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Single Idea 5741

[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models ]

Full Idea

In first-order predicate calculus validity is defined thus: an argument is valid iff every model that makes the premises of the argument true also makes the conclusion of the argument true.

Clarification

'Iff' means if and only if

Gist of Idea

If every model that makes premises true also makes conclusion true, the argument is valid

Source

Joseph Melia (Modality [2003], Ch.2)

Book Ref

Melia,Joseph: 'Modality' [Acumen 2003], p.48

A Reaction

See Melia Ch. 2 for an explanation of a 'model'. Traditional views of validity tend to say that if the premises are true the conclusion has to be true (necessarily), but this introduces the modal term 'necessarily', which is controversial.

The 35 ideas with the same theme [general features of logical models]:

22294 We can show that a concept is consistent by producing something which falls under it [Frege]
13343 A 'model' is a sequence of objects which satisfies a complete set of sentential functions [Tarski]
16323 The object language/ metalanguage distinction is the basis of model theory [Tarski, by Halbach]
13826 Model theory looks at valid sentences and consequence, but not how we know these things [Prawitz]
10473 Model theory studies formal or natural language-interpretation using set-theory [Hodges,W]
10475 A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W]
10481 Models in model theory are structures, not sets of descriptions [Hodges,W]
9968 A model is 'fundamental' if it contains only concrete entities [Jubien]
10827 Model theory is unusual in restricting the range of the quantifiers [Field,H]
13505 Model theory studies how set theory can model sets of sentences [Hart,WD]
13511 Model theory is mostly confined to first-order theories [Hart,WD]
13513 Models are ways the world might be from a first-order point of view [Hart,WD]
13512 Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD]
13644 Semantics for models uses set-theory [Shapiro]
10240 Model theory deals with relations, reference and extensions [Shapiro]
10239 The central notion of model theory is the relation of 'satisfaction' [Shapiro]
15411 We only need to study mathematical models, since all other models are isomorphic to these [Burgess]
15412 Models leave out meaning, and just focus on truth values [Burgess]
15416 We aim to get the technical notion of truth in all models matching intuitive truth in all instances [Burgess]
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]
13724 In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation [Sider]
10129 A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman]
5741 If every model that makes premises true also makes conclusion true, the argument is valid [Melia]
19207 Sentence logic maps truth values; predicate logic maps objects and sets [Merricks]
10158 A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman]
10162 Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman]
10693 Models are mathematical structures which interpret the non-logical primitives [Beall/Restall]
13519 Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
13531 Model theory reveals the structures of mathematics [Wolf,RS]
13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
13533 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
18694 Permutation Theorem: any theory with a decent model has lots of models [Button]
10756 A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg]
18744 Models are sets with functions and relations, and truth built up from the components [Horsten/Pettigrew]
17747 A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]