To get an idea of how esoteric New Math curricula have been, consider the topic of
material implication. This topic deals with conditional sentences, which are statements in the form of "if ..., then."
Material implication obviates any causal relationship in these statements.
The notion of
material implication is the source for the semantic definition of the conditional constructions.
The heterogeneity in generators' responses of fuel consumption to fuel price has
material implication for carbon dioxide emissions.
This paper extends the defense of a simple theory of indicative conditionals previously proposed by the author, in which the truth conditions are material, and Grice-style assertability conditions are given to explain the paradoxes of
material implication. The paper discusses various apparent counterexamples to the material account in which conditionals are not asserted, and so the original theory cannot be applied; it is argued that, nevertheless, the material theory can be defended.
To avoid some pitfalls of Lukasiewicz's construction Field introduces a special conditional beside
material implication. The conditional is true at a stage if there is an ordinal (in the preceding iteration process) starting from which the antecedent always has a lower semantic value than the consequent; it is false if, starting from some ordinal, it always has a higher semantic value; otherwise it is neither true nor false.
Drawing from the methodical structures of empiricism, "
Material Implication" begins each poem with an "if-then" clause.
Starting from the observation that the consequential relation does not happen when the antecedent is true and the consequent is false, and from the tendency to reduce the relations between sentences to truth functions, the consequential relation is represented through the truth function of
material implication. According to this analysis, if the
material implication from a sentence "p" to the sentence "q" is a tautology, then the fact q is a consequence of the fact p and the other way around.
Some paradoxes require the revision of their intuitive conception (Russell's paradox, Cantor's paradox), others depend on the inadmissibility of their description (Grelling's paradox), others show counter-intuitive features of formal theories (
Material implication paradox, Skolem Paradox), others are self-contradictory--Smarandache Paradox: "All is <A> the <Non-A> too!", where <A> is an attribute and <Non-A> its opposite; for example "All is possible the impossible too!" (Weisstein, 1998 [2]).
A distinction is made between
material implication, represented by the symbol [contains], and strict implication, represented by [right arrow].
It is noteworthy that he succeeds in introducing implication as
material implication ([right arrow]) in a very convincing way.
To derive the instances, an additional principle is needed, such as McGrath's principle of
material implication (PMI),
Rosenthal discusses Lewis' groundbreaking work in logic, which originated from his objections to the concept of
material implication and its development in the extensional logic of Russell and Whitehead's Principia Mathematica.