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Single Idea 9718
[filed under theme 5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
]
Full Idea
The point of logic is to give an account of the notion of validity,..in two standard ways: the semantic way says that a valid inference preserves truth (symbol |=), and the proof-theoretic way is defined in terms of purely formal procedures (symbol |-).
Gist of Idea
Validity is either semantic (what preserves truth), or proof-theoretic (following procedures)
Source
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.3..)
Book Ref
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.1
A Reaction
This division can be mirrored in mathematics, where it is either to do with counting or theorising about things in the physical world, or following sets of rules from axioms. Language can discuss reality, or play word-games.
The
38 ideas
from Herbert B. Enderton
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13200
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Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ
[Enderton]
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13199
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The empty set may look pointless, but many sets can be constructed from it
[Enderton]
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13201
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∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
[Enderton]
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13202
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Fraenkel added Replacement, to give a theory of ordinal numbers
[Enderton]
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13203
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The singleton is defined using the pairing axiom (as {x,x})
[Enderton]
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13204
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The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}
[Enderton]
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13205
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We can only define functions if Choice tells us which items are involved
[Enderton]
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13206
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A 'linear or total ordering' must be transitive and satisfy trichotomy
[Enderton]
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9718
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Validity is either semantic (what preserves truth), or proof-theoretic (following procedures)
[Enderton]
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9719
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A proof theory is 'sound' if its valid inferences entail semantic validity
[Enderton]
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9720
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A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
[Enderton]
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9724
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Until the 1960s the only semantics was truth-tables
[Enderton]
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9994
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A truth assignment to the components of a wff 'satisfy' it if the wff is then True
[Enderton]
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9721
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A logical truth or tautology is a logical consequence of the empty set
[Enderton]
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9723
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Sentences with 'if' are only conditionals if they can read as A-implies-B
[Enderton]
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9996
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Expressions are 'decidable' if inclusion in them (or not) can be proved
[Enderton]
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9722
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Inference not from content, but from the fact that it was said, is 'conversational implicature'
[Enderton]
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9995
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Proof in finite subsets is sufficient for proof in an infinite set
[Enderton]
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9997
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For a reasonable language, the set of valid wff's can always be enumerated
[Enderton]
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9712
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A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
[Enderton]
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9713
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A relation is 'transitive' if it can be carried over from two ordered pairs to a third
[Enderton]
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9703
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'dom R' indicates the 'domain' of objects having a relation
[Enderton]
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9705
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'fld R' indicates the 'field' of all objects in the relation
[Enderton]
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9704
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'ran R' indicates the 'range' of objects being related to
[Enderton]
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9710
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We write F:A→B to indicate that A maps into B (the output of F on A is in B)
[Enderton]
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9707
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'F(x)' is the unique value which F assumes for a value of x
[Enderton]
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9715
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An 'equivalence relation' is a reflexive, symmetric and transitive binary relation
[Enderton]
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9716
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We 'partition' a set into distinct subsets, according to each relation on its objects
[Enderton]
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9699
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The 'powerset' of a set is all the subsets of a given set
[Enderton]
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9700
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Two sets are 'disjoint' iff their intersection is empty
[Enderton]
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9702
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A 'domain' of a relation is the set of members of ordered pairs in the relation
[Enderton]
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9701
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A 'relation' is a set of ordered pairs
[Enderton]
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9706
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A 'function' is a relation in which each object is related to just one other object
[Enderton]
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9708
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A function 'maps A into B' if the relating things are set A, and the things related to are all in B
[Enderton]
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9709
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A function 'maps A onto B' if the relating things are set A, and the things related to are set B
[Enderton]
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9711
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A relation is 'reflexive' on a set if every member bears the relation to itself
[Enderton]
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9714
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A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects
[Enderton]
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9717
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A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second
[Enderton]
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