34 ideas
| 9376 | A sentence may simultaneously define a term, and also assert a fact [Boghossian] |
| Full Idea: It doesn't follow from the fact that a given sentence is being used to implicitly define one of its ingredient terms, that it is not a factual statement. 'This stick is a meter long at t' may define an ingredient terms and express something factual. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: This looks like a rather good point, but it is tied in with a difficulty about definition, which is deciding which sentences are using a term, and which ones are defining it. If I say 'this stick in Paris is a meter long', I'm not defining it. |
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| Full Idea: A proof of the consistency of propositional logic was given by Emil Post in 1921. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2.1) |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| Full Idea: Propositional language is very rudimentary and has limited powers of expression. The only relation between various statements it can handle is that of identity and difference. As are all the same, but Bs can be different from As. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 7 Intro) | |
| A reaction: [second sentence a paraphrase] In predicate logic you could represent two statements as being the same except for one element (an object or predicate or relation or quantifier). |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| Full Idea: Boolean connectives are interpreted as functions on the set {1,0}. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 5.1) | |
| A reaction: 1 and 0 are normally taken to be true (T) and false (F). Thus the functions output various combinations of true and false, which are truth tables. |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1) | |
| A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set. |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level. |
| 17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
| Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro) | |
| A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything. |
| 9375 | Conventionalism agrees with realists that logic has truth values, but not over the source [Boghossian] |
| Full Idea: Conventualism is a factualist view: it presupposes that sentences of logic have truth values. It differs from a realist view in its conception of the source of those truth values, not on their existence. I call the denial of truths Non-Factualism. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: It barely seems to count as truth is we say 'p is true because we say so'. It is a truth about an agreement, not a truth about logic. Driving on the left isn't a truth about which side of the road is best. |
| 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
| Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3) | |
| A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going. |
| 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
| Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2) | |
| A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh????? |
| 17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
| Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
| Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
| Full Idea: An ordinal can be defined as a transitive set of transitive sets, or else, as a transitive set totally ordered by set inclusion. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
| Full Idea: The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: [symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set. |
| 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
| Full Idea: We can form infinite ordinals by taking unions of ordinals. We can thus form 'limit ordinals', which have no immediate predecessor. ω is the first (the union of all finite ordinals), ω + ω = sω is second, 3ω the third.... | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
| Full Idea: There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) | |
| A reaction: This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different. |
| 17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
| Full Idea: Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal). | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3) |
| 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
| Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1) |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1) | |
| A reaction: There has to be an well-founded ordering for inductive proofs to be possible. |
| 15201 | That Queen Anne is dead is a 'general fact', not a fact about Queen Anne [Prior,AN] |
| Full Idea: The fact that Queen Anne has been dead for some years is not, in the strict sense of 'about', a fact about Queen Anne; it is not a fact about anyone or anything - it is a general fact. | |
| From: Arthur N. Prior (Changes in Events and Changes in Things [1968], p.13), quoted by Robin Le Poidevin - Past, Present and Future of Debate about Tense 1 b | |
| A reaction: He distinguishes 'general facts' (states of affairs, I think) from 'individual facts', involving some specific object. General facts seem to be what are expressed by negative existential truths, such as 'there is no Loch Ness Monster'. Useful. |
| 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
| Full Idea: The link between time and modality was severed by Duns Scotus, who proposed a notion of possibility based purely on the notion of semantic consistency. 'Possible' means for him logically possible, that is, not involving contradiction. | |
| From: Michal Walicki (Introduction to Mathematical Logic [2012], History B.4) |
| 9369 | 'Snow is white or it isn't' is just true, not made true by stipulation [Boghossian] |
| Full Idea: Isn't it overwhelmingly obvious that 'Either snow is white or it isn't' was true before anyone stipulated a meaning for it, and that it would have been true even if no one had thought about it, or chosen it to be expressed by one of our sentences? | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §I) | |
| A reaction: Boghossian would have to believe in propositions (unexpressed truths) to hold this - which he does. I take the notion of truth to only have relevance when there are minds around. Otherwise the so-called 'truths' are just the facts. |
| 9367 | The a priori is explained as analytic to avoid a dubious faculty of intuition [Boghossian] |
| Full Idea: The central impetus behind the analytic explanation of the a priori is a desire to explain the possibility of a priori knowledge without having to postulate a special evidence-gathering faculty of intuition. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §I) | |
| A reaction: I don't see at all why one has to postulate a 'faculty' in order to talk about intuition. I take an intuition to be an apprehension of a probable truth, combined with an inability to articulate how the conclusion was arrived at. |
| 9373 | That logic is a priori because it is analytic resulted from explaining the meaning of logical constants [Boghossian] |
| Full Idea: The analytic theory of the apriority of logic arose indirectly, as a by-product of the attempt to explain in what a grasp of the meaning of the logical constants consists. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: Preumably he is referring to Wittgenstein's anguish over the meaning of the word 'not' in his World War I notebooks. He first defined the constants by truth tables, then asserted that they were purely conventional - so logic is conventional. |
| 9380 | We can't hold a sentence true without evidence if we can't agree which sentence is definitive of it [Boghossian] |
| Full Idea: If there is no sentence I must hold true if it is to mean what it does, then there is no basis on which to argue that I am entitled to hold it true without evidence. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: He is exploring Quine's view. Truth by convention depends on agreeing which part of the usage of a term constitutes its defining sentence(s), and that may be rather tricky. Boghossian says this slides into the 'dreaded indeterminacy of meaning'. |
| 9384 | We may have strong a priori beliefs which we pragmatically drop from our best theory [Boghossian] |
| Full Idea: It is consistent with a belief's being a priori in the strong sense that we should have pragmatic reasons for dropping it from our best overall theory. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], n 6) | |
| A reaction: Does 'dropping it' from the theory mean just ignoring it, or actually denying it? C.I. Lewis is the ancestor of this view. Could it be our 'best' theory, while conflicting with beliefs that were strongly a priori? Pragmatism can embrace falsehoods. |
| 9374 | If we learn geometry by intuition, how could this faculty have misled us for so long? [Boghossian] |
| Full Idea: If we learn geometrical truths by intuition, how could this faculty have misled us for so long? | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: This refers to the development of non-Euclidean geometries, though the main misleading concerns parallels, which involves infinity. Boghossian cites 'distance' as a concept the Euclideans had misunderstood. Why shouldn't intuitions be wrong? |
| 9378 | If meaning depends on conceptual role, what properties are needed to do the job? [Boghossian] |
| Full Idea: Conceptual Role Semantics must explain what properties an inference or sentence involving a logical constant must have, if that inference or sentence is to be constitutive of its meaning. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: This is my perennial request that if something is to be defined by its function (or role), we must try to explain what properties it has that make its function possible, and those properties will be the more basic explanation. |
| 9377 | 'Conceptual role semantics' says terms have meaning from sentences and/or inferences [Boghossian] |
| Full Idea: 'Conceptual role semantics' says the logical constants mean what they do by virtue of figuring in certain inferences and/or sentences involving them and not others, ..so some inferences and sentences are constitutive of an expression's meaning. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §III) | |
| A reaction: If the meaning of the terms derives from the sentences in which they figure, that seems to be meaning-as-use. The view that it depends on the inferences seems very different, and is a more interesting but more risky claim. |
| 9372 | Could expressions have meaning, without two expressions possibly meaning the same? [Boghossian] |
| Full Idea: Could there be a fact of the matter about what each expression means, but no fact of the matter about whether they mean the same? | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §II) | |
| A reaction: He is discussing Quine's attack on synonymy, and his scepticism about meaning. Boghossian and I believe in propositions, so we have no trouble with two statements having the same meaning. Denial of propositions breeds trouble. |
| 17721 | There are no truths in virtue of meaning, but there is knowability in virtue of understanding [Boghossian, by Jenkins] |
| Full Idea: Boghossian distinguishes metaphysical analyticity (truth purely in virtue of meaning, debunked by Quine, he says) from epistemic analyticity (knowability purely in virtue of understanding - a notion in good standing). | |
| From: report of Paul Boghossian (Analyticity Reconsidered [1996]) by Carrie Jenkins - Grounding Concepts 2.4 | |
| A reaction: [compressed] This fits with Jenkins's claim that we have a priori knowledge just through understanding and relating our concepts. She, however, rejects that idea that a priori is analytic. |
| 9368 | Epistemological analyticity: grasp of meaning is justification; metaphysical: truth depends on meaning [Boghossian] |
| Full Idea: The epistemological notion of analyticity: a statement is 'true by virtue of meaning' provided that grasp of its meaning alone suffices for justified belief in its truth; the metaphysical reading is that it owes its truth to its meaning, not to facts. | |
| From: Paul Boghossian (Analyticity Reconsidered [1996], §I) | |
| A reaction: Kripke thinks it is neither, but is a purely semantic notion. How could grasp of meaning alone be a good justification if it wasn't meaning which was the sole cause of the statement's truth? I'm not convinced by his distinction. |
| 22899 | 'Thank goodness that's over' is not like 'thank goodness that happened on Friday' [Prior,AN] |
| Full Idea: One says 'thank goodness that is over', ..and it says something which it is impossible which any use of any tenseless copula with a date should convey. It certainly doesn't mean the same as 'thank goodness that occured on Friday June 15th 1954'. | |
| From: Arthur N. Prior (Changes in Events and Changes in Things [1968]), quoted by Adrian Bardon - Brief History of the Philosophy of Time 4 'Pervasive' | |
| A reaction: [Ref uncertain] This seems to be appealing to ordinary usage, in which tenses have huge significance. If we take time (with its past, present and future) as primitive, then tenses can have full weight. Did tenses mean anything at all to Einstein? |