display all the ideas for this combination of texts
11 ideas
18793 | Material implication (and classical logic) considers nothing but truth values for implications [Mares] |
Full Idea: The problem with material implication, and classical logic more generally, is that it considers only the truth value of formulas in deciding whether to make an implication stand between them. It ignores everything else. | |
From: Edwin D. Mares (Negation [2014], 7.1) | |
A reaction: The obvious problem case is conditionals, and relevance is an obvious extra principle that comes to mind. |
18784 | In classical logic the connectives can be related elegantly, as in De Morgan's laws [Mares] |
Full Idea: Among the virtues of classical logic is the fact that the connectives are related to one another in elegant ways that often involved negation. For example, De Morgan's Laws, which involve negation, disjunction and conjunction. | |
From: Edwin D. Mares (Negation [2014], 2.2) | |
A reaction: Mares says these enable us to take disjunction or conjunction as primitive, and then define one in terms of the other, using negation as the tool. |
18786 | Excluded middle standardly implies bivalence; attacks use non-contradiction, De M 3, or double negation [Mares] |
Full Idea: On its standard reading, excluded middle tells us that bivalence holds. To reject excluded middle, we must reject either non-contradiction, or ¬(A∧B) ↔ (¬A∨¬B) [De Morgan 3], or the principle of double negation. All have been tried. | |
From: Edwin D. Mares (Negation [2014], 2.2) |
18780 | Standard disjunction and negation force us to accept the principle of bivalence [Mares] |
Full Idea: If we treat disjunction in the standard way and take the negation of a statement A to mean that A is false, accepting excluded middle forces us also to accept the principle of bivalence, which is the dictum that every statement is either true or false. | |
From: Edwin D. Mares (Negation [2014], 1) | |
A reaction: Mates's point is to show that passively taking the normal account of negation for granted has important implications. |
17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa. |
18782 | The connectives are studied either through model theory or through proof theory [Mares] |
Full Idea: In studying the logical connectives, philosophers of logic typically adopt the perspective of either model theory (givng truth conditions of various parts of the language), or of proof theory (where use in a proof system gives the connective's meaning). | |
From: Edwin D. Mares (Negation [2014], 1) | |
A reaction: [compressed] The commonest proof theory is natural deduction, giving rules for introduction and elimination. Mates suggests moving between the two views is illuminating. |
18783 | Many-valued logics lack a natural deduction system [Mares] |
Full Idea: Many-valued logics do not have reasonable natural deduction systems. | |
From: Edwin D. Mares (Negation [2014], 1) |
18792 | Situation semantics for logics: not possible worlds, but information in situations [Mares] |
Full Idea: Situation semantics for logics consider not what is true in worlds, but what information is contained in situations. | |
From: Edwin D. Mares (Negation [2014], 6.2) | |
A reaction: Since many theoretical physicists seem to think that 'information' might be the most basic concept of a natural ontology, this proposal is obviously rather appealing. Barwise and Perry are the authors of the theory. |
17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) |
17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic. | |
From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) | |
A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'. |
18785 | Consistency is semantic, but non-contradiction is syntactic [Mares] |
Full Idea: The difference between the principle of consistency and the principle of non-contradiction is that the former must be stated in a semantic metalanguage, whereas the latter is a thesis of logical systems. | |
From: Edwin D. Mares (Negation [2014], 2.2) |