Implicational Propositional Calculus

In the desolate landscape of mathematical logic, where precision is a currency and ambiguity a capital offense, one occasionally stumbles upon stark, minimalist constructions. Among these is the implicational propositional calculus—a pared-down, almost ascetic, version of classical propositional calculus. Its defining characteristic, and indeed its only functional component, is a singular connective: the venerable implication or conditional. This binary operation, often denoted by the deceptively simple "implies," "if ..., then ...," "→," or its more formal typographic sibling "

→

{\displaystyle \rightarrow }

," forms the entire expressive power of this particular logical system. It posits a relationship where the truth of the first proposition necessarily leads to the truth of the second, or, more accurately, it simply asserts that it's not the case that the first is true and the second is false. A rather limited palette, one might observe, but occasionally, limitations force a certain... clarity.

Functional (in)completeness

Now, one might be tempted to assume that such a fundamental connective could, by itself, build the entire edifice of two-valued logic. A rather naive assumption, if you're paying attention. Implication alone is, regrettably, not functionally complete as a logical operator. This isn't a flaw; it's a statement of fact, a stark reflection of its singular nature. You simply cannot construct all other fundamental two-valued truth functions from it. It lacks the raw expressive power to represent the full spectrum of logical relationships.

Consider, for instance, the most basic of logical absurdities: the two-place truth function that perpetually returns false. A rather straightforward concept, yet utterly elusive within the confines of pure implication. Any formula meticulously constructed solely from the implication connective and a collection of arbitrary propositional variables will invariably yield the value 'true' if all its constituent variables are themselves assigned 'true'. This is not a profound insight; it's merely how implication behaves. If 'true' implies 'true', the result is 'true'. If all components are 'true', the overall structure remains 'true'. Ergo, since it cannot produce a 'false' output under an all-'true' input scenario, it follows with the crushing weight of inevitability that the set {→} is emphatically not functionally complete. A rather obvious conclusion, really, if you bother to trace the implications.

However, the situation is not entirely hopeless, assuming one is willing to introduce a single, non-negotiable concession: a nullary connective, ⊥, representing absolute falsity. With this humble addition, the landscape shifts. Suddenly, the full expressive power of classical logic becomes accessible. Formulas constructed over this expanded set of connectives, {→, ⊥}, are, rather aptly, termed f-implicational. The introduction of ⊥ acts as a logical anchor, providing the necessary 'false' starting point from which all other logical constructs can be precisely defined. For any two propositions, P and Q, the transformation is quite elegant:

The negation of P, written as ¬ P, achieves logical equivalence with the implicational statement P → ⊥. If P implies falsity, then P must itself be false.
The conjunction P ∧ Q, which asserts that both P and Q are true, can be precisely expressed as (P → (Q → ⊥)) → ⊥. This rather convoluted structure effectively states that if P implies that Q implies falsity, then that entire proposition must itself imply falsity, meaning P and Q must both be true.
The disjunction P ∨ Q, asserting that at least one of P or Q is true, finds its equivalent in (P → Q) → Q. This translates to: if P implies Q, and that implication itself implies Q, then P or Q must be true.
Finally, the biconditional P ↔ Q, signifying that P and Q share the same truth value, can be formulated as ((P → Q) → ((Q → P) → ⊥)) → ⊥. A mouthful, certainly, but it logically captures the essence of mutual implication leading to non-falsity.

Given that these derived operators (negation, conjunction, disjunction, and biconditional) are themselves known to constitute a functionally complete set, it becomes undeniably clear that any conceivable truth function within classical two-valued logic can be articulated using only the implication connective and the constant for falsity, ⊥. A rather neat trick, if you can appreciate the economy of it.

Axiom system

To establish a formal system, one requires a bedrock of undeniable truths and a set of rules for constructing new truths. In the realm of the implicational propositional calculus, these foundational statements are designated as tautologies—propositions inherently true, demanding no external validation. They are, by their very definition, irreducible and intuitively self-evident, though 'intuitive' often means 'obvious to anyone who has stared at this for a sufficient number of hours'.

The classical presentation of this calculus typically includes the following axiom schemas, where P, Q, and R are placeholders for any valid formulas constructed solely using the implication connective:

Axiom schema 1: P → (Q → P). This axiom, often called 'positive paradox' or 'weakening', essentially states that if P is true, then it is true regardless of Q. Or, more simply, anything implies a truth. A rather low bar for implication, if you ask me.
Axiom schema 2: (P → (Q → R)) → ((P → Q) → (P → R)). This is a form of self-distributivity, often referred to as the 'exportation law' or 'Frege's axiom'. It asserts that if P implies that Q implies R, then if P implies Q, it must also imply R. It's a statement about the transitivity of implication under certain conditions, a property one might expect.
Axiom schema 3 (Peirce's law): ((P → Q) → P) → P. This one is a bit more intriguing, and certainly less 'intuitive' in the immediate sense. It essentially states that if assuming P implies Q leads to P, then P must be true. It's a powerful tool, particularly in proofs by contradiction, and its presence is what ensures the system's classical nature. Without it, you're venturing into intuitionistic territory, and frankly, who has the time for that?

Alongside these axiomatic declarations, there is precisely one non-nullary rule of inference, the venerable modus ponens. This rule is the engine of deduction, the mechanism by which new theorems are forged from existing ones. It states: from P and P → Q, one may logically infer Q. Simple, efficient, and utterly indispensable.

In this formal framework, if Γ represents a set of formulas (our initial hypotheses or known theorems) and A is a single formula, the notation

Γ ⊢ A

{\displaystyle \Gamma \vdash A}

signifies that A can be derived from the set Γ using only the aforementioned axioms and the rule of modus ponens. It's the formal declaration of provability within this system.

It's worth noting the historical contributions to this minimalist system. Łukasiewicz, in 1948, embarked on a quest for ultimate conciseness, discovering an axiom system for the implicational calculus that elegantly replaced the three schemas above with a single, remarkably dense schema:

((P → Q) → R) → ((R → P) → (S → P)).

He didn't just find it; he meticulously argued for its minimality, asserting that no shorter single axiom could achieve the same logical completeness. A testament to his dedication, or perhaps a demonstration of just how much one can pack into a few symbols.

Basic properties of derivation

Within any formal logical system, the process of derivation often exhibits certain predictable and useful characteristics. The implicational propositional calculus, despite its spartan nature, is no exception.

One such fundamental property is that derivation is robustly closed under substitution. This means that if a formula A can be derived from a set of formulas Γ (i.e.,

Γ ⊢ A ,

{\displaystyle \Gamma \vdash A,}

), then applying any consistent substitution σ (where σ replaces propositional variables with other formulas, again, using only implication as a connective) to all formulas in Γ and to A will result in a new valid derivation (i.e.,

σ ( Γ ) ⊢ σ ( A ) ,

{\displaystyle \sigma (\Gamma )\vdash \sigma (A),}

). This property is not merely convenient; it's a critical aspect of how axiom schemas function, allowing a single axiomatic form to represent an infinite number of specific instances. It’s the logical equivalent of a template, endlessly reusable.

Perhaps even more crucial for practical proof construction is the satisfaction of the deduction theorem. This meta-theorem provides a direct and powerful link between hypothetical reasoning and implicational statements within the system. It states: if a formula B can be derived from a set of hypotheses Γ and an additional hypothesis A (i.e.,

Γ , A ⊢ B

{\displaystyle \Gamma ,A\vdash B}

), then B can be derived from Γ alone, with the implication A → B as the conclusion (i.e.,

Γ ⊢ A → B .

{\displaystyle \Gamma \vdash A\to B.}

). This theorem essentially formalizes the common logical intuition that if assuming A allows you to prove B, then you have effectively shown that A implies B. As the deduction theorem article itself elaborates, this property holds for any axiomatic extension of a system that includes the first two axiom schemas (P → (Q → P) and (P → (Q → R)) → ((P → Q) → (P → R))) and modus ponens. Its presence simplifies proof construction immensely, transforming multi-premise arguments into single implicational statements, a rather elegant reduction of complexity.

Completeness

A logical system's true utility is often measured by its completeness. For the implicational propositional calculus, this means it is semantically complete with respect to the standard two-valued semantics of classical propositional logic. In plain terms, if a set of implicational formulas Γ logically entails an implicational formula A (meaning A is true whenever all formulas in Γ are true), then A is also derivable from Γ within the formal axiomatic system (i.e.,

Γ ⊢ A

{\displaystyle \Gamma \vdash A}

). This is a profound statement: it ensures that the axiomatic system is robust enough to capture every logically valid implicational truth. No tautology is left behind, unprovable.

Proof

The demonstration of this completeness theorem is an exercise in logical fortitude, often outlined as follows. Initially, one can simplify the task by leveraging the compactness theorem (which, in essence, states that if a conclusion follows from an infinite set of premises, it must follow from a finite subset) and the aforementioned deduction theorem. These powerful tools allow us to reduce the scope of the completeness theorem to a more manageable special case: we only need to prove that every tautology (a formula true under all possible truth assignments) is derivable from an empty set of hypotheses (i.e., it is a theorem of the system).

The core of the proof bears a resemblance to the completeness proofs for full propositional logic, but it cleverly navigates the inherent functional incompleteness of implication alone. The key insight involves defining two specific forms related to an arbitrary fixed formula F: for any formula A, we define A⁰ = (A → F) and A¹ = ((A → F) → F). These constructions become substitutes for expressing that A* is false or A* is true, respectively, under certain conditions, where A* is a modified version of A.

Before diving into the main inductive argument, some basic facts about derivability must be established, serving as lemmas:

Transitivity of Implication:

A → B , B → C ⊢ A → C

{\displaystyle A\to B,B\to C\vdash A\to C}

Indeed, this can be derived. First, one can obtain A → (B → C) using Axiom 1 (P → (Q → P)), by substituting P with A and Q with (B → C). Then, applying Axiom 2 ((P → (Q → R)) → ((P → Q) → (P → R))) twice, combined with modus ponens, allows for the derivation of A → C. This establishes a fundamental chain of implication.
Conditional Transitivity:

A → B ⊢ ( B → C ) → ( A → C )

{\displaystyle A\to B\vdash (B\to C)\to (A\to C)}

This follows directly from the first point by a single application of the deduction theorem. If assuming A → B allows us to derive (B → C) → (A → C), then A → B implies that whole statement.
A Form of Modus Ponens for Consequence:

A → C , ( A → B ) → C ⊢ C

{\displaystyle A\to C,(A\to B)\to C\vdash C}

To show this, let's assume C → B as an additional hypothesis. From this, and A → C, we can derive A → B using point (1) (substituting C for B, and B for C, then applying transitivity). With A → B and (A → B) → C, we can then derive C by modus ponens. This demonstrates that

A → C , ( A → B ) → C , C → B ⊢ C

{\displaystyle A\to C,(A\to B)\to C,C\to B\vdash C}

. Applying the deduction theorem to remove C → B as a hypothesis yields

A → C , ( A → B ) → C ⊢ ( C → B ) → C

{\displaystyle A\to C,(A\to B)\to C\vdash (C\to B)\to C}

. Finally, applying Axiom 3 (Peirce's law) to the conclusion, we obtain the desired C.

Now, let F be an arbitrary fixed formula. For any formula A, we define A⁰ = (A → F) and A¹ = ((A → F) → F). We then consider formulas in propositional variables p₁, ..., pₙ. The claim is that for every formula A in these variables and every truth assignment e, the following holds:

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{\displaystyle p_{1}^{e(p_{1})},\dots ,p_{n}^{e(p_{n})}\vdash A^{e(A)}.}

(4)

This statement is proven by induction on the structure of A.

Base case: If A = pᵢ (a propositional variable), the claim is trivially true. If e(pᵢ) = 1, then pᵢ¹ is ((pᵢ → F) → F), which is derived from pᵢ. If e(pᵢ) = 0, then pᵢ⁰ is (pᵢ → F), which is derived from pᵢ.
Inductive step: Assume A = (B → C). We distinguish three cases based on the truth values of B and C under assignment e:
- Case 1: e(C) = 1. In this scenario, e(A) = 1 as well (since anything implied by C is true if C is true). By the induction hypothesis, we have derived C¹ (i.e., (C → F) → F). We need to derive A¹ (i.e., ((B → C) → F) → F). We can apply the derived fact (2) (A → B ⊢ (B → C) → (A → C)) twice, substituting carefully, to the axiom C → (B → C) (which is an instance of Axiom 1). This allows us to establish that (C → F) → F ⊢ ((B → C) → F) → F. Since we have derived (C → F) → F by the induction hypothesis, we can infer ((B → C) → F) → F (which is A¹) using modus ponens.
- Case 2: e(B) = 0. Here, e(A) = 1 (since if B is false, B implies anything, so A is true). By the induction hypothesis, we have derived B⁰ (i.e., B → F). We need to derive A¹ (i.e., ((B → C) → F) → F). The deduction theorem applied to the derived fact (3) (A → C, (A → B) → C ⊢ C) yields B → F ⊢ ((B → C) → F) → F. Since we have derived B → F by the induction hypothesis, we can infer ((B → C) → F) → F (A¹) using modus ponens.
- Case 3: e(B) = 1 and e(C) = 0. This is the only case where e(A) = 0. By the induction hypothesis, we have derived B¹ (i.e., (B → F) → F) and C⁰ (i.e., C → F). We need to derive A⁰ (i.e., (B → C) → F). We start with the premises (B → F) → F and C → F, and (B → C). We can show that:
  
  ( B → F ) → F , C → F , B → C
  
  ⊢ B → F
  
  by (1) (transitivity, using (B → C) and (C → F) to get (B → F)).
  
  ⊢ F
  
  by modus ponens (from (B → F) and (B → F) → F, which is B¹).
  
  {\displaystyle {\begin{aligned}(B\to F)\to F,C\to F,B\to C&\vdash B\to F&&{\text{by (1)}}\&\vdash F&&{\text{by modus ponens,}}\end{aligned}}}
  
  Thus, we have shown that if we assume B¹, C⁰, and (B → C), we can derive F. Applying the deduction theorem to remove (B → C) as a hypothesis, we get:
  
  ( B → F ) → F , C → F ⊢ ( B → C ) → F
  
  {\displaystyle (B\to F)\to F,C\to F\vdash (B\to C)\to F}
  
  . Since we have derived (B → F) → F (B¹) and C → F (C⁰) by the induction hypothesis, we can infer (B → C) → F (A⁰) using modus ponens. This completes the proof of (4).

Now, to prove that every tautology is derivable, let F be an arbitrary tautology in variables p₁, ..., pₙ. We proceed by reverse induction on k, ranging from n down to 0. The goal is to show that for every assignment e:

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{\displaystyle p_{1}^{e(p_{1})},\dots ,p_{k}^{e(p_{k})}\vdash F.}

(5)

Base case (k = n): This follows directly from a special instance of (4). Since F is a tautology, e(F) = 1 for any assignment e. Therefore, we need to show that from p₁^(e(p₁)), ..., pₙ^(e(pₙ)), we can derive F¹. F¹ is ((F → F) → F). The formula F → F is itself a theorem (derivable from Axiom 1, P → (Q → P), by setting P = F and Q = F, then applying the deduction theorem to P → Q → P ⊢ P → P). Since F → F is a theorem, and F¹ is ((F → F) → F), and F → F is equivalent to True, then (True → F) → F is equivalent to F → F, which is True. Thus F¹ is provable.
Inductive step: Assume that (5) holds for k + 1. We aim to show it holds for k. By applying the deduction theorem to the induction hypothesis, we obtain two crucial derivations:
- First, by setting e(p_{k+1}) = 0 in the (k+1)-th hypothesis: p₁^(e(p₁)), ..., pₖ^(e(pₖ)) ⊢ (p_{k+1} → F) → F.
- Second, by setting e(p_{k+1}) = 1 in the (k+1)-th hypothesis: p₁^(e(p₁)), ..., pₖ^(e(pₖ)) ⊢ ((p_{k+1} → F) → F) → F. From these two derivations, we can then derive F using modus ponens and Peirce's Law (Axiom 3), by making appropriate substitutions. Specifically, if we can prove (X → F) → F and ((X → F) → F) → F, then by Peirce's law, we can prove F. This is precisely what the two derivations provide when X is p_{k+1}. This completes the proof of (5).

Finally, when k = 0, we arrive at the conclusion that the tautology F is provable without any initial assumptions. This is precisely what the completeness theorem asserts.

While this proof is, in theory, constructive—meaning one could meticulously follow its instructions to generate a formal proof for any given tautology—its practical application is severely limited. The length of such a proof tends to grow exponentially with the number of propositional variables present in the tautology. Consequently, it remains a theoretical assurance rather than a practical method, suitable only for the most trivially short tautologies. For anything of significant complexity, one would be better off simply accepting its truth and moving on.

The Bernays–Tarski axiom system

While the previously discussed axiom system serves its purpose, an alternative, widely recognized set of axioms for the implicational calculus is the Bernays–Tarski system. It often appears in literature and provides a slightly different flavor of derivation. Notably, Łukasiewicz's seminal paper on his single axiom meticulously demonstrates its completeness by deriving the Bernays–Tarski axioms from his own axiom, thus proving their equivalence.

The primary distinction between the Bernays–Tarski system and the one previously detailed lies in its second axiom schema. It replaces the original Axiom schema 2, (P → (Q → R)) → ((P → Q) → (P → R)), with a subtly different, yet equally powerful, formulation:

Axiom schema 2': (P → Q) → ((Q → R) → (P → R)). This is classically known as the hypothetical syllogism. It directly formalizes the transitive nature of implication: if P implies Q, and Q implies R, then P must imply R. This axiom feels, to some, more immediately intuitive than the original Axiom 2, reflecting a chain of reasoning.

The adoption of Axiom 2' does, however, introduce a minor complication: the derivation of the deduction meta-theorem becomes a little more intricate than with the original Axiom 2. However, this is merely an increase in complexity, not an insurmountable barrier; the meta-theorem can still be proven within this system.

To illustrate how the system functions, particularly how it compensates for the change in Axiom 2, let's demonstrate a key derivation. We show that from P → (Q → R) and P → Q, one can still derive P → R. This particular fact is crucial because it effectively serves the same purpose as the original Axiom 2 in establishing the deduction theorem.

P → (Q → R) (Given hypothesis)
P → Q (Given hypothesis)
(P → Q) → ((Q → R) → (P → R)) (Instance of Axiom schema 2' hypothetical syllogism)
(Q → R) → (P → R) (From steps 2 and 3 by modus ponens)
(P → (Q → R)) → (((Q → R) → (P → R)) → (P → (P → R))) (Another instance of Axiom schema 2', substituting carefully)
((Q → R) → (P → R)) → (P → (P → R)) (From steps 1 and 5 by modus ponens)
P → (P → R) (From steps 4 and 6 by modus ponens)
(P → (P → R)) → (((P → R) → R) → (P → R)) (Instance of Axiom schema 2', quite a mouthful)
((P → R) → R) → (P → R) (From steps 7 and 8 by modus ponens)
(((P → R) → R) → (P → R)) → (P → R) (Instance of Axiom schema 3 Peirce's law)
P → R (From steps 9 and 10 by modus ponens) qed

This rather involved chain demonstrates that even with the altered second axiom, the essential logical power—the ability to establish transitivity of implication in a way that supports the deduction theorem—is retained. It's a longer path, perhaps, but it ultimately arrives at the same destination.

Satisfiability and validity

When dealing with logical formulas, two fundamental questions always arise: is a formula satisfiable, and is it valid? In the context of the implicational propositional calculus, these questions yield answers that are either trivially simple or surprisingly complex.

Satisfiability in this calculus is, frankly, a non-issue. It's utterly trivial. Every formula within the implicational propositional calculus is inherently satisfiable. The reason is straightforward: simply assign 'true' to all propositional variables. As established earlier, any formula constructed solely with implication will evaluate to 'true' if all its variables are 'true'. Thus, finding a valuation that makes a formula true is as simple as setting everything to true. No deep philosophical insights required; just a brute-force assignment that always works.

Falsifiability, however, is where the system reveals its teeth. Determining whether a given implicational formula can be made false—or, conversely, whether it is a tautology (always true)—is a significantly more challenging computational problem. It is classified as NP-complete. This means that, while if a solution (a falsifying assignment) exists, it can be verified quickly, finding such an assignment in the first place can take a prohibitively long time, scaling exponentially with the size of the formula in the worst case. Consequently, the problem of determining validity (i.e., whether a formula is a tautology) is co-NP-complete. This implies that proving a formula is a tautology is equally difficult; while a valid proof can be quickly checked, finding that proof is hard.

Given this computational complexity, a practical and often useful technique for validity checking is to adopt a strategy of refutation. Instead of trying to prove a formula is a tautology, one presumes it is not a tautology and then attempts to construct a valuation that renders it false.

If this attempt succeeds—if a specific assignment of truth values makes the formula false—then the initial presumption is confirmed, and the formula is indeed not a tautology.
If, however, all attempts to find such a falsifying valuation fail, systematically ruling out every possibility, then the formula must, by logical exhaustion, be a tautology. This method transforms the positive search for proof into a negative search for counterexample.

Let's illustrate with a couple of examples:

Example of a non-tautology: Consider the rather unwieldy formula: [(A → B) → ((C → A) → E)] → ([F → ((C → D) → E)] → [(A → F) → (D → E)])

To check if this is a tautology, we assume it is false. For an implication P → Q to be false, P must be true and Q must be false. So, we assume:

(A → B) → ((C → A) → E) is true.
[F → ((C → D) → E)] → [(A → F) → (D → E)] is false.

From (2), for the implication to be false: 3. F → ((C → D) → E) is true. 4. (A → F) → (D → E) is false.

From (4), for the implication to be false: 5. A → F is true. 6. D → E is false.

From (6), for the implication to be false: 7. D is true. 8. E is false.

Now we begin to propagate these derived values:

Since D is true (7) and E is false (8), the subformula C → D must be true (a true consequent means the implication is true regardless of the antecedent).
Substitute these into (3): F → ((C → D) → E) becomes F → (true → false), which simplifies to F → false.
Since F → false is true (from 3) and false is indeed false, F itself must be false. (If true implies false, then the antecedent must be false).

Continue propagating:

Since F is false, and A → F is true (from 5), A must be false. (If true implies false, then the antecedent must be false).

Now back to (1): (A → B) → ((C → A) → E) is true.

Since A is false, A → B is true (a false antecedent means the implication is true regardless of the consequent).
Also, since A is false, C → A is false (for C → A to be true, C would have to be false, but here A is the false consequent). So C must be true for C → A to be false.
Now, look at ((C → A) → E). Since C → A is false, and E is false, then (false → false) is true.
So, (A → B) → ((C → A) → E) becomes (true → true), which is true. This is consistent with our initial assumption (1).

We have successfully constructed a valuation:

A is false
C is true
D is true
E is false
F is false
B can be either true or false; its value doesn't affect the final outcome. Let's arbitrarily set B to true.

This specific assignment (A=false, B=true, C=true, D=true, E=false, F=false) makes the original formula false. Therefore, it is definitively not a tautology. A rather tedious process, but effective.

Example of a tautology: Suppose we test the formula: ((A → B) → C) → ((C → A) → (D → A))

Again, we assume it is false.

(A → B) → C is true.
(C → A) → (D → A) is false.

From (2), for the implication to be false: 3. C → A is true. 4. D → A is false.

From (4), for the implication to be false: 5. D is true. 6. A is false.

Now, propagate these derived values:

Since A is false (6), and C → A is true (3), then C must be false. (If true implies false, then the antecedent must be false).

Now, substitute A=false, C=false into (1): (A → B) → C is true.

Since A is false, A → B is true (false antecedent always makes implication true).
So, (A → B) → C becomes (true → false).
For (true → false) to be true, this is a contradiction! (True implies false is always false).

We have reached a contradiction. Our initial assumption that the formula could be false has led to an impossible scenario. Therefore, there exists no valuation that can make ((A → B) → C) → ((C → A) → (D → A)) false. Consequently, it must be a tautology. This method, while still requiring careful step-by-step analysis, provides a concrete path to determine validity or non-validity.

Adding an axiom schema

The delicate balance of an axiomatic system can be profoundly altered by the introduction of new axioms. When considering the implicational propositional calculus and the potential addition of another axiom schema to its established set, two distinct outcomes emerge, each with significant ramifications for the system's nature.

Case 1: The new axiom schema is a tautology. If the newly introduced axiom schema is itself a tautology—meaning it is always true under any valuation—then the fundamental set of theorems (all formulas derivable within the system) remains precisely the same: the set of all implicational tautologies. The system's completeness is preserved, as it still proves exactly what it should. However, this addition is not entirely without consequence. In certain instances, the new tautologous axiom might provide a shortcut, allowing for significantly shorter proofs for some existing theorems. It could effectively encapsulate a complex chain of inference into a single, readily available step. Despite this potential for efficiency, a crucial characteristic remains unchanged: the minimum length of proofs for theorems will still be unbounded. This means that for any arbitrarily large natural number n, there will always exist theorems that cannot be proven in n or fewer steps. The system, even with its new shortcut, continues to contain arbitrarily complex truths that demand extensive derivation. It's like adding a new tool to a toolkit; it might make some jobs easier, but it doesn't fundamentally change the nature of the work or the existence of truly difficult tasks.

Case 2: The new axiom schema is not a tautology. This is where things become significantly more disruptive. If the new axiom schema is not a tautology—meaning there exists at least one valuation under which it evaluates to false—then the entire system collapses into triviality. In such a scenario, every single formula becomes a theorem. This renders the concept of a "theorem" utterly meaningless, as everything is provable, and the system loses all discriminatory power. It ceases to be a useful tool for distinguishing valid inferences from invalid ones. Moreover, this collapse also brings an unexpected consequence regarding proof length: there will then be an upper bound on the minimum length of a proof for every formula. This is because a common, systematic method for proving any formula becomes available. For example, suppose we were to add the non-tautologous axiom schema ((B → C) → C) → B. Consider an instance of this new axiom: ((A → (A → A)) → (A → A)) → A. This particular instance is itself not a tautology. However, the implication [((A → (A → A)) → (A → A)) → A] → A is a tautology. This is provable using the original, sound axioms (thanks to the completeness result discussed earlier). Now, by applying modus ponens (from the new non-tautologous axiom instance and the tautology that implies A), we can derive A as a theorem within this extended, now inconsistent, system. Once A is established as a theorem (where A is a simple propositional variable), then any arbitrary formula X can be proven by simply replacing every occurrence of A in the proof of A with the desired formula X. This transformed proof will have the exact same number of steps as the original proof of A. Thus, a fixed upper bound on proof length is established for all formulas, making the system both universally provable and utterly useless. It's a logical wasteland where truth and falsehood become indistinguishable.

An alternative axiomatization

Beyond the standard axiom systems, alternative approaches exist, sometimes aiming for directness over elegance, or to highlight different aspects of logical derivation. One such alternative axiomatization for the implicational propositional calculus explicitly targets completeness without relying as heavily on the deduction meta-theorem as the prior system. It takes a more constructive, perhaps even brute-force, path towards ensuring all tautologies are provable.

This system begins with a set of axiom schemas meticulously designed to efficiently prove the subset of tautologies that contain only a single propositional variable. This foundational layer ensures that the simplest truths are readily available.

aa 1: ꞈ A → A. This is the axiom of identity, stating that A implies A. It's a fundamental tautology. The symbol "ꞈ" is merely a comment, indicating where the conclusion used in the completeness proof effectively begins; it is not a part of the formal formula itself.
aa 2: (A → B) → ꞈ(A → (C → B)). This schema shows how an implication can be 'weakened' by adding an irrelevant antecedent (C) to the consequent.
aa 3: A → ((B → C) → ꞈ((A → B) → C)). This axiom, reminiscent of a converse hypothetical syllogism or a form of contraction, is crucial for managing nested implications.
aa 4: A → ꞈ(B → A). This is another form of 'weakening', asserting that a true proposition (A) is implied by anything (B).

The strategy here is to build up proofs for single-variable tautologies systematically. A proof would typically start with an identity (A → A), then strategically insert additional hypotheses, perhaps by introducing new variables, or by adding tautological hypotheses that remain true even when the sole variable is false. This procedure, guided by these axioms, is designed to quickly yield every tautology that contains only one variable.

The real ingenuity of this alternative system emerges when dealing with formulas containing multiple variables. For any formula Φ that may contain variables A, B, C₁, ..., Cₙ and culminates with A as its final conclusion, we introduce a powerful, recursive axiom schema:

aa 5: Φ⁻ → (Φ⁺ → ꞈΦ).

Here, Φ⁻ is the result of uniformly replacing every occurrence of B with A throughout Φ. Conversely, Φ⁺ is the result of uniformly replacing every occurrence of B with (A → A) throughout Φ. This is a schema for axiom schemas, operating on two levels of substitution: first, Φ itself is substituted (with variations); second, any of the variables (including both A and B) may be replaced by arbitrary formulas of the implicational propositional calculus.

This schema, aa 5, provides a powerful mechanism for proving tautologies with more than one variable by effectively reducing the problem to cases. It works by considering two scenarios for the variable B: the case when B is considered 'false' (represented by replacing B with A, thus Φ⁻) and the case when B is considered 'true' (represented by replacing B with (A → A), thus Φ⁺). The axiom states that if Φ holds when B is false, and it holds when B is true, then Φ must hold universally.

Let's consider the mechanics: If the variable that serves as the final conclusion of a formula (say, A) happens to take the value 'true', then the entire formula Φ will evaluate to 'true', irrespective of the truth values assigned to any other variables. Consequently, if A is true, then Φ, Φ⁻, Φ⁺, and even Φ⁻ → (Φ⁺ → Φ) are all true. This allows us, without loss of generality, to simplify the problem by assuming that A is false. The elegance lies in the reduction: Φ is a tautology if and only if both Φ⁻ and Φ⁺ are tautologies. While Φ might have n + 2 distinct variables, both Φ⁻ and Φ⁺ will have n + 1 variables. This recursive reduction allows us to systematically diminish the number of variables until we are left with single-variable tautologies, which are efficiently handled by axioms aa 1-4. It's a divide-and-conquer strategy applied to logical proof. Furthermore, the schema Φ⁻ → (Φ⁺ → Φ) is itself always a tautology, regardless of whether Φ itself is. This holds because if Φ happens to be false, then either B must be false (making Φ⁻ false) or B must be true (making Φ⁺ false). In either scenario, the antecedent of the main implication (Φ⁻ → (Φ⁺ → Φ)) becomes false, making the entire implication true. This guarantees the consistency of the schema itself.

Examples:

Let's see this alternative system in action, deriving some well-known theorems. These derivations, while formally correct, highlight the painstaking nature of working within a purely formal system, contrasting sharply with the often-quicker methods of semantic truth tables.

Deriving Peirce's law ((P → Q) → P) → P:

[((P → P) → P) → P] → ([((P → (P → P)) → P) → P] → [((P → Q) → P) → P]) (Instance of aa 5, with A=P, B=Q)
P → P (Instance of aa 1)
(P → P) → ((P → P) → (((P → P) → P) → P)) (Instance of aa 3, with A=P→P, B=P, C=P)
(P → P) → (((P → P) → P) → P) (From 2 and 3 by modus ponens)
((P → P) → P) → P (From 2 and 4 by modus ponens)
[((P → (P → P)) → P) → P] → [((P → Q) → P) → P] (From 5 and 1 by modus ponens)
P → (P → P) (Instance of aa 4)
(P → (P → P)) → ((P → P) → (((P → (P → P)) → P) → P)) (Instance of aa 3)
(P → P) → (((P → (P → P)) → P) → P) (From 7 and 8 by modus ponens)
((P → (P → P)) → P) → P (From 2 and 9 by modus ponens)
((P → Q) → P) → P (From 10 and 6 by modus ponens) qed

A rather circuitous route to a truth, wouldn't you agree?

Deriving Łukasiewicz's sole axiom ((P → Q) → R) → ((R → P) → (S → P)):

[((P → Q) → P) → ((P → P) → (S → P))] → ([((P → Q) → (P → P)) → (((P → P) → P) → (S → P))] → [((P → Q) → R) → ((R → P) → (S → P))]) (Instance of aa 5, with A=(P→Q), B=R, C=S)
[((P → P) → P) → ((P → P) → (S → P))] → ([((P → (P → P)) → P) → ((P → P) → (S → P))] → [((P → Q) → P) → ((P → P) → (S → P))]) (Instance of aa 5, with A=P, B=Q, C=S)
P → (S → P) (Instance of aa 4)
(P → (S → P)) → (P → ((P → P) → (S → P))) (Instance of aa 2)
P → ((P → P) → (S → P)) (From 3 and 4 by modus ponens)
P → P (Instance of aa 1)
(P → P) → ((P → ((P → P) → (S → P))) → [((P → P) → P) → ((P → P) → (S → P))]) (Instance of aa 3)
(P → ((P → P) → (S → P))) → [((P → P) → P) → ((P → P) → (S → P))] (From 6 and 7 by modus ponens)
((P → P) → P) → ((P → P) → (S → P)) (From 5 and 8 by modus ponens)
[((P → (P → P)) → P) → ((P → P) → (S → P))] → [((P → Q) → P) → ((P → P) → (S → P))] (From 9 and 2 by modus ponens)
P → (P → P) (Instance of aa 4)
(P → (P → P)) → ((P → ((P → P) → (S → P))) → [((P → (P → P)) → P) → ((P → P) → (S → P))]) (Instance of aa 3)
(P → ((P → P) → (S → P))) → [((P → (P → P)) → P) → ((P → P) → (S → P))] (From 11 and 12 by modus ponens)
((P → (P → P)) → P) → ((P → P) → (S → P)) (From 5 and 13 by modus ponens)
((P → Q) → P) → ((P → P) → (S → P)) (From 14 and 10 by modus ponens)
[((P → Q) → (P → P)) → (((P → P) → P) → (S → P))] → [((P → Q) → R) → ((R → P) → (S → P))] (From 15 and 1 by modus ponens)
(P → P) → ((P → (S → P)) → [((P → P) → P) → (S → P)]) (Instance of aa 3)
(P → (S → P)) → [((P → P) → P) → (S → P)] (From 6 and 17 by modus ponens)
((P → P) → P) → (S → P) (From 3 and 18 by modus ponens)
(((P → P) → P) → (S → P)) → [((P → Q) → (P → P)) → (((P → P) → P) → (S → P))] (Instance of aa 4)
((P → Q) → (P → P)) → (((P → P) → P) → (S → P)) (From 19 and 20 by modus ponens)
((P → Q) → R) → ((R → P) → (S → P)) (From 21 and 16 by modus ponens) qed

Verifying Łukasiewicz's sole axiom using a simple truth table would necessitate evaluating 2⁴ = 16 distinct cases, given its four propositional variables (P, Q, R, S). In stark contrast, this formal derivation, while achieving the same goal, required 22 steps. The recursive nature of axiom aa 5 did allow us to restrict our consideration to merely three cases—R is false and Q is false, R is false and Q is true, and R is true—but because we are operating within the stringent confines of a formal logical system (rather than the more flexible, informal meta-level), each of those cases demanded a significantly greater expenditure of effort and meticulous symbolic manipulation. It's a stark reminder that what is intuitively 'obvious' can be excruciatingly complex to formally demonstrate.