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18781 | Inconsistency doesn't prevent us reasoning about some system |
Full Idea: We are able to reason about inconsistent beliefs, stories, and theories in useful and important ways | |||
From: Edwin D. Mares (Negation [2014], 1) |
18789 | Intuitionist logic looks best as natural deduction |
Full Idea: Intuitionist logic appears most attractive in the form of a natural deduction system. | |||
From: Edwin D. Mares (Negation [2014], 5.5) |
18790 | Intuitionism as natural deduction has no rule for negation |
Full Idea: In intuitionist logic each connective has one introduction and one elimination rule attached to it, but in the classical system we have to add an extra rule for negation. | |||
From: Edwin D. Mares (Negation [2014], 5.5) | |||
A reaction: How very intriguing. Mares says there are other ways to achieve classical logic, but they all seem rather cumbersome. |
18787 | Three-valued logic is useful for a theory of presupposition |
Full Idea: One reason for wanting a three-valued logic is to act as a basis of a theory of presupposition. | |||
From: Edwin D. Mares (Negation [2014], 3.1) | |||
A reaction: [He cites Strawson 1950] The point is that you can get a result when the presupposition does not apply, as in talk of the 'present King of France'. |
18793 | Material implication (and classical logic) considers nothing but truth values for implications |
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 |
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 |
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 |
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. |
18782 | The connectives are studied either through model theory or through proof theory |
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 |
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 |
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. |
18785 | Consistency is semantic, but non-contradiction is syntactic |
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) |
18788 | For intuitionists there are not numbers and sets, but processes of counting and collecting |
Full Idea: For the intuitionist, talk of mathematical objects is rather misleading. For them, there really isn't anything that we should call the natural numbers, but instead there is counting. What intuitionists study are processes, such as counting and collecting. | |||
From: Edwin D. Mares (Negation [2014], 5.1) | |||
A reaction: That is the first time I have seen mathematical intuitionism described in a way that made it seem attractive. One might compare it to a metaphysics based on processes. Apparently intuitionists struggle with infinite sets and real numbers. |
18791 | In 'situation semantics' our main concepts are abstracted from situations |
Full Idea: In 'situation semantics' individuals, properties, facts, and events are treated as abstractions from situations. | |||
From: Edwin D. Mares (Negation [2014], 6.1) | |||
A reaction: [Barwise and Perry 1983 are cited] Since I take the process of abstraction to be basic to thought, I am delighted to learn that someone has developed a formal theory based on it. I am immediately sympathetic to situation semantics. |