This is the eleventh part of a series developed through the Department of Philosophical Theology at the Institute of Lutheran Theology’s Christ School of Theology, exploring major results and developments in modern logic and their significance for philosophical and systematic theology.
Classical logic is extraordinarily powerful, so powerful in fact that one can easily begin to speak simply of 'logic' as though the classical system exhausted the possibilities of valid inference. The propositional calculus gives us familiar principles governing negation, conjunction, disjunction, and implication; first-order logic adds quantification and identity; model theory then permits us to specify satisfaction, consequence, validity, and interpretation with great precision. Much of the preceding series has depended upon precisely this framework, and nothing in what follows should be taken as withdrawing the confidence we have repeatedly placed in it.
Yet the development of logic in the twentieth century made clear that one need not abandon rigor in order to ask whether every feature of classical consequence is appropriate to every domain of reasoning. One can instead ask which principles are being used, what assumptions support them, and what changes when one of those assumptions is altered. This is the setting within which the various nonclassical logics emerged, not as one unified rebellion against classical logic, but as a family of formally disciplined attempts to revise particular features of consequence for particular purposes.
The plurality is important. Intuitionistic logic alters what may count as sufficient warrant for assertion and consequently declines to validate some classical principles. Relevant logics require a stronger connection between antecedent and consequent than material implication ordinarily supplies. Paraconsistent logics deny that contradiction must entail everything whatsoever, while many-valued logics permit semantic values other than the classical pair of truth and falsity. These systems do not all solve the same problem, and theology gains nothing by treating them as though they were variations upon one general theme called 'nonclassical logic'.
The more fruitful question is why theology might care about any of them.
Intuitionistic Logic: What Warrants the Assertion?
Classical logic validates the law of excluded middle:
P ∨ ¬P.
Either P or not-P.
It also validates double-negation elimination:
¬¬P → P.
If it is not the case that P is false, then P.
For classical reasoning these principles are familiar enough that one may hardly notice when they are being used. Intuitionistic logic, arising from Brouwer's philosophy of mathematics and subsequently formalized especially by Heyting, does not accept them unrestrictedly, not because the intuitionist is somehow more tolerant of contradiction, but because the standards governing assertion have changed. Under the constructive interpretation, asserting P requires an appropriate construction or proof of P, while asserting P ∨ Q requires having grounds for one disjunct or the other; consequently, the impossibility of ¬P need not itself amount to a constructive establishment of P.
The theological temptation here is obvious, and it should be resisted. It would be careless to claim that theological propositions are intuitionistic simply because faith is not mathematical proof, or to suppose that intuitionistic logic somehow captures religious trust better than classical logic. The connection is much more modest, but also more interesting, because intuitionistic logic forces us to ask a question theological argument often leaves implicit: What exactly warrants the assertion being made?
Consider the difference between
¬¬P
and
P.
Classical logic permits the passage from the former to the latter, whereas intuitionistic logic does not generally permit that inference. The distinction is useful because it forces us to ask whether showing that the denial of a proposition is untenable amounts to positively establishing the proposition itself. A theologian may successfully argue that a particular denial of divine action produces contradiction, but it remains a further question whether this alone establishes the particular positive account of divine action that the theologian wishes to defend.
One need not become an intuitionist in order to profit from the distinction. The formal system is valuable here because it makes visible an inferential step that ordinary theological prose can conceal, namely, the transition from the failure of a denial to the warrant for an affirmation.
Relevant Logic: What Has the Premise to Do with the Conclusion?
Classical material implication produces results that can initially seem peculiar. Since
P → Q
is classically equivalent to
¬P ∨ Q,
the conditional is true whenever P is false or Q is true, and consequently classical logic validates forms such as
P → (Q → P)
and
¬P → (P → Q).
These are often called paradoxes of material implication, though they are not contradictions within classical logic; they follow directly from the truth-functional definition of the conditional.
Relevant logicians ask whether a genuine relation of implication should require more than this. If we say that one proposition follows from another, should there not be some appropriate connection between the content of premise and conclusion? Relevant logics attempt to build such a requirement into the consequence relation itself, so that implication is not secured merely by the falsity of an antecedent or the independent truth of a consequent.
For theology the question is hardly peripheral, because theological discourse is saturated with conditionals. We say:
If Christ is risen, then …
If God creates ex nihilo, then …
If justification is by faith, then …
If God is immutable, then …
In such cases the theological force of the conditional ordinarily depends upon some intelligible relation between what is asserted in the antecedent and what is claimed in the consequent. We do not usually mean merely that the conditional happens to receive the value true under the truth table for material implication.
This is not yet an argument for replacing classical implication. It may instead be an argument for recognizing that many theological uses of 'if … then …' express more than the material conditional was ever intended to capture. The important point is therefore methodological: before formalizing a theological conditional, one must determine what sort of inferential relation the natural-language formulation is attempting to express.
Relevant logic helps precisely because it refuses to allow us to ignore that question.
Paraconsistent Logic: What Follows from Contradiction?
Perhaps no family of nonclassical logics is more immediately attractive to theologians, and perhaps none is more easily abused, than paraconsistent logic. Classical logic validates the principle commonly called explosion:
P, ¬P ⊢ Q.
From a contradiction, anything follows.
The principle does not mean that Q bears some hidden relation to P. Rather, once both P and ¬P have been admitted into a classical theory, every sentence becomes derivable, and the theory consequently loses its ability to discriminate among conclusions. In that technical sense, contradiction produces triviality.
Paraconsistent logics reject explosion. In a paraconsistent consequence relation, it is not generally the case that
P, ¬P ⊨ Q
for arbitrary Q, and therefore inconsistent information can be reasoned from without permitting every proposition to follow. What must be emphasized, however, is that paraconsistency does not by itself entail that contradictions are true; it entails only that contradiction need not produce inferential collapse.
This distinction is especially important in theology, where doctrines are often said loosely to be 'paradoxical' or even 'contradictory'. Christ is divine and human; God is one and three; the believer is righteous and sinful; God acts while creatures genuinely act. Yet none of these formulations has the form
P ∧ ¬P
unless one has first identified 'human' with 'not divine', 'three' with 'not one', or otherwise made the predicates contradictory in the same respect and under the same description.
Indeed, much of the history of Christian doctrine can be read as sustained resistance to exactly such conflations. Chalcedonian Christology does not say that Christ is finite and not finite in the same respect; Trinitarian doctrine distinguishes essence from person; the Lutheran formula simul iustus et peccator does not require that righteousness and sin be predicated univocally in the same respect. The logical discipline here lies not in invoking paraconsistency too quickly, but in determining first whether a genuine contradiction exists.
Paraconsistent logic becomes genuinely interesting when we confront a theological corpus, a historical tradition, or a developing theory that actually contains inconsistent commitments. Must everything then follow? A paraconsistent framework says no, and that can be useful when analyzing historically layered materials, competing doctrinal formulations, or theories under revision, because one can study the consequences of inconsistency without first pretending that the inconsistency is absent and without allowing the system to become trivial.
It is therefore essential to distinguish paraconsistency from dialetheism. The former concerns the behavior of consequence in the presence of contradiction; the latter is the metaphysical thesis that some contradictions are in fact true. One may use paraconsistent logic as a formal tool without thereby committing oneself to the reality of true contradictions, and theology should preserve that distinction with some care.
Many-Valued Logic: Must Every Proposition Be Simply True or False?
Classical propositional logic operates with two truth values, true and false. Many-valued logics generalize this architecture by permitting additional semantic values, though the significance of those additional values varies considerably from system to system. Some contain three values, others finitely many, and still others infinitely many; moreover, the extra values need not always be understood as degrees of truth, since they may instead represent indeterminacy, lack of information, semantic defect, or some other feature of the evaluation.
The theological temptation must again be controlled. The existence of many-valued logics does not establish that theological truth itself comes in degrees, nor does it show that mystery or doctrinal controversy requires intermediate truth values. What these logics do show is that bivalence is a semantic choice that can be examined rather than silently presupposed in every domain.
Suppose, for example, that we consider a predicate such as
x is mature in faith.
At what precise point does this predicate become true? If there is no sharp boundary, the issue may concern vagueness rather than either contradiction or theological confusion. Similar difficulties arise with predicates such as 'orthodox', 'responsible', 'culpable', 'spiritually mature', and even, in some contexts, 'alive' and 'dead', where biological or conceptual boundaries may be difficult to draw sharply.
A many-valued semantics offers one family of ways of representing such cases. It is not the only family, since supervaluationism, epistemicism, contextualism, and other theories compete with it, but the formal possibility is philosophically useful because it prevents us from assuming without argument that every semantically difficult case must still admit a sharp classical assignment of exactly one of two values.
The lesson for theology is therefore not that truth is fuzzy. It is that the semantics appropriate to a theological predicate must be investigated rather than assumed.
Which Logic for Theology?
At this point one might ask which logic theology should use, but the question is too coarse if it is understood as demanding one system for every theological task. There is no reason to suppose that theology needs a single nonclassical logic to replace classical logic across the board, and there is every reason to retain classical first-order logic for the enormous range of theological reasoning for which its proof theory, semantics, and inferential behavior are entirely adequate.
The existence of nonclassical logics does not overthrow classical logic. What it does overthrow is the assumption that every feature of classical consequence lies beyond philosophical examination. Intuitionistic logic asks what licenses assertion; relevant logic asks what connection implication should require between premise and conclusion; paraconsistent logic asks whether inconsistency must entail triviality; many-valued logic asks whether every semantic domain is adequately represented by exactly two truth values.
What has happened, accordingly, is not an abandonment of logic but a deepening of the philosophy of logic, because logical consequence itself has become an object of investigation. Earlier in this series we asked what follows from a theory, what structures satisfy it, whether the intended structure can be characterized, whether truth can be defined within the relevant language, whether consequences can be mechanically decided, and whether possible-world semantics supplies a sufficiently fine-grained account of content. Nonclassical logic now asks a question prior to many of those questions: Which relation of consequence are we employing when we say that one proposition follows from another?
The answer cannot simply be read off from the theological subject matter. The doctrine of the Trinity does not announce that its proper formal reconstruction must be classical, relevant, paraconsistent, or intuitionistic, nor does the Incarnation tell us in advance what sort of logical system best represents the relations among its propositions. One must first determine what the doctrine actually asserts, whether its apparent tensions are genuine contradictions or merely differences of respect, what kinds of conditionals occur within the argument, and what semantic distinctions the doctrine itself requires.
Only after that work has been done does the choice of formal machinery become philosophically responsible.
There is a danger in both directions. One can force every theological claim into classical form and conclude that whatever does not fit must be confused, or one can invoke a nonclassical logic whenever a doctrine appears difficult and thereby protect a defective formulation from criticism by simply changing the consequence relation. Neither procedure is satisfactory, because in both cases logic is being selected before the theological and semantic work has been done.
The choice of logic should instead follow from an analysis of the inferential phenomena one is attempting to represent. If the problem is vagueness, paraconsistency may be beside the point; if the problem is inconsistent information, many-valuedness may not address it; if the issue concerns the relation between antecedent and consequent, intuitionistic logic does not automatically solve it. Different logics revise different structures, and therefore there is no generic escape hatch labeled 'nonclassical'.
Why It Matters for Theology
Nonclassical logic matters for theology because it reveals that our conception of consequence already contains philosophical commitments. Classical logic gives powerful and often entirely appropriate accounts of theological reasoning, but its principles are better understood when we know what alternatives would look like and which features of inference those alternatives modify.
The discipline imposed by nonclassical logic is therefore double. We must specify exactly which classical principle appears inadequate for the task before us, and we must also identify what is gained and what is lost when that principle is revised. Merely changing the logic does not settle the theological question, since every alteration in consequence brings with it new semantic and proof-theoretic commitments of its own.
For theology this discipline can be salutary, because it forces distinctions that theological rhetoric too easily obscures. Apparent contradiction must be distinguished from genuine contradiction; material implication from explanatory or relevant connection; lack of proof from falsity; vagueness from inconsistency; mystery from contradiction; and formal tolerance of inconsistency from the metaphysical claim that some contradiction is actually true.
These are not logical niceties added after the theological work is complete. They belong to the conditions under which theology can say clearly what it means to say.
The most important lesson, therefore, is not that theology needs another logic, but that theology should know what its logic is doing, what it permits, what it forbids, and why. Once that much has been learned, the existence of alternative logics becomes less threatening and more useful, because each can be treated not as a rival worldview but as an instrument for testing the assumptions built into a particular account of consequence.
This brings us naturally to the final installment of the series. We began with Frege, Peirce, and Cantor, where modern logic dramatically enlarged the resources available for formal expression; we moved through model theory, completeness, incompleteness, compactness, truth, computability, modality, second-order logic, and hyperintensionality; and we have now reached the point at which consequence itself can be formally varied.
It is therefore fitting to return at the end to Gödel. His ontological argument brings together much of the machinery accumulated along the way: quantified modal logic, higher-order resources, necessity, possibility, formal derivation, and the distinction between the validity of an argument and the truth or adequacy of the axioms from which it proceeds. The final question will not be whether logic can 'prove God', but something more precise and, I think, more interesting: what exactly has been established when a theological argument has been successfully formalized and proved valid?
Bibliographical Note
The modern study of intuitionistic logic grows from L. E. J. Brouwer's philosophy of mathematics and Arend Heyting's subsequent formalization of intuitionistic reasoning. Alan Ross Anderson and Nuel Belnap's work on entailment became foundational for relevance logic. Stanisław Jaśkowski and Newton da Costa were among the major pioneers of paraconsistent logic, while subsequent work by Graham Priest and others developed its philosophical implications. Jan Łukasiewicz's work on three-valued logic, initially associated especially with future contingents, helped initiate the systematic study of many-valued logics. These traditions should not be treated as a single alternative to classical logic, since each revises different features of classical consequence for different formal and philosophical purposes.
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