Consequently, there was no rigorous account of what it means for a sentence in those languages to be true and, hence, no account of the critical semantic notions of validity and logical consequence to underwrite the corresponding deductive notions of theoremhood and provability.
Now, the deductive-logical notions of entailment and refutation (contradiction) exhibit the following well-known properties: Contraposition of entailment Entailment is contrapositive, but not commutative.
In doing so, it can be regarded also as generalizing deductive logic and its notion of implication, to a complete theory of inference equipped with the notion of 'degree of implication' that relates E to H.
The constraint of necessity is not sufficient to settle the notion of deductive validity, for the notion of necessity may also be fleshed out in a number of ways.
Let us briefly consider each axiom, 1-5, to see how plausible it is as a constraint on a quantitative measure of inductive support, and how it extends the notion of deductive entailment.
Perhaps this should be supplemented by the notion of a Deductive Slip, meaning something a politician says, however inadvertently, that can be shoehorned into a pre-existing "narrative".
The idea was to extend the deductive entailment relation to a notion of probabilistic entailment for cases where premises provide less than conclusive support for conclusions.
First, we need a laxer notion of prediction than deductive entailment.
They do not investigate formal logic or deductive inference, nor formulate an explicit notion of logical consequence.
In order to give an account of this notion we wil introduce a deductive system (a natural deduction system), which explicate the intuitive notion of 'follow' in terms of derivational rules in a calculus.
