37 ideas
| 6947 | Metaphysics does not rest on facts, but on what we are inclined to believe [Peirce] |
| Full Idea: Metaphysical systems have not usually rested upon any observed facts, or not in any great degree. They are chiefly adopted because their fundamental propositions seem 'agreeable to reason', which means that which we find ourselves inclined to believe. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.15) | |
| A reaction: This leads to Peirce's key claim - that we should allow our beliefs to be formed by something outside of ourselves. I don't share Peirce's contempt for metaphysics, which I take to be about the most abstract presuppositions of our ordinary beliefs. |
| 6937 | Reason aims to discover the unknown by thinking about the known [Peirce] |
| Full Idea: The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 7) | |
| A reaction: I defy anyone to come up with a better definition of reasoning than that. The emphasis is on knowledge rather than truth, which you would expect from a pragmatist. …Actually the definition doesn't cover conditional reasoning terribly well. |
| 10888 | Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo] |
| Full Idea: We can define a set by 'enumeration' (by listing the items, within curly brackets), or by 'abstraction' (by specifying the elements as instances of a property), pretending that they form a determinate totality. The latter is written {x | x is P}. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) |
| 10889 | The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo] |
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Full Idea:
The 'Cartesian Product' of two sets, written A x B, is the relation which pairs every element of A with every element of B. So A x B = { |
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| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) |
| 10890 | A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo] |
| Full Idea: A binary relation in a set is a 'partial ordering' just in case it is reflexive, antisymmetric and transitive. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6) |
| 10886 | Determinacy: an object is either in a set, or it isn't [Zalabardo] |
| Full Idea: Principle of Determinacy: For every object a and every set S, either a is an element of S or a is not an element of S. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.2) |
| 10887 | Specification: Determinate totals of objects always make a set [Zalabardo] |
| Full Idea: Principle of Specification: Whenever we can specify a determinate totality of objects, we shall say that there is a set whose elements are precisely the objects that we have specified. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3) | |
| A reaction: Compare the Axiom of Specification. Zalabardo says we may wish to consider sets of which we cannot specify the members. |
| 10897 | A first-order 'sentence' is a formula with no free variables [Zalabardo] |
| Full Idea: A formula of a first-order language is a 'sentence' just in case it has no free variables. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.2) |
| 10893 | Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo] |
| Full Idea: A propositional logic sentence is a 'logical consequence' of a set of sentences (written Γ |= φ) if for every admissible truth-assignment all the sentences in the set Γ are true, then φ is true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) | |
| A reaction: The definition is similar for predicate logic. |
| 10899 | Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo] |
| Full Idea: A formula is the 'logical consequence' of a set of formulas (Γ |= φ) if for every structure in the language and every variable interpretation of the structure, if all the formulas within the set are true and the formula itself is true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10896 | Propositional logic just needs ¬, and one of ∧, ∨ and → [Zalabardo] |
| Full Idea: In propositional logic, any set containing ¬ and at least one of ∧, ∨ and → is expressively complete. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.8) |
| 10898 | The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo] |
| Full Idea: The semantic pattern of a first-order language is the ways in which truth values depend on which individuals instantiate the properties and relations which figure in them. ..So we pair a truth value with each combination of individuals, sets etc. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.3) | |
| A reaction: So truth reduces to a combination of 'instantiations', which is rather like 'satisfaction'. |
| 10902 | We can do semantics by looking at given propositions, or by building new ones [Zalabardo] |
| Full Idea: We can look at semantics from the point of view of how truth values are determined by instantiations of properties and relations, or by asking how we can build, using the resources of the language, a proposition corresponding to a given semantic pattern. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) | |
| A reaction: The second version of semantics is model theory. |
| 10892 | We make a truth assignment to T and F, which may be true and false, but merely differ from one another [Zalabardo] |
| Full Idea: A truth assignment is a function from propositions to the set {T,F}. We will think of T and F as the truth values true and false, but for our purposes all we need to assume about the identity of these objects is that they are different from each other. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) | |
| A reaction: Note that T and F are 'objects'. This remark is important in understanding modern logical semantics. T and F can be equated to 1 and 0 in the language of a computer. They just mean as much as you want them to mean. |
| 10895 | 'Logically true' (|= φ) is true for every truth-assignment [Zalabardo] |
| Full Idea: A propositional logic sentence is 'logically true', written |= φ, if it is true for every admissible truth-assignment. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) |
| 10900 | Logically true sentences are true in all structures [Zalabardo] |
| Full Idea: In first-order languages, logically true sentences are true in all structures. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10894 | A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true [Zalabardo] |
| Full Idea: A propositional logic set of sentences Γ is 'satisfiable' if there is at least one admissible truth-assignment that makes all of its sentences true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4) |
| 10901 | Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true [Zalabardo] |
| Full Idea: A set of formulas of a first-order language is 'satisfiable' if there is a structure and a variable interpretation in that structure such that all the formulas of the set are true. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5) |
| 10903 | A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model [Zalabardo] |
| Full Idea: A structure is a model of a sentence if the sentence is true in the model; a structure is a model of a set of sentences if they are all true in the structure. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.6) |
| 10891 | If a set is defined by induction, then proof by induction can be applied to it [Zalabardo] |
| Full Idea: Defining a set by induction enables us to use the method of proof by induction to establish that all the elements of the set have a certain property. | |
| From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.3) |
| 21492 | Realism is basic to the scientific method [Peirce] |
| Full Idea: The fundamental hypothesis of the method of science is this: There are real things, whose characters are entirely independent of our opinion of them. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877]), quoted by Albert Atkin - Peirce 3 'method' | |
| A reaction: He admits later that this is only a commitment and not a fact. It seems to me that when you combine this idea with the huge success of science, the denial of realism is crazy. Philosophy has a lot to answer for. |
| 6949 | If someone doubted reality, they would not actually feel dissatisfaction [Peirce] |
| Full Idea: Nobody can really doubt that there are Reals, for, if he did, doubt would not be a source of dissatisfaction. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.19) | |
| A reaction: This rests on Peirce's view that all that really matters is a sense of genuine dissatisfaction, rather than a theoretical idea. So even at the end of Meditation One, Descartes isn't actually worried about whether his furniture exists. |
| 13768 | Validity can preserve certainty in mathematics, but conditionals about contingents are another matter [Edgington] |
| Full Idea: If your interest in logic is confined to applications to mathematics or other a priori matters, it is fine for validity to preserve certainty, ..but if you use conditionals when arguing about contingent matters, then great caution will be required. | |
| From: Dorothy Edgington (Conditionals [2001], 17.2.1) |
| 13770 | There are many different conditional mental states, and different conditional speech acts [Edgington] |
| Full Idea: As well as conditional beliefs, there are conditional desires, hopes, fears etc. As well as conditional statements, there are conditional commands, questions, offers, promises, bets etc. | |
| From: Dorothy Edgington (Conditionals [2001], 17.3.4) |
| 13764 | Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? [Edgington] |
| Full Idea: Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? Are they non-truth-functional, like 'because' or 'before'? Do the values of A and B, in some cases, leave open the value of 'If A,B'? | |
| From: Dorothy Edgington (Conditionals [2001], 17.1) | |
| A reaction: I would say they are not truth-functional, because the 'if' asserts some further dependency relation that goes beyond the truth or falsity of A and B. Logical ifs, causal ifs, psychological ifs... The material conditional ⊃ is truth-functional. |
| 13765 | 'If A,B' must entail ¬(A & ¬B); otherwise we could have A true, B false, and If A,B true, invalidating modus ponens [Edgington] |
| Full Idea: If it were possible to have A true, B false, and If A,B true, it would be unsafe to infer B from A and If A,B: modus ponens would thus be invalid. Hence 'If A,B' must entail ¬(A & ¬B). | |
| From: Dorothy Edgington (Conditionals [2001], 17.1) | |
| A reaction: This is a firm defence of part of the truth-functional view of conditionals, and seems unassailable. The other parts of the truth table are open to question, though, if A is false, or they are both true. |
| 6940 | The feeling of belief shows a habit which will determine our actions [Peirce] |
| Full Idea: The feeling of believing is a more or less sure indication of there being established in our nature some habit which will determine our actions. Doubt never has such an effect. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.10) | |
| A reaction: It is one thing to assert this fairly accurate observation, and another to assert that this is the essence or definition of a belief. Perhaps it is the purpose of belief, without being the phenomenological essence of it. We act in states of uncertainty. |
| 6941 | We are entirely satisfied with a firm belief, even if it is false [Peirce] |
| Full Idea: As soon as a firm belief is reached we are entirely satisfied, whether the belief be true or false. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.10) | |
| A reaction: This does not deny that the truth or falsehood of a belief is independent of whether we are satisfied with it. It is making a fair point, though, about why we believe things, and it can't be because of truth, because we don't know how to ensure that. |
| 6942 | We want true beliefs, but obviously we think our beliefs are true [Peirce] |
| Full Idea: We seek for a belief that we shall think to be true; but we think each one of our beliefs to be true, and, indeed, it is mere tautology to say so. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
| A reaction: If, as I do, you like to define belief as 'commitment to truth', Peirce makes a rather startling observation. You are rendered unable to ask whether your beliefs are true, because you have defined them as true. Nice point… |
| 6943 | A mere question does not stimulate a struggle for belief; there must be a real doubt [Peirce] |
| Full Idea: The mere putting of a proposition into the interrogative form does not stimulate the mind to any struggle after belief; there must be a real and living doubt. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
| A reaction: This the attractive aspect of Peirce's pragmatism, that he is always focusing on real life rather than abstract theory or pure logic. |
| 6598 | We need our beliefs to be determined by some external inhuman permanency [Peirce] |
| Full Idea: It is necessary that a method should be found by which our beliefs be determined by nothing human, but by some external permanency - by something upon which our thinking has no effect. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877]), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
| A reaction: This very sensible and interesting remark hovers somewhere between empiricism and pragmatism. Fogelin very persuasively builds his account of knowledge on it. The key point is that we hardly ever choose what to believe. See Idea 2454. |
| 6944 | Demonstration does not rest on first principles of reason or sensation, but on freedom from actual doubt [Peirce] |
| Full Idea: It is a common idea that demonstration must rest on indubitable propositions, either first principles of a general nature, or first sensations; but actual demonstration is completely satisfactory if it starts from propositions free from all actual doubt. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
| A reaction: Another nice example of Peirce focusing on the practical business of thinking, rather than abstract theory. I agree with this approach, that explanation and proof do not aim at perfection and indubitability, but at what satisfies a critical mind. |
| 6948 | Doubts should be satisfied by some external permanency upon which thinking has no effect [Peirce] |
| Full Idea: To satisfy our doubts it is necessary that a method should be found by which our beliefs may be determined by nothing human, but by some external permanency - by something upon which our thinking has no effect. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.18) | |
| A reaction: This may be the single most important idea in pragmatism and in the philosophy of science. See Fodor on experiments (Idea 2455). Put the question to nature. The essential aim is to be passive in our beliefs - just let reality form them. |
| 6945 | Once doubt ceases, there is no point in continuing to argue [Peirce] |
| Full Idea: Some people seem to love to argue a point after all the world is fully convinced of it. But no further advance can be made. When doubt ceases, mental action on the subject comes to an end; and, if it did go on, it would be without purpose. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
| A reaction: This is the way Peirce's pragmatism, which deals with how real thinking actually works (rather than abstract logic), deals with scepticism. However, there is a borderline where almost everyone is satisfied, but the very wise person remains sceptical. |
| 6939 | What is true of one piece of copper is true of another (unlike brass) [Peirce] |
| Full Idea: The guiding principle is that what is true of one piece of copper is true of another; such a guiding principle with regard to copper would be much safer than with regard to many other substances - brass, for example. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 8) | |
| A reaction: Peirce is so beautifully simple and sensible. This gives the essential notion of a natural kind, and is a key notion in our whole understanding of physical reality. |
| 6938 | Natural selection might well fill an animal's mind with pleasing thoughts rather than true ones [Peirce] |
| Full Idea: It is probably of more advantage to an animal to have his mind filled with pleasing and encouraging visions, independently of their truth; and thus, upon unpractical subjects, natural selection might occasion a fallacious tendency of thought. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 8) | |
| A reaction: Note that this is a pragmatist saying that a set of beliefs might work fine but be untrue. So Peirce does not have the highly relativistic notion of truth of some later pragmatists. Good for him. Note the early date to be thinking about Darwin. |
| 6946 | If death is annihilation, belief in heaven is a cheap pleasure with no disappointment [Peirce] |
| Full Idea: If death is annihilation, then the man who believes that he will certainly go straight to heaven when he dies, provided he have fulfilled certain simple observances in this life, has a cheap pleasure which will not be followed by the least disappointment. | |
| From: Charles Sanders Peirce (The Fixation of Belief [1877], p.12) | |
| A reaction: This is a nicely wicked summary of one side of Pascal's options. All the problems of the argument are built into Peirce's word "cheap". Peirce goes on to talk about ostriches burying their heads. |