It would be impossible to restrict these formulas in advance, thus no subformula property can hold. ❋ Unknown (2009)
Consistency would then follow from the normalization of derivations and the subformula property. ❋ Unknown (2009)
Prawitz gave first a normalization theorem and subformula property for a system of natural deduction for classical logic. ❋ Unknown (2009)
Formulas in derivations are arranged in a linear succession, but Jaskowski's paper of 1934 remained fragmentary and without substantial results such as a subformula property. ❋ Unknown (2009)
The standard terminology today is “cut elimination theorem” All of the logical rules of SC have the subformula property in a very immediate sense: each formula in a premiss is a formula or subformula in the conclusion. ❋ Unknown (2009)
Disjunction and conjunction differ only by the player who has the choice of the immediate subformula with which the game will proceed: in a conjunction, the challenger may choose, confident that either disjunct can be refuted; in a disjunction the choice lies with the defender. ❋ Keiff, Laurent (2009)
In contrast, an application of detachment results in a formula that is a proper subformula of the major premise (and in the exceptional case when the major premise is an instance of self-identity it is identical to the minor premise). ❋ Bimbó, Katalin (2008)
The axioms are all formulas of the following forms, where in the last two schemas the subformula A (t) is the result of substituting an occurrence of the term t for every free occurrence of x in A (x), and no variable free in t becomes bound in A (t) as a result of the substitution. ❋ Moschovakis, Joan (2007)
Fortunately, NK enjoys the subformula property in the sense that each formula entering into a natural deduction proof can be restricted to being a subformula of Γ ª Δ ª {α}, where Δ is the set of auxiliary assumptions made by the ~-Elimination rule. ❋ Portoraro, Frederic (2005)
By exploiting the subformula property a natural deduction automated theorem prover can drastically reduce its search space and bring the backward application of the elimination rules under control (Portoraro 1998, Sieg and Byrnes ❋ Portoraro, Frederic (2005)
