display all the ideas for this combination of texts
26 ideas
| 14187 | If logic is topic-neutral that means it delves into all subjects, rather than having a pure subject matter [Read] |
| Full Idea: The topic-neutrality of logic need not mean there is a pure subject matter for logic; rather, that the logician may need to go everywhere, into mathematics and even into metaphysics. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Logic') |
| 14188 | Not all arguments are valid because of form; validity is just true premises and false conclusion being impossible [Read] |
| Full Idea: Belief that every valid argument is valid in virtue of form is a myth. ..Validity is a question of the impossibility of true premises and false conclusion for whatever reason, and some arguments are materially valid and the reason is not purely logical. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Logic') | |
| A reaction: An example of a non-logical reason is the transitive nature of 'taller than'. Conceptual connections are the usual example, as in 'it's red so it is coloured'. This seems to be a defence of the priority of semantic consequence in logic. |
| 14182 | If the logic of 'taller of' rests just on meaning, then logic may be the study of merely formal consequence [Read] |
| Full Idea: In 'A is taller than B, and B is taller than C, so A is taller than C' this can been seen as a matter of meaning - it is part of the meaning of 'taller' that it is transitive, but not of logic. Logic is now seen as the study of formal consequence. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Reduct') | |
| A reaction: I think I find this approach quite appealing. Obviously you can reason about taller-than relations, by putting the concepts together like jigsaw pieces, but I tend to think of logic as something which is necessarily implementable on a machine. |
| 14183 | Maybe arguments are only valid when suppressed premises are all stated - but why? [Read] |
| Full Idea: Maybe some arguments are really only valid when a suppressed premise is made explicit, as when we say that 'taller than' is a transitive concept. ...But what is added by making the hidden premise explicit? It cannot alter the soundness of the argument. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Suppress') |
| 13678 | The most popular account of logical consequence is the semantic or model-theoretic one [Sider] |
| Full Idea: On the question of the nature of genuine logical consequence, ...the most popular answer is the semantic, or model-theoretic one. | |
| From: Theodore Sider (Logic for Philosophy [2010], 1.5) | |
| A reaction: Reading the literature, one might be tempted to think that this is the only account that anyone takes seriously. Substitutional semantics seems an interesting alternative. |
| 13679 | Maybe logical consequence is more a matter of provability than of truth-preservation [Sider] |
| Full Idea: Another answer to the question about the nature of logical consequence is a proof-theoretic one, according to which it is more a matter of provability than of truth-preservation. | |
| From: Theodore Sider (Logic for Philosophy [2010], 1.5) | |
| A reaction: I don't like this, and prefer the model-theoretic or substitutional accounts. Whether you can prove that something is a logical consequence seems to me entirely separate from whether you can see that it is so. Gödel seems to agree. |
| 13682 | Maybe logical consequence is impossibility of the premises being true and the consequent false [Sider] |
| Full Idea: The 'modal' account of logical consequence is that it is not possible for the premises to be true and the consequent false (under some suitable notion of possibility). | |
| From: Theodore Sider (Logic for Philosophy [2010], 1.5) | |
| A reaction: Sider gives a nice summary of five views of logical consequence, to which Shapiro adds substitutional semantics. |
| 13680 | Maybe logical consequence is a primitive notion [Sider] |
| Full Idea: There is a 'primitivist' account, according to which logical consequence is a primitive notion. | |
| From: Theodore Sider (Logic for Philosophy [2010], 1.5) | |
| A reaction: While sympathetic to substitutional views (Idea 13674), the suggestion here pushes me towards thinking that truth must be at the root of it. The trouble, though, is that a falsehood can be a good logical consequence of other falsehoods. |
| 13722 | A 'theorem' is an axiom, or the last line of a legitimate proof [Sider] |
| Full Idea: A 'theorem' is defined as the last line of a proof in which each line is either an axiom or follows from earlier lines by a rule. | |
| From: Theodore Sider (Logic for Philosophy [2010], 9.7) | |
| A reaction: In other words, theorems are the axioms and their implications. |
| 14184 | In modus ponens the 'if-then' premise contributes nothing if the conclusion follows anyway [Read] |
| Full Idea: A puzzle about modus ponens is that the major premise is either false or unnecessary: A, If A then B / so B. If the major premise is true, then B follows from A, so the major premise is redundant. So it is false or not needed, and contributes nothing. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Repres') | |
| A reaction: Not sure which is the 'major premise' here, but it seems to be saying that the 'if A then B' is redundant. If I say 'it's raining so the grass is wet', it seems pointless to slip in the middle the remark that rain implies wet grass. Good point. |
| 14186 | Logical connectives contain no information, but just record combination relations between facts [Read] |
| Full Idea: The logical connectives are useful for bundling information, that B follows from A, or that one of A or B is true. ..They import no information of their own, but serve to record combinations of other facts. | |
| From: Stephen Read (Formal and Material Consequence [1994], 'Repres') | |
| A reaction: Anyone who suggests a link between logic and 'facts' gets my vote, so this sounds a promising idea. However, logical truths have a high degree of generality, which seems somehow above the 'facts'. |
| 13696 | When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity [Sider] |
| Full Idea: When a variable is not combined with a quantifier (and so is 'free'), the result is, intuitively, semantically incomplete, and incapable of truth or falsity. | |
| From: Theodore Sider (Logic for Philosophy [2010], 4.2) |
| 13700 | A 'total' function must always produce an output for a given domain [Sider] |
| Full Idea: Calling a function a 'total' function 'over D' means that the function must have a well-defined output (which is a member of D) whenever it is given as inputs any n members of D. | |
| From: Theodore Sider (Logic for Philosophy [2010], 5.2) |
| 13703 | λ can treat 'is cold and hungry' as a single predicate [Sider] |
| Full Idea: We might prefer λx(Fx∧Gx)(a) as the symbolization of 'John is cold and hungry', since it treats 'is cold and hungry' as a single predicate. | |
| From: Theodore Sider (Logic for Philosophy [2010], 5.5) |
| 13688 | Good axioms should be indisputable logical truths [Sider] |
| Full Idea: Since they are the foundations on which a proof rests, the axioms in a good axiomatic system ought to represent indisputable logical truths. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.6) |
| 13687 | No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider] |
| Full Idea: Axiomatic systems do not allow reasoning with assumptions, and therefore do not allow conditional proof or reductio ad absurdum. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.6) | |
| A reaction: Since these are two of the most basic techniques of proof which I have learned (in Lemmon), I shall avoid axiomatic proof systems at all costs, despites their foundational and Ockhamist appeal. |
| 13690 | Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms [Sider] |
| Full Idea: The style of proof called 'induction on formula construction' (or 'on the number of connectives', or 'on the length of the formula') rest on the fact that all formulas are built up from atomic formulas according to strict rules. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.7) | |
| A reaction: Hence the proof deconstructs the formula, and takes it back to a set of atomic formulas have already been established. |
| 13691 | Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider] |
| Full Idea: A proof by induction starts with a 'base case', usually that an atomic formula has some property. It then assumes an 'inductive hypothesis', that the property is true up to a certain case. The 'inductive step' then says it will be true for the next case. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.7) | |
| A reaction: [compressed] |
| 13685 | Natural deduction helpfully allows reasoning with assumptions [Sider] |
| Full Idea: The method of natural deduction is popular in introductory textbooks since it allows reasoning with assumptions. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.5) | |
| A reaction: Reasoning with assumptions is generally easier, rather than being narrowly confined to a few tricky axioms, You gradually show that an inference holds whatever the assumption was, and so end up with the same result. |
| 13686 | We can build proofs just from conclusions, rather than from plain formulae [Sider] |
| Full Idea: We can construct proofs not out of well-formed formulae ('wffs'), but out of sequents, which are some premises followed by their logical consequence. We explicitly keep track of the assumptions upon which the conclusion depends. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.5.1) | |
| A reaction: He says the method of sequents was invented by Gerhard Gentzen (the great nazi logician) in 1935. The typical starting sequents are the introduction and elimination rules. E.J. Lemmon's book, used in this database, is an example. |
| 13697 | Valuations in PC assign truth values to formulas relative to variable assignments [Sider] |
| Full Idea: A valuation function in predicate logic will assign truth values to formulas relative to variable assignments. | |
| From: Theodore Sider (Logic for Philosophy [2010], 4.2) | |
| A reaction: Sider observes that this is a 'double' relativisation (due to Tarski), since propositional logic truth was already relative to an interpretation. Now we are relative to variable assignments as well. |
| 13684 | The semantical notion of a logical truth is validity, being true in all interpretations [Sider] |
| Full Idea: The semantical notion of a logical truth is that of a valid formula, which is true in all interpretations. In propositional logic they are 'tautologies'. | |
| From: Theodore Sider (Logic for Philosophy [2010], 2.3) | |
| A reaction: This implies that there is a proof-theoretic account of logical truth as well. Intuitively a logical truth is a sequent which holds no matter which subject matter it refers to, so the semantic view sounds OK. |
| 13704 | It is hard to say which are the logical truths in modal logic, especially for iterated modal operators [Sider] |
| Full Idea: It isn't clear which formulas of modal propositional logic are logical truths, ...especially for sentences that contain iterations of modal operators. Is □P→□□P a logical truth? It's hard to say. | |
| From: Theodore Sider (Logic for Philosophy [2010], 6.3) | |
| A reaction: The result, of course, is that there are numerous 'systems' for modal logic, so that you can choose the one that gives you the logical truths you want. His example is valid in S4 and S5, but not in the others. |
| 13724 | In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation [Sider] |
| Full Idea: In model theory one normally defines some notion of truth in a model, and then uses it to define validity as truth in all models, and semantic consequence as the preservation of truth in models. | |
| From: Theodore Sider (Logic for Philosophy [2010], 10.1) |
| 13698 | In a complete logic you can avoid axiomatic proofs, by using models to show consequences [Sider] |
| Full Idea: You can establish facts of the form Γ|-φ while avoiding the agonies of axiomatic proofs by reasoning directly about models to conclusions about semantic consequence, and then citing completeness. | |
| From: Theodore Sider (Logic for Philosophy [2010], 4.5) | |
| A reaction: You cite completeness by saying that anything which you have shown to be a semantic consequence must therefore be provable (in some way). |
| 13699 | Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider] |
| Full Idea: Compactness is intuitively surprising, ..because one might have thought there could be some contradiction latent within some infinite set, preventing it from being satisfiable, only discovered when you consider the whole set. But this can't happen. | |
| From: Theodore Sider (Logic for Philosophy [2010], 4.5) |