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5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence

[different modes of logical consequence]

6 ideas
Syntactic consequence is positive, for validity; semantic version is negative, with counterexamples [Dummett]
     Full Idea: A plausible account is that the syntactic notion of consequence is for positive results, that some form of argument is valid; the semantic notion is required for negative results, that some argument is invalid, because a counterexample can be found.
     From: Michael Dummett (The Justification of Deduction [1973], p.292)
     A reaction: This rings true for the two strategies of demonstration, the first by following the rules in steps, the second by using your imagination (or a tableau) to think up problems.
Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton]
     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 |-).
     From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.3..)
     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 two standard explanations of consequence are semantic (in models) and deductive [Shapiro]
     Full Idea: The two best historical explanations of consequence are the semantic (model-theoretic), and the deductive versions.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2)
     A reaction: Shapiro points out the fictionalists are in trouble here, because the first involves commitment to sets, and the second to the existence of deductions.
Logical consequence needs either proofs, or absence of counterexamples [Beall/Restall]
     Full Idea: Technical work on logical consequence has either focused on proofs, where validity is the existence of a proof of the conclusions from the premises, or on models, which focus on the absence of counterexamples.
     From: JC Beall / G Restall (Logical Consequence [2005], 3)
There are several different consequence relations [Beall/Restall]
     Full Idea: We are pluralists about logical consequence because we take there to be a number of different consequence relations, each reflecting different precisifications of the pre-theoretic notion of deductive logical consequence.
     From: JC Beall / G Restall (Logical Pluralism [2006], 8)
     A reaction: I don't see how you avoid the slippery slope that leads to daft logical rules like Prior's 'tonk' (from which you can infer anything you like). I say that nature imposes logical conquence on us - but don't ask me to prove it.
Logical consequence is intuitively semantic, and captured by model theory [Rossberg]
     Full Idea: Logical consequence is intuitively taken to be a semantic notion, ...and it is therefore the formal semantics, i.e. the model theory, that captures logical consequence.
     From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
     A reaction: If you come at the issue from normal speech, this seems right, but if you start thinking about the necessity of logical consequence, that formal rules and proof-theory seem to be the foundation.