Valid Logical statements not only establish the validity of other statements within that Domain, they establish the Boundaries of Domains by the transformation of Logical Formulations of Validity. ("IT IS VERY BAD TO PULL THE RIPCORD. INSIDE THE PLANE. BUT NOT AFTER YOU JUMP OUT. BUT NOT TOO SOON. WAITING UNTIL AFTER YOU HIT THE GROUND IS NOT TERRIFIC EITHER.")
The Five Axioms of Principia Mathematica
Explanatory notes copied from:Wikiproofs.org
- The next axiom, called Taut for tautology, asserts the idempotency of disjunction in one direction:
- stmt (Taut () () ((p ∨ p) → p)) # *1.2
- A1. CAppp
- (One of the Five Necessary Tests of Validity is Reproducibility.)
- ((D1 --> p) v (D2 --> p)) --> (p --> D1 v D2)
- (Either David takes the Red Pill OR Donna takes the Red Pill OR both implies that a missing Red Pill implies either David OR Donna OR both, has taken that Red Pill.) ___ (All ur 1(one) Red Pills is mine!) ___ P;D
- The Add axiom, for addition, asserts that a disjunction may be added to a wff.
- stmt (Add () () (q → (p ∨ q))) # *1.3
- A2. CqApq
- (Every Valid Statement Poses the (Possibility? Certainty?) of its (Absence? Lack of Confirmation?) in a Valid Domain.)
- (D1 --> (q --> ~p)) --> (D2 --> q v p)
- The Perm axiom, for permutation, asserts the commutativity of disjunction.
- stmt (Perm () () ((p ∨ q) → (q ∨ p))) # *1.4
- A3. CApqAqp
- (((p v q) --> p) --> D1) --> (((q v p) --> q) --> D2)
- The next axiom, Assoc for associativity, asserts the associative law for disjunction. However, there is a twist. A commutation is built in the axiom as well. Thus, the axiom obtains more deductive power than plain associativity alone.
- stmt (Assoc () () ((p ∨ (q ∨ r)) → (q ∨ (p ∨ r)))) # *1.5
- A4. CApAqrAqApr
- (Rhetoric:The comparison of an alternative Argument to a Binary Set of Arguments implies it can be compared to either of the two Arguments of that Set.)
- The final axiom, Sum for summation, provides the basic building block for assembling complex formulas. It states that , meaning that an arbitrary formula may be added to both antecedent and consequent of an implication.
- stmt (Sum () () ((q → r) → ((p ∨ q) → (p ∨ r)))) # *1.6
- A5. CCqrCApqApr
- (Rhetoric: Denial of the Argument forgoes the proffered Consequence.)
- If one may assert p, and also that p implies q (i.e., ), then one may infer q.
- This rule has become known as modus ponens. Since the corresponding JHilbert statement requires hypotheses, we formulate it in imperative form:
- stmt (applyModusPonens () (p (p → q)) q) # *1.1
The Derivation of "Modus Ponens" from the Five Axioms alone
Proof of Modus Ponens
- Propositional logic can get founded in many different ways using different rules of inference and different axioms. Most formulations of propositional calculus have modus ponens as a basic rule of inference. That is, "From p, and Cpq, infer q". In fact, some systems of propositional calculus have only modus ponens as a rule of inference. All of these systems have other rules of inference, such as modus tollens, as derivable from modus ponens. Can we have a formulation of propositional calculus without modus ponens? If we formulate a set of inference rules which have modus ponens as a derivable rule of inference, then it would at least seem to follow that we don't need to have modus ponens as a basic rule of inference for all formulations of propositional calculus.
- Suppose we have Cpq defined as NKpNq. Let our rules of inference go 1. negation introduction: if from a wff q, we can derive a wff of the form KpNp, we can infer Nq. 2. negation omission: from a wff of the form NNp, we may infer p. 3. conjunction introduction: from a wff p, and from a wff q, we may infer Kpq. With these rules in place we can derive modus ponens. We need to have p, and Cpq as hypotheses, and derive q. So
1 p assumption
2 Cpq assumption
3 |Nq assumption
4 |NKpNq 2, Cpq==NKpNq definition
5 |KpNq 1, 3 conjunction introduction
6 |KKpNqNKpNq 4, 5 conjunction introduction
7 NNq 3-6 negation introduction
8 q 7 negation omission
Paradoxes of material implication
Doug Spoonwood also introduces us to the "paradoxes of material implication", giving us a link to a Wikipedia entry: Paradoxes of material implication
The paradoxes are logical statements which are true but whose truth is intuitively surprising to people who are not familiar with them. If the terms 'p', 'q' and 'r' stand for arbitrary propositions then the main paradoxes are given formally as follows:
- (P & ~P -> Q)
- p and its negation imply q. This is the paradox of entailment.
- P -> (Q -> P)
- if p is true then it is implied by every q.
- ~P -> (P -> Q)
- if p is false then it implies every q. This is referred to as 'explosion'.
- [I believe that it is this little gem that makes Godel's Theorem what it is. PAB]
- P -> (Q V ~Q)
- either q or its negation is true, so their disjunction is implied by every p.
- [Every feature or property comes into existence in terms of a want or need? PAB]
- (P -> Q) V (Q -> R)
- if p, q and r are three arbitrary propositions, then either p implies q or q implies r. This is because if q is true then p implies it, and if it is false then q implies any other statement. Since r can be p, it follows that given two arbitrary propositions, one must imply the other, even if they are mutually contradictory. For instance, "Nadia is in Barcelona implies Nadia is singing, or Nadia is singing implies Nadia is in Barcelona." This is the rather conventional question of Correlation and Causation within a Rhetorical Argument.
- ~(P -> Q) -> (P & ~Q)
- if p does not imply q then p is true and q is false. NB if p were false then it would imply q, so p is true. If q were also true then p would imply q, hence q is false. This paradox is particularly surprising because it tells us that if one proposition does not imply another then the first is true and the second false.
- The paradoxes of material implication arise because of the truth-functional definition of material implication, which is said to be true merely because the antecedent is false or the consequent is true. By this criterion, "If the moon is made of green cheese, then the world is coming to an end," is true merely because the moon isn't made of green cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true. (All paraconsistent logics must, by definition, reject (1) as false.) Also, any tautology is implied by anything whatsoever, since a tautology is always true.
- To sum up, although it is deceptively similar to what we mean by "logically follows" in ordinary usage, material implication does not capture the meaning of "if... then".
((Q -> P) & (P -> Q) -> (P -> Q))