysical basis for logic, but equally there is no logical basis for metaphysics, if that implies that we can settle the choice of logic in advance of settling any seriously contested metaphysical-cum-semantic issues.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.5)
     A reaction: Is this aimed at Tim Williamson's book on treating modal logic as metaphysics? I agree with the general idea that logic won't deliver a metaphysics. I might want to defend a good metaphysics giving rise to a good logic.
3. Truth / A. Truth Problems / 1. Truth
The idea that there are unrecognised truths is basic to our concept of truth
     Full Idea: The realist principle that a statement may be true even though no one is able to recognise its truth is so deeply embedded in our ordinary conception of truth that any account that flouts it is liable to engender confusion.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 5.1)
3. Truth / B. Truthmakers / 7. Making Modal Truths
'True at a possibility' means necessarily true if what is said had obtained
     Full Idea: A statement is 'true at a possibility' if, necessarily, things would have been as the statement (actually) says they are, had the possibility obtained.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.6)
     A reaction: This is deliberately vague about what a 'possibility' is, but it is intended to be more than a property instantiation, and less than a possible world.
4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
Semantics for propositions: 1) validity preserves truth 2) non-contradition 3) bivalence 4) truth tables
     Full Idea: The classical semantics of natural language propositions says 1) valid arguments preserve truth, 2) no statement is both true and false, 3) each statement is either true or false, 4) the familiar truth tables.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
The logic of metaphysical necessity is S5
     Full Idea: It is a widely accepted thesis that the logic of metaphysical necessity is S5.
     From: Ian Rumfitt (Logical Necessity [2010], 5)
     A reaction: Rumfitt goes on to defend this standard view (against Dummett's defence of S4). The point, I take it, is that one can only assert that something is 'true in all possible worlds' only when the worlds are all accessible to one another.
'Absolute necessity' would have to rest on S5
     Full Idea: If there is such a notion as 'absolute necessity', its logic is surely S5.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3)
     A reaction: There are plenty of people (mainly in the strict empiricist tradition) who don't believe in 'absolute' necessity.
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
It is the second-order part of intuitionistic logic which actually negates some classical theorems
     Full Idea: Although intuitionistic propositional and first-order logics are sub-systems of the corresponding classical systems, intuitionistic second-order logic affirms the negations of some classical theorems.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
Intuitionists can accept Double Negation Elimination for decidable propositions
     Full Idea: Double Negation Elimination is a rule of inference which the classicist accepts without restriction, but which the intuitionist accepts only for decidable propositions.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
     A reaction: This cures me of my simplistic understanding that intuitionists just reject the rules about double negation.
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic
     Full Idea: Many set theorists doubt if the Generalised Continuum Hypothesis must be either true or false; certainly, its bivalence is far from obvious. All the same, almost all set theorists use classical logic in their proofs.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2)
     A reaction: His point is that classical logic is usually taken to rest on bivalence. He offers the set theorists a helping hand, by defending classical logic without resorting to bivalence.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
The iterated conception of set requires continual increase in axiom strength
     Full Idea: We are doomed to postulate an infinite sequence of successively stronger axiom systems as we try to spell out what is involved in iterating the power set operation 'as far as possible'.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.3)
     A reaction: [W.W. Tait is behind this idea] The problem with set theory, then, especially as a foundation of mathematics, is that it doesn't just expand, but has to keep reinventing itself. The 'large cardinal axioms' are what is referred to.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
A set may well not consist of its members; the empty set, for example, is a problem
     Full Idea: There seem strong grounds for rejecting the thesis that a set consists of its members. For one thing, the empty set is a perpetual embarrassment for the thesis.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4)
     A reaction: Rumfitt also says that if 'red' has an extension, then membership of that set must be vague. Extensional sets are precise because their objects are decided in advance, but intensional (or logical) sets, decided by a predicate, can be vague.
A set can be determinate, because of its concept, and still have vague membership
     Full Idea: Vagueness in respect of membership is consistent with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of the concept A.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4)
     A reaction: To be determinate, it must be presumed that there is some test which will decide what falls under the concept. The rule can say 'if it is vague, reject it' or 'if it is vague, accept it'. Without one of those, how could the set have a clear identity?
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom
     Full Idea: Someone who is sympathetic to the thesis that the totality of sets is not well-defined ought to concede that we have no reason to think that the Power Set Axiom is true.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6)
     A reaction: The point is that it is only this Axiom which generates the vast and expanding totality. In principle it is hard, though, to see what is intrinsically wrong with the operation of taking the power set of a set. Hence 'limitation of size'?
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
If a sound conclusion comes from two errors that cancel out, the path of the argument must matter
     Full Idea: If a designated conclusion follows from the premisses, but the argument involves two howlers which cancel each other out, then the moral is that the path an argument takes from premisses to conclusion does matter to its logical evaluation.
     From: Ian Rumfitt ("Yes" and "No" [2000], II)
     A reaction: The drift of this is that our view of logic should be a little closer to the reasoning of ordinary language, and we should rely a little less on purely formal accounts.
Logic is higher-order laws which can expand the range of any sort of deduction
     Full Idea: On the conception of logic recommended here, logical laws are higher-order laws that can be applied to expand the range of any deductive principles.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3)
     A reaction: You need the concept of a 'deductive principle' to get this going, but I take it that might be directly known, rather than derived from a law.
5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
Logic guides thinking, but it isn't a substitute for it
     Full Idea: Logic is part of a normative theory of thinking, not a substitute for thinking.
     From: Ian Rumfitt (The Logic of Boundaryless Concepts [2007], p.13)
     A reaction: There is some sort of logicians' dream, going back to Leibniz, of a reasoning engine, which accepts propositions and outputs inferences. I agree with this idea. People who excel at logic are often, it seems to me, modest at philosophy.
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
Classical logic rules cannot be proved, but various lines of attack can be repelled
     Full Idea: There is not the slightest prospect of proving that the rules of classical logic are sound. .All that the defender of classical logic can do is scrutinize particular attacks and try to repel them.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
     A reaction: This is the agenda for Rumfitt's book.
If truth-tables specify the connectives, classical logic must rely on Bivalence
     Full Idea: If we specify the senses of the connectives by way of the standard truth-tables, then we must justify classical logic only by appeal to the Principle of Bivalence.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7)
     A reaction: Rumfitt proposes to avoid the truth-tables, and hence not to rely on Bivalence for his support of classical logic. He accepts that Bivalence is doubtful, citing the undecidability of the Continuum Hypothesis as a problem instance.
The case for classical logic rests on its rules, much more than on the Principle of Bivalence
     Full Idea: I think it is a strategic mistake to rest the case for classical logic on the Principle of Bivalence: the soundness of the classical logic rules is far more compelling than the truth of Bivalence.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
     A reaction: The 'rules' to which he is referring are those of 'natural deduction', which make very few assumptions, and are intended to be intuitively appealing.
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Soundness in argument varies with context, and may be achieved very informally indeed
     Full Idea: Our ordinary standards for deeming arguments to be sound vary greatly from context to context. Even the package tourist's syllogism ('It's Tuesday, so this is Belgium') may meet the operative standards for soundness.
     From: Ian Rumfitt (Logical Necessity [2010], Intro)
     A reaction: No doubt one could spell out the preconceptions of package tourist reasoning, and arrive at the logical form of the implication which is being offered.
There is a modal element in consequence, in assessing reasoning from suppositions
     Full Idea: There is a modal element in consequence, in its applicability to assessing reasoning from suppositions.
     From: Ian Rumfitt (Logical Necessity [2010], 2)
We reject deductions by bad consequence, so logical consequence can't be deduction
     Full Idea: A rule is to be rejected if it enables us to deduce from some premisses a purported conclusion that does not follow from them in the broad sense. The idea that deductions answer to consequence is incomprehensible if consequence consists in deducibility.
     From: Ian Rumfitt (Logical Necessity [2010], 2)
Logical consequence is a relation that can extended into further statements
     Full Idea: Logical consequence, I argue, is distinguished from other implication relations by the fact that logical laws may be applied in extending any implication relation so that it applies among some complex statements involving logical connectives.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.3)
     A reaction: He offers implication in electronics as an example of a non-logical implication relation. This seems to indicate that logic must be monotonic, that consequence is transitive, and that the Cut Law always applies.
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
Normal deduction presupposes the Cut Law
     Full Idea: Our deductive practices seem to presuppose the Cut Law.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3)
     A reaction: That is, if you don't believe that deductions can be transitive (and thus form a successful chain of implications), then you don't really believe in deduction. It remains a well known fact that you can live without the Cut Law.
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
When faced with vague statements, Bivalence is not a compelling principle
     Full Idea: I do not regard Bivalence, when applied to vague statements, as an intuitively compelling principle which we ought to try to preserve.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.7)
     A reaction: The point of Rumfitt's book is to defend classical logic despite failures of bivalence. He also cites undecidable concepts such as the Continuum Hypothesis.
5. Theory of Logic / D. Assumptions for Logic / 3. Contradiction
Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B'
     Full Idea: Overt contradictions include formal contradictions of form 'B and not B', but I also take them to include 'This is red all over and green all over' and 'This is red and not coloured'.
     From: Ian Rumfitt (Logical Necessity [2010], Intro)
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Standardly 'and' and 'but' are held to have the same sense by having the same truth table
     Full Idea: If 'and' and 'but' really are alike in sense, in what might that likeness consist? Some philosophers of classical logic will reply that they share a sense by virtue of sharing a truth table.
     From: Ian Rumfitt ("Yes" and "No" [2000])
     A reaction: This is the standard view which Rumfitt sets out to challenge.
The sense of a connective comes from primitively obvious rules of inference
     Full Idea: A connective will possess the sense that it has by virtue of its competent users' finding certain rules of inference involving it to be primitively obvious.
     From: Ian Rumfitt ("Yes" and "No" [2000], III)
     A reaction: Rumfitt cites Peacocke as endorsing this view, which characterises the logical connectives by their rules of usage rather than by their pure semantic value.
In specifying a logical constant, use of that constant is quite unavoidable
     Full Idea: There is no prospect whatever of giving the sense of a logical constant without using that very constant, and much else besides, in the metalinguistic principle that specifies that sense.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths)
     Full Idea: The geometrical style of formalization of logic is now little more than a quaint anachronism, largely because it fails to show logical truths for what they are: simply by-products of rules of inference that are applicable to suppositions.
     From: Ian Rumfitt (Logical Necessity [2010], 1)
     A reaction: This is the rejection of Russell-style axiom systems in favour of Gentzen-style natural deduction systems (starting from rules). Rumfitt quotes Dummett in support.
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Introduction rules give deduction conditions, and Elimination says what can be deduced
     Full Idea: 'Introduction rules' state the conditions under which one may deduce a conclusion whose dominant logical operator is the connective. 'Elimination rules' state what may be deduced from some premises, where the major premise is dominated by the connective.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 1.1)
     A reaction: So Introduction gives conditions for deduction, and Elimination says what can actually be deduced. If my magic wand can turn you into a frog (introduction), and so I turn you into a frog, how does that 'eliminate' the wand?
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Logical truths are just the assumption-free by-products of logical rules
     Full Idea: Gentzen's way of formalising logic has accustomed people to the idea that logical truths are simply the by-products of logical rules, that arise when all the assumptions on which a conclusion rests have been discharged.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.5)
     A reaction: This is the key belief of those who favour the natural deduction account of logic. If you really believe in separate logic truths, then you can use them as axioms.
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
Monotonicity means there is a guarantee, rather than mere inductive support
     Full Idea: Monotonicity seems to mark the difference between cases in which a guarantee obtains and those where the premises merely provide inductive support for a conclusion.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3)
     A reaction: Hence it is plausible to claim that 'non-monotonic logic' is a contradiction in terms.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set
     Full Idea: Menzel proposes that an ordinal is something isomorphic well-ordered sets have in common, so while an ordinal can be represented as a set, it is not itself a set, but a 'property' of well-ordered sets.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.2)
     A reaction: [C.Menzel 1986] This is one of many manoeuvres available if you want to distance mathematics from set theory.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
A single object must not be counted twice, which needs knowledge of distinctness (negative identity)
     Full Idea: One requirement for a successful count is that double counting should be avoided: a single object should not be counted twice. ...but that is to make a knowledgeable judgement of distinctness - to resolve a question of identity in the negative.
     From: Ian Rumfitt (Concepts and Counting [2002], III)
     A reaction: He also notes later (p.65) that you must count all and only the right things.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
Infinitesimals do not stand in a determinate order relation to zero
     Full Idea: Infinitesimals do not stand in a determinate order relation to zero: we cannot say an infinitesimal is either less than zero, identical to zero, or greater than zero. .Infinitesimals are so close to zero as to be theoretically indiscriminable from it.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.4)
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
     Full Idea: One of the motivations behind Cantor's and Dedekind's pioneering explorations in the field was the ambition to give real analysis a new foundation in set theory - and hence a foundation independent of geometry.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6)
     A reaction: Rumfitt is inclined to think that the project has failed, although a weaker set theory than ZF might do the job (within limits).
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
Some 'how many?' answers are not predications of a concept, like 'how many gallons?'
     Full Idea: We hit trouble if we hear answers to some 'How many?' questions as predications about concepts. The correct answer to 'how many gallons of water are in the tank?' may be 'ten', but that doesn''t mean ten things instantiate 'gallon of water in the tank'.
     From: Ian Rumfitt (Concepts and Counting [2002], I)
     A reaction: Rumfitt makes the point that a huge number of things instantiate that concept in a ten gallon tank of water. No problem, says Rumfitt, because Frege wouldn't have counted that as a statement of number.
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
Vague membership of sets is possible if the set is defined by its concept, not its members
     Full Idea: Vagueness in respect of membership is consistency with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of a concept.
     From: Ian Rumfitt (The Logic of Boundaryless Concepts [2007], p.5)
     A reaction: I find this view of sets much more appealing than the one that identifies a set with its members. The empty set is less of a problem, as well as non-existents. Logicians prefer the extensional view because it is tidy.
An object that is not clearly red or orange can still be red-or-orange, which sweeps up problem cases
     Full Idea: A borderline red-orange object satisfies the disjunctive predicate 'red or orange', even though it satisfies neither 'red' or 'orange'. When applied to adjacent bands of colour, the disjunction 'sweeps up' objects which are reddish-orange.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.5)
     A reaction: Rumfitt offers a formal principle in support of this. There may be a problem with 'adjacent'. Different colour systems will place different colours adjacent to red. In other examples the idea of 'adjacent' may make no sense. Rumfitt knows this!
The extension of a colour is decided by a concept's place in a network of contraries
     Full Idea: On Sainsbury's picture, a colour has an extension that it has by virtue of its place in a network of contrary colour classifications. Something is determined to be 'red' by being a colour incompatible with orange, yellow, green, blue, indigo and violet.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.5)
     A reaction: Along with Idea 18839, this gives quite a nice account of vagueness, by requiring a foil to the vague predicate, and using the disjunction of the predicate and its foil to handle anything caught in between them.
10. Modality / A. Necessity / 3. Types of Necessity
A distinctive type of necessity is found in logical consequence
     Full Idea: Rumfitt argues that there is a distinctive notion of necessity implicated in the notion of logical consequence.
     From: report of Ian Rumfitt (Logical Necessity [2010]) by Bob Hale/ Aviv Hoffmann - Introduction to 'Modality' 2
10. Modality / A. Necessity / 5. Metaphysical Necessity
Metaphysical modalities respect the actual identities of things
     Full Idea: The central characteristic mark of metaphysical necessity is that a metaphysical possibility respects the actual identities of things - in a capacious sense of 'thing'.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 3.4)
     A reaction: He contrast this with logical necessity, and concludes that some truths are metaphysically but not logically necessary, such as 'Hesperus is identical with Phosphorus'. Personally I like the idea of a 'necessity-maker', so that fits.
10. Modality / A. Necessity / 6. Logical Necessity
Logical necessity is when 'necessarily A' implies 'not-A is contradictory'
     Full Idea: By the notion of 'logical necessity' I mean that there is a sense of 'necessary' for which 'It is necessary that A' implies and is implied by 'It is logically contradictory that not A'. ...From this, logical necessity is implicated in logical consequence.
     From: Ian Rumfitt (Logical Necessity [2010], Intro)
     A reaction: Rumfitt expresses a commitment to classical logic at this point. We will need to be quite sure what we mean by 'contradiction', which will need a clear notion of 'truth'....
Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not
     Full Idea: While Fine suggests defining a narrow notion of logical necessity in terms of metaphysical necessity by 'restriction' (to logical truths that can be defined in non-modal terms), this seems unpromising for broad logical necessity, which is modal.
     From: Ian Rumfitt (Logical Necessity [2010], 2)
     A reaction: [compressed] He cites Kit Fine 2002. Rumfitt glosses the non-modal definitions as purely formal. The metaphysics lurks somewhere in the proof.
A logically necessary statement need not be a priori, as it could be unknowable
     Full Idea: There is no reason to suppose that any statement that is logically necessary (in the present sense) is knowable a priori. ..If a statement is logically necessary, its negation will yield a contradiction, but that does not imply that someone could know it.
     From: Ian Rumfitt (Logical Necessity [2010], 2)
     A reaction: This remark is aimed at Dorothy Edgington, who holds the opposite view. Rumfitt largely defends McFetridge's view (q.v.).
S5 is the logic of logical necessity
     Full Idea: I accept the widely held thesis that S5 is the logic of logical necessity.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.4 n16)
     A reaction: It seems plausible that S5 is also the logic of metaphysical necessity, but that does not make them the same thing. The two types of necessity have two different grounds.
10. Modality / B. Possibility / 1. Possibility
If two possibilities can't share a determiner, they are incompatible
     Full Idea: Two possibilities are incompatible when no possibility determines both.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.1)
     A reaction: This strikes me as just the right sort of language for building up a decent metaphysical picture of the world, which needs to incorporate possibilities as well as actualities.
Since possibilities are properties of the world, calling 'red' the determination of a determinable seems right
     Full Idea: Some philosophers describe the colour scarlet as a determination of the determinable red; since the ways the world might be are naturally taken to be properties of the world, it helps to bear this analogy in mind.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6.4)
     A reaction: This fits nicely with the disposition accounts of modality which I favour. Hence being 'coloured' is a real property of objects, even in the absence of the name of its specific colour.
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
If a world is a fully determinate way things could have been, can anyone consider such a thing?
     Full Idea: A world is usually taken to be a fully determinate way that things could have been; but then one might seriously wonder whether anyone is capable of 'considering' such a thing at all.
     From: Ian Rumfitt (Logical Necessity [2010], 4)
     A reaction: This has always worried me. If I say 'maybe my coat is in the car', I would hate to think that I had to be contemplating some entire possible world (including all the implications of my coat not being on the hat stand).
Possibilities are like possible worlds, but not fully determinate or complete
     Full Idea: Possibilities are things of the same general character as possible worlds, on one popular conception of the latter. They differ from worlds, though, in that they are not required to be fully determinate or complete.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 6)
     A reaction: A rather promising approach to such things, even though a possibility is fairly determinate at its core, but very vague at the edges. It is possible that the UK parliament might be located in Birmingham, for example. Is this world 'complete'?
11. Knowledge Aims / A. Knowledge / 2. Understanding
Medieval logicians said understanding A also involved understanding not-A
     Full Idea: Mediaeval logicians had a principle, 'Eadem est scientia oppositorum': in order to attain a clear conception of what it is for A to be the case, one needs to attain a conception of what it is for A not to be the case.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.2)
     A reaction: Presumably 'understanding' has to be a fairly comprehensive grasp of the matter, so understanding the negation sounds like a reasonable requirement for the real thing.
13. Knowledge Criteria / B. Internal Justification / 3. Evidentialism / a. Evidence
In English 'evidence' is a mass term, qualified by 'little' and 'more'
     Full Idea: In English, the word 'evidence' behaves as a mass term: we speak of someone's having little evidence for an assertion, and of one thinker's having more evidence than another for a claim. One the other hand, we also speak of 'pieces' of evidence.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 5.2)
     A reaction: And having 'more' evidence does not mean having a larger number of pieces of evidence, so it really is like an accumulated mass.
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
We understand conditionals, but disagree over their truth-conditions
     Full Idea: It is striking that our understanding of conditionals is not greatly impeded by widespread disagreement about their truth-conditions.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 4.2)
19. Language / F. Communication / 3. Denial
We learn 'not' along with affirmation, by learning to either affirm or deny a sentence
     Full Idea: The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A.
     From: Ian Rumfitt ("Yes" and "No" [2000], IV)
     A reaction: [compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role).
The truth grounds for 'not A' are the possibilities incompatible with truth grounds for A
     Full Idea: The truth-grounds of 'A' are precisely those possibilities that are incompatible with any truth-ground of A.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.1)
     A reaction: This is Rumfitt's proposal for the semantics of 'not', based on the central idea of a possibility, rather than a possible world. The incompatibility tracks back to an absence of shared grounding.