display all the ideas for this combination of texts
13 ideas
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) |
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) |
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. |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
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. |
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) |
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) |
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) |
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) |
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) |