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6.2 Conditional derivation. As a handy rule of thumb, we can think of the inference rules as providing a way to either show a kind of sentence, or to make use of a kind of sentence. For example, adjunction allows us to show a conjunction. Simplification allows us to make use of a conjunction.
The above examples involve two show-lines; each one involves a direct derivation inside a conditional derivation. The following examples introduce a new twist – three show-lines in the same derivation, with a conditional derivation inside a conditional derivation. Example 4 (1) (P & Q) † R Pr (2) ›: P † (Q † R) CD (3) |P As
These examples suggest that we ought to also add the following two rules, to deal with ∨, which represents “or”: Modus Tollendo Ponens and Addition. Modus Tollendo Ponens (abbreviated MTP), the argument which takes the form. ϕ ∨ ψ, ¬ϕ ⊢ ψ. or. ϕ ∨ ψ, ¬ψ ⊢ ϕ. is a rule of direct inference.
As this example illustrates, there are three basic operations involved in creating useful subproofs - (1) making assumptions, (2) using ordinary rules of inference to derive conclusions, and (3) using conditional rules of inference to derive conclusions outside of subproofs. Let's look at each of these operations in turn.
Conditional elimination. This method of proof is also known by its Latin name, modus ponens (literally, “method of affirming”—roughly, having affirmed the antecedent of a conditional, you may affirm the consequent). From P and P → Q , you may infer Q. Biconditional elimination. This is sometimes called “modus ponens for the biconditional.”
22 lis 2023 · Example derivation. [ | ] We begin with an example derivation which illustrates Conditional Introduction, then follow with an explanation. A derivation for the argument. ∧ ∴ →. is as follows: 1. ∧. Premise. 2. Assumption. 3. 1 KE.
