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Intuitionistic Logic (Wikipedia [accessed 19Jul2015], Stanford Encyclopedia of Philosophy [accessed 19Jul2015]) 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, 7May2018)
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 (axia1, axia2, axia3, axio, axin1 and axin2), when added to ax1, ax2 and axmp, allow for the development of intuitionistic propositional logic. We omit the classical axiom ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) (which is ax3 in set.mm). Each of our new axioms is a theorem of classical propositional logic, but ax3 cannot be derived from them. Similarly, other basic classical theorems, like the third middle excluded or the equvalence 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 (axial, axi5r, axie1 and axie2) together with the set.mm axioms ax4, ax5, ax7 and axgen do not mention equality or distinct variables.
The axi9 axiom is just a slight variation of the classical ax9. The classical axiom ax12 is strengthened into first axi12 and then axbnd (two results which would be fairly readily equivalent to ax12 classically but which do not follow from ax12, at least not in an obvious way, in intuitionistic logic). The substitution of axi9, axi12, and axbnd for ax9 and ax12 and the inclusion of ax8, ax10, ax11, ax13, ax14 and ax17 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 ax3, 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  ax1  ⊢ (φ → (ψ → φ)) 
Axiom Frege  ax2  ⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ))) 
Rule of Modus Ponens  axmp  ⊢ φ & ⊢ φ → ψ => ⊢ψ 
Left 'and' elimination  axia1  ⊢ ((φ ∧ ψ) → φ) 
Right 'and' elimination  axia2  ⊢ ((φ ∧ ψ) → ψ) 
'And' introduction  axia3  ⊢ (φ → (ψ → (φ ∧ ψ))) 
Definition of 'or'  axio  ⊢ (((φ ∨ χ) → ψ) ↔ ((φ → ψ) ∧ (χ → ψ))) 
'Not' introduction  axin1  ⊢ ((φ → ¬ φ) → ¬ φ) 
'Not' elimination  axin2  ⊢ (¬ 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 axin1 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 dfbi. Other ways of understanding the biconditional, such as dfbi1 or dfbi3, however, are classical rather than intuitionistic results (following the proofs on those pages shows they depend on ax3).
Predicate logic adds set variables (individual variables) and the ability to quantify them with ∀ (forall) and ∃ (thereexists). 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  ax4  ⊢ (∀xφ → φ) 
Axiom of Quantified Implication  ax5  ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ)) 
The converse of ax5o  axi5r  ⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ)) 
Axiom of Quantifier Commutation  ax7  ⊢ (∀x∀yφ → ∀y∀xφ) 
Rule of Generalization  axgen  ⊢ φ => ⊢ ∀xφ 
x is bound in ∀xφ  axial  ⊢ (∀xφ → ∀x∀xφ) 
x is bound in ∃xφ  axie1  ⊢ (∃xφ → ∀x∃xφ) 
Define existential quantification  axie2  ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ))) 
Axiom of Equality  ax8  ⊢ (x = y → (x = z → y = z)) 
Axiom of Existence  axi9  ⊢ ∃x x = y 
Axiom of Quantifier Substitution  ax10  ⊢ (∀x x = y → ∀y y = x) 
Axiom of Variable Substitution  ax11  ⊢ (x = y → (∀yφ → ∀x(x = y → φ))) 
Axiom of Quantifier Introduction  axi12  ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) 
Axiom of Bundling  axbnd  ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀x∀z(x = y → ∀z x = y))) 
Left Membership Equality  ax13  ⊢ (x = y → (x ∈ z → y ∈ z)) 
Right Membership Equality  ax14  ⊢ (x = y → (z ∈ x → z ∈ y)) 
Distinctness  ax17  ⊢ (φ → ∀xφ), where x does not occur in φ 
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page was last updated on 22Jul2015. Your comments are welcome: Norman Megill Copyright terms: Public domain 
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