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Intuitionistic Logic (Wikipedia [accessed 19-Jul-2015], Stanford Encyclopedia of Philosophy [accessed 19-Jul-2015]) can be thought of as a constructive logic in which we must build and exhibit concrete examples of objects before we can accept their existence. Unproved statements in intuitionistic logic are not given an intermediate truth value, instead, they remain of unknown truth value until they are either proved or disproved. Intuitionist logic can also be thought of as a weakening of classical logic such that the law of excluded middle (LEM), (φ ∨ ¬ φ), doesn't always hold. Specifically, it holds if we have a proof for φ or we have a proof for ¬ φ, but it doesn't necessarily hold if we don't have a proof of either one. There is also no rule for double negation elimination. Brouwer observed in 1908 that LEM was abstracted from finite situations, then extended without justification to statements about infinite collections.
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(By Gérard Lang, 7-May-2018)
Mario Carneiro's work (Metamath database) "iset.mm" provides in Metamath a development of "set.mm" whose eventual aim is to show how many of the theorems of set theory and mathematics that can be derived from classical first order logic can also be derived from a weaker system called "intuitionistic logic." To achieve this task, iset.mm adds (or substitutes) intuitionistic axioms for a number of the classical logical axioms of set.mm.
Among these new axioms, the 6 first (ax-ia1, ax-ia2, ax-ia3, ax-io, ax-in1 and ax-in2), when added to ax-1, ax-2 and ax-mp, allow for the development of intuitionistic propositional logic. We omit the classical axiom ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) (which is ax-3 in set.mm). Each of our new axioms is a theorem of classical propositional logic, but ax-3 cannot be derived from them. Similarly, other basic classical theorems, like the third middle excluded or the equivalence of a proposition with its double negation, cannot be derived in intuitionistic propositional calculus. Glivenko showed that a proposition φ is a theorem of classical propositional calculus if and only if ¬¬φ is a theorem of intuitionistic propositional calculus.
The next 4 new axioms (ax-ial, ax-i5r, ax-ie1 and ax-ie2) together with the set.mm axioms ax-4, ax-5, ax-7 and ax-gen do not mention equality or distinct variables.
The ax-i9 axiom is just a slight variation of the classical ax-9. The classical axiom ax-12 is strengthened into first ax-i12 and then ax-bnd (two results which would be fairly readily equivalent to ax-12 classically but which do not follow from ax-12, at least not in an obvious way, in intuitionistic logic). The substitution of ax-i9, ax-i12, and ax-bnd for ax-9 and ax-12 and the inclusion of ax-8, ax-10, ax-11, ax-13, ax-14 and ax-17 allow for the development of the intuitionistic predicate calculus.
Each of the new axioms is a theorem of classical first order logic with equality. But some axioms of classical first order logic with equality, like ax-3, cannot be derived in the intuitionistic predicate calculus.
One of the major interests of the intuitionistic predicate calculus is that its use can be considered as a realization of the program of the constructivist philosophical view of mathematics.
As with the classical axioms we have propositional logic and predicate logic.
The axioms of intuitionistic propositional logic consist of some of the axioms from classical propositional logic, plus additional axioms for the operation of the 'and', 'or' and 'not' connectives.
|Axiom Simp||ax-1||⊢ (φ → (ψ → φ))|
|Axiom Frege||ax-2||⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))|
|Rule of Modus Ponens||ax-mp||⊢ φ & ⊢ φ → ψ => ⊢ψ|
|Left 'and' elimination||ax-ia1||⊢ ((φ ∧ ψ) → φ)|
|Right 'and' elimination||ax-ia2||⊢ ((φ ∧ ψ) → ψ)|
|'And' introduction||ax-ia3||⊢ (φ → (ψ → (φ ∧ ψ)))|
|Definition of 'or'||ax-io||⊢ (((φ ∨ χ) → ψ) ↔ ((φ → ψ) ∧ (χ → ψ)))|
|'Not' introduction||ax-in1||⊢ ((φ → ¬ φ) → ¬ φ)|
|'Not' elimination||ax-in2||⊢ (¬ φ → (φ → ψ))|
Unlike in classical propositional logic, 'and' and 'or' are not readily defined in terms of implication and 'not'. In particular, φ ∨ ψ is not equivalent to ¬ φ → ψ, nor is φ → ψ equivalent to ¬ φ ∨ ψ, nor is φ ∧ ψ equivalent to ¬ (φ → ¬ ψ).
The ax-in1 axiom is a form of proof by contradiction which does hold intuitionistically. That is, if φ implies a contradiction (such as its own negation), then one can conclude ¬ φ. By contrast, assuming ¬ φ and then deriving a contradiction only serves to prove ¬ ¬ φ, which in intuitionistic logic is not the same as φ.
The biconditional can be defined as the conjunction of two implications, as in dfbi2 and df-bi.
Predicate logic adds set variables (individual variables) and the ability to quantify them with ∀ (for-all) and ∃ (there-exists). Unlike in classical logic, ∃ cannot be defined in terms of ∀. As in classical logic, we also add = for equality (which is key to how we handle substitution in metamath) and ∈ (which for current purposes can just be thought of as an arbitrary predicate, but which will later come to mean set membership).
Our axioms are based on the classical set.mm predicate logic axioms, but adapted for intuitionistic logic, chiefly by adding additional axioms for ∃ and also changing some aspects of how we handle negations.
|Axiom of Specialization||ax-4||⊢ (∀xφ → φ)|
|Axiom of Quantified Implication||ax-5||⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))|
|The converse of ax-5o||ax-i5r||⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ))|
|Axiom of Quantifier Commutation||ax-7||⊢ (∀x∀yφ → ∀y∀xφ)|
|Rule of Generalization||ax-gen||⊢ φ => ⊢ ∀xφ|
|x is bound in ∀xφ||ax-ial||⊢ (∀xφ → ∀x∀xφ)|
|x is bound in ∃xφ||ax-ie1||⊢ (∃xφ → ∀x∃xφ)|
|Define existential quantification||ax-ie2||⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ)))|
|Axiom of Equality||ax-8||⊢ (x = y → (x = z → y = z))|
|Axiom of Existence||ax-i9||⊢ ∃x x = y|
|Axiom of Quantifier Substitution||ax-10||⊢ (∀x x = y → ∀y y = x)|
|Axiom of Variable Substitution||ax-11||⊢ (x = y → (∀yφ → ∀x(x = y → φ)))|
|Axiom of Quantifier Introduction||ax-i12||⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))|
|Axiom of Bundling||ax-bnd||⊢ (∀z z = x ∨ (∀z z = y ∨ ∀x∀z(x = y → ∀z x = y)))|
|Left Membership Equality||ax-13||⊢ (x = y → (x ∈ z → y ∈ z))|
|Right Membership Equality||ax-14||⊢ (x = y → (z ∈ x → z ∈ y))|
|Distinctness||ax-17||⊢ (φ → ∀xφ), where x does not occur in φ|
Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be contained in another set, and this relationship is indicated by the symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. These axioms were developed in response to Russell's Paradox, a discovery that revolutionized the foundations of mathematics and logic.
In the IZF axioms that follow, all set variables are assumed to be distinct. If you click on their links you will see the explicit distinct variable conditions.
|Axiom of Extensionality||ax-ext||⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)|
|...more to follow as we develop set theory|
Listed here are some examples of starting points for your journey through the maze. You should study some simple proofs from propositional calculus until you get the hang of it. Then try some predicate calculus and finally set theory. The Theorem List shows the complete set of theorems in the database. You may also find the Bibliographic Cross-Reference useful.
|Propositional calculus Predicate calculus Set theory|
page was last updated on 15-Aug-2015.
Your comments are welcome: Norman Megill
Copyright terms: Public domain
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