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Preface

The reflection over the role of logic in theology has a long and rich tradition. Medieval thinkers investigated complex theological problems, utilizing the best logical tools at their disposal and often inventing new ones. The evidence is to be found in the writings of almost all major medieval thinkers, from St. Anselm to John Buridan, dealing with various theological issues, such as existence of God, His omniscience or the mystery of the Holy Trinity. In the 20th century, two different approaches were developed to applying logical tools to theological problems. The first may be called ‘incidental’, and includes numerous attempts to logically analyze some particular theological ideas, such as the ontological argument or the properties of God. This trend includes works by famous logicians and philosophers such as Kurt Gödel or Alvin Plantinga. The second strategy is ‘systematic’; it is an attempt to reconstruct the entirety of theological discourse with the use of modern logical techniques. It was the goal of the Kraków Circle, a group of theologians, philosophers and logicians (including Jan Salamucha, Józef Bocheński, Jan Drewnowski and Bolesław Sobociński), which was formed in the 1930s. After World War II, the programme of the Circle was kept alive by J. Bocheński, and culminated with his The Logic of Religion.

Following in the footsteps of the Kraków Circle, we believe that there are a number of intriguing theological problems which may be analyzed, and illuminated, from the logical perspective. On a minimal reading, theology is any theory which embraces the thesis that God (or the sphere of the divine) exists and provides us with some description of God (or the sphere of the divine) and the relationship between God (the sphere of the divine) and human beings. In some theologies, e.g. in Catholicism, things are more complicated, as they include the content of the revelation – a body of knowledge which has a special epistemic status – as well as all the theories that serve to explain and develop it in more detail. Crucially, the assumption here is that both revelation and theological theories are expressed in human language and, as such, may be subject to formal scrutiny. As in the case of any set of sentences, one can investigate logically a number of theological issues, including – but not being limited to – syntactic (the structure of theological sentences and theological theories), semantic (truth, evidence in theology), pragmatic (analysis of the propositional attitudes in theology), conceptual (definitions of and interrelations between theological concepts) and methodological (types of arguments and criteria of justification in theology) ones. Thus, we understand logic broadly to include not only semantics, syntax and pragmatics, but also the study of the theological conceptual scheme, as well as its methodological aspects. Importantly, in all these dimensions one can analyze not only theological discourse per se, but also its relations with other discourses, such as scientific, philosophical, etc. Our second assumption concerns logic: we believe that a fruitful formal study of theological discourse cannot be limited to the application of classical logic. Rather, any logical tool may prove useful, and examples include many-valued logics, modal logics, non-monotonic logics, and other formal mechanisms such as belief revision or the theory of circular definitions.

Given this broad scope of the theological problems accessible with logical tools, and the number of logical techniques one can utilize while analyzing theological discourse, we prefer to speak of logic in theology rather than logic of theology. The latter expression seems to suggest that there is one and only logical system suitable for theological reflection or that the logical aspect of theological thinking is somehow external to the content of theology. In our eyes, both these statements are false. On the one hand, there is no reason to believe that the different theological problems can be handled with one formal recipe; on the other, many theological questions are so intertwined with the underlying logical problems that they cannot be properly comprehended without some logical analysis. Thus, we believe that speaking of logic in theology is more appropriate. This small change of prepositions means an important change of approach: one is not limited to the question of what is the ‘proper’ logic of theology, but is free to investigate any theological problem which gives rise to logical analysis, or to consider loci theologici which may inspire the development of new logical instrumentarium.

Thus, the present volume is intended as an exercise in logic in theology. It puts together contributions pertaining to various aspects of the relationship between logic and theology, from historical essays, through the formal reconstruction of theological concepts and the structure of theological discourse, to more methodologically oriented papers, examining the criteria of theological justification and the interplay between theology and other disciplines. The papers collected here have been written within the research project entitled The Limits of Scientific Explanation, carried out at the Copernicus Center for Interdisciplinary Studies in Kraków, and sponsored by the John Templeton Foundation.

Bartosz Brożek

Adam Olszewski

Mateusz Hohol

Jan Woleński

University of Information, Technology and Management, Rzeszów

Theology and Logic*

Is logic applicable to theology? If so, is logic basically relevant for theology or does only the former have a secondary importance for the latter? Answers to both questions require several preliminary explanations concerning logic as well as theology. Let me begin with the scientia divina. First of all, there are many theologies, not only because many religions exist. Even if we restrict our attention to three great monotheistic religions (Islam, Judaism, Christianity), we do not find a uniform idea of theology. The situation does not change very much if we make further restrictions, for example to Christianity only. The theological tradition of this confession does not appear as uniform. On the contrary, we encounter various conceptions of theology. Two of them are particularly relevant for my further considerations. Firstly, we have the negative or aphophantic theology as represented by Nicolaus Cusanus (Nicholas of Cusa), which is very popular in the Orthodox Church. This theology says that we cannot say anything positive about God and His attributes. We should abstain from positive assertions and limit ourselves to statements like ‘I do not know what God is like or God is not...’. According to this kind of theology, the cognitive gap stemming from such assertions is sufficiently filled by belief as faith. If we believe, we do not need to be bothered by apparent inconsistencies in the body of theological statements. As Cusanus said, docta ignorantia sufficies and coicidentia oppositorum are to be accepted. Clearly, logic plays no essential role in negative theology, which is not particularly interested in arguments.

Secondly, there is theology, which is called natural, rational or philosophical (this last label is not quite correct, because negative theology has an explicit philosophical dimension). This version of the scientia divina[1] admits and recommends logical arguments in theological debates. Although its particular representatives, for example, Augustine of Hippo, Anselm of Canterbury, Thomas Aquinas, Duns Scotus or Leibniz accepted the formula fides quarens intellectum, they differ considerably as far as the matter concerns the scope of logic as a tool of theology. The standpoint of the official Catholic theology, confirmed by decisions of several councils, follows Thomas Aquinas, who distinguished theological truths (for example, the existence of God) accessible to human reason and subjected to rational or logical argumentations and theological truths, which exceed human understanding and are grounded in the revelation (for example, the doctrine creatio ex nihilo). Moreover, there cannot be any inconsistency between both kinds of truths, because both are given by God. Thomas’ picture, for its compromising character, provides a very good point of orientation. Augustine ascribed more to faith than Thomas did, Anselm and Scotus were more rationalistic than the Doctor Angelicus, but Leibniz and the Deists of the 18th century became the extreme rationalists in theology.

The rational scientia divina forms the proper environment for investigating the role of logic in theology. But what is logic? The noun ‘logic’ co-occurs with various and mutually different qualifications. Omitting several metaphorical denominations, like ‘the logic of history’ or ‘the logic of politics’, we read about formal logic, transcendental logic, hermeneutic logic, dialectical logic, material logic or philosophical logic. My further considerations are directed to formal logic, because I think that it is the only legitimate kind of logic. We can eventually distinguish logic in the narrow sense and the logic in the broad sense. The latter covers semantics or semiotics (roughly speaking, logical theory of language), formal logic and the methodology of science, but the former restricts the scope of logic to its formal part. Since I will touch upon semantic, formal-logical and methodological problems, it is convenient to work with logic in the broad sense. However, formal logic is the core of logic in any legitimate sense, because it provides the basic criterion of logical correctness. Assume that A₁, ..., An, A is an argument with A₁, ..., An, as its premises and A as its conclusion. We say that this argument is formally correct or valid, when A logically follows from the set of premises consisting of A₁, ..., An (symbolically A₁, ..., An ├ A), but it is materially correct, when its premises A₁, ..., An are true. This definition entails that if A₁, ..., An ├ A and premises are true, and the conclusion must be true as well. This statement exhibits the main property of deduction, namely its soundness or truth-preserving. In general, formal logic is a collection of systems consisting of principles being special cases of logical entailment. Of course, we can try to weaken the concept of logical validity and admit partial or inductive correctness, but it has to be done very carefully. In particular, clear features of generalized validity are required. In fact, logic called dialectical, material or transcendental completely lacks, at least until now, precise criteria of the validity of its arguments.

I include a logical analysis of various problems into the scope of logic. Logical analysis has a special importance for philosophy. In my view, rational theology is more a kind of philosophy than the scientia divina in the traditional sense; this opinion holds to some parts of theology at least, but I will not enter into details. Consequently, natural theology is comparable with philosophy with respect to their status as academic disciplines. This statement has very far-reaching consequences, because if a field has an academic character, it has to fulfil several methodological requirements. Thus, the applications of logic to theology should be considered as analogous to those performed in general philosophy, if one is inclined to agree that both are similar. I do not claim that logical analysis exhausts the entire scope of philosophy or per analogiam theology. Yet several theological problems are suitable for logical treatment. Anyway, the phrase ‘Logical analysis in theology’ could be another title of this paper. Some authors, like Jan Salamucha or Joseph M. Bocheński, Polish theologian-philosophers, claimed that rigorous logical analysis is indispensable in order to modernize theology;[2] other were less ambitious, but there is no doubt that the place of logic in rational theology is important. Thus, I do not make programmatic postulates, but assert the existing state of affairs.

I am fully conscious that there are several specific problems in theology, which should be taken into account, when methods of theology become a discussed topic. The typical list of problems included into a survey of rational philosophy[3] covers the following problems: (i) Theological Prolegomena; (ii) Divine Attributes; (iii) God and Creation; (iv) Topics in Christian Philosophical Theology; (v) Non-Christian Philosophical Theology. Logical analysis can be used in dealing with any topic taken from the above list. The point (i) has a special character. It concerns, among others, the authority of the Scripture, tradition and the Church, and the revelation and inspiration. Although particular Christian confessions differ in their relation to the Scripture, the revelation, the Tradition and the Church as authorities, we can assume for the sake of argument that there is the ultimate core of theological opinions. They concern further problems, like divine attributes, God and creation, and several other theological questions. Believing Christians are obliged to accept theological preliminaries and their consequences for special issues. A secular thinker can complain about this situation and say that theological limitations are at odds with rationality of arguments, particularly, because these restrictions unjustifiably mix natural and supra-natural factors. On the other hand, a Christian thinker might argue that if someone does not accept theological preliminaries, he or she unjustly excludes supra-natural entities from the scope of sound argumentations. This controversy results in the conclusion that any rational exchange of views between believers and unbelievers is impossible. I do not agree with this diagnosis. First of all, it appears as unclear why the line of division separates believers and unbelievers. In fact, although there are various dissimilarities between Catholics and some Protestants as far as the matter is concerned, theological preliminaries force completely different views concerning divine issues, but this fact does not exclude discussions between rival Christian confessions.

I would like to propose a debate without special assumptions, in particular, accusing the opposite side of irrationality. In order to give a concrete example, I employ a well-known meta-logical fact. Assume that X is a consistent set of sentences. By the Gödel-Malcev completeness theorem, X has a model in which its elements (sentences belonging to it) are true. We do not need to think that these sentences are true in the real world. The model of X is a mathematical entity, philosophically speaking, an intentional object correlated with this set by the content of propositions in question. Assume now that these sentences are about God and follow theological preliminaries. According to our assumption, we initially consider X as a consistent set of sentences. These assumptions can be correct or not and one of the uses of logic in theology consists in checking which answer should be accepted. Even if both parties of the debate, that is, believers and unbelievers, propose conflicting answers to the problem of consistency of theological discourse, they can understand each other perfectly. Similarly, we can investigate several other questions, for example, the problem of the form of propositions about God, the logical status of the term ‘God’ or the relation between naturalistic and supra-naturalistic language. Step by step, our theo-logic, to use a convenient label, might be extended to include discussions about the argument invented to demonstrate the existence of God (are they proofs or only persuasive strategies?) or the relation of God to evil (do the attributes of God cohere with the existence of God?). A skeptic might observe that such debates are dramatically inconclusive. Well, but this remark concerns every philosophical topic logically analyzed. Thus, philosophical theology is not an exception in this respect and the opposite situation would be surprising. Similarly, an argument that some theological problems, for instance, the Resurrection or the Incarnation exceed the tools of logical analysis does not seem to be as convincing, because this is typical for philosophers to maintain that some topics transcend given methods. My position is rather modest and consists in a claim that logical analysis can help in clarifying one’s own positions, defending them and arguing against opposite views. I will give some examples of such analytic arguments. And one additional remark is in order. I think that any participant of a discussion concerning theology should openly declare their own relation to religion. I consider myself as an unbeliever (this is temporary denomination). This attitude determines to some extent that I am interested in the place of logic in arguments for or against God’s existence. Otherwise speaking, I consider this problem as the most central question of religion and theology.

The first problem related to the ‘model-theoretic’ account of theological discourse pertains to the status of the term ‘God’. Assume that we work in the standard first-order logic. If we take the sentence (*) ‘God exists’, its subject-word looks like a proper name (individual constant). Now, first-order logic excludes empty proper names. This is the reason why some believers say that unbelievers cannot discuss the main theological issue. However, it is not true. The unbeliever has no reason to maintain a priori that theology is inconsistent, he or she can, without violating own convictions, accept models in which (a) is true, that is, God exists. Yet it must be emphasized that ‘to exist in a model’ does not mean ‘to really exist ‘. If a believer and his/her opponent agree at this point, they can argue for their views. We have another device enabling us to achieve the same task. In fact, the logical form of (*) exceeds beyond the syntax of first order-logic, because the verb ‘exists’ functions as a predicate. In order to work with a form which is coherent with first-order syntax, we can employ:

(**) ∃x(x = God),

in which the word ‘God’ serves as an individual constant, but the existential mark is hidden in the quantificational prefix. Yet (**) assumes that the term ‘God’ is non-empty. This situation seems be problematic for some unbelievers, because they deny that God exists. Fortunately, there is a very simple manoeuvre in order to read (**) in a way which satisfies all parties. We can change (**) into:

(***) ∃x(x is God).

The word ‘God’ occurs in (***) as a part of the predicate ‘is God’. This predicate has a set as its denotation. Since denotations of predicates can be empty sets, the difficulty noted above simply disappears. Denote the reference of ‘is God’ by G. Roughly speaking, believers maintain that G is a non-empty set, but unbelievers think that G = ∅. Obviously, the believer should justify that there is an x such that x ∈ G, but the unbeliever ought to argue that there is no x such that x ∈ G.

The division into believers and unbelievers is too simple in order to cover all religious (theological) possibilities. Some people declare that they know that God exists, other say that they know the opposite. The statements ‘I believe that God exists’ and its negation ‘I do not believe that God exists’ also render attitudes towards the existence of God. The notorious ambiguity of ‘I know’ and ‘I believe’ causes considerable difficulties in establishing a map of possible positions in religious matters. Ordinarily speaking, any motivated belief can be considered to be a piece of knowledge. Motives for believing can vary from astrology and superstitions to mature arguments taken from science. They are psychological, cultural, practical, political, etc. This approach to knowledge makes it indistinguishable from merely having beliefs. On the other hand, we have the definition of knowledge as justified true belief. For many reasons (see, the Gettier problem, for example), it claims too much. Let us take scientific knowledge as a paradigmatic case of knowledge. We can say that I know that A if and only if (a) I believe that A; (b) A is true according to the present shape of science; (c) I am ready to defend A using arguments taken from science. The point (a) treats belief as a mental state, the point (b) suggests that truth is approximate, not ultimate, but the point (c) treats defending as something dispositional. I assume that common knowledge about empirical matters can be regarded as satisfying points (a)-(c) and call both specimens of knowledge cognitively rational. The last qualification has nothing to do with evaluation, positive or negative. Rational knowledge in the outlined sense can be much less valuable from a practical point of view. Rational cognitive knowledge is to be contrasted with beliefs based on the act of faith. Religious beliefs are paradigmatic in this respect. Some believers say that they believe in God and they do not need any rational arguments for their faith. I will consider beliefs based on faith and knowledge (or cognitive beliefs, if you like) as ideal types. Actual mental states are usually a mixture of various and different factors.

I propose a map of attitudes towards religion.[4] It consists of theism, fideism, atheism and agnosticism as attitudes related to the statement (***) (or (*) or (*). Theism and fideism are frequently considered equivalent (or almost equivalent) standpoints and the same concerns atheism and agnosticism. My analysis proposes to distinguish theism and fideism on the one hand, and atheism and agnosticism on the other hand. I guess that the distinction proposed below might help in clarifying discussions between philosophers and ordinary people affirming or denying the existence of God.

Using contemporary terminology, the theist in Aquinas’ style considers (***) as a piece of knowledge, that is, the statement satisfying (a)-(c). Hence, the theist of this kind can express his or her position by:

(1) I know that God exists.

What should the atheist say? If he or she replies ‘God does not exist’, the theist might ask ‘Well, but do you know that?’. The atheist has a problem, because if his/her position is considered the negation of theism, the answer should be expressed by:

(2) I do not know that God exists,

but this is too weak. The atheist is rather inclined to say:

(3) I know that God does not exist.

However, (2) and (3) are not mutually contradictory, but contrary. This simple reasoning shows that there are other possibilities in attitudes towards (***), then (1) and (2), if the latter is taken as an adequate expression of atheism. The matter is still more complicated if we observe that:

(4) Although I do not know that God exists, I believe that He exists.

Since many historical fideists accepted (4), this possibility is not abstract. Thus, a closer analysis of theism, atheism, etc. is required. In particular, we should take into account knowledge and belief as parameters important in attitudes related to (***). How are they related? I accept the following dependence (KA – I know that A, BA – I believe that A):

(5) KA ⇒ BA.

This postulate is not a logical axiom but rather the result of an analysis of the meaning of ‘to know’ and ‘to believe’. It can be justified by the account of knowledge as justified true belief. Even if we consider ‘true’ and ‘justified’ in the above definition of knowledge to be somehow problematic, (5) can be taken as fully granted for any kind of belief, including religious faith. The reverse dependence, that is BA ⇒ KA, does not hold. Speaking metaphorically, belief and knowledge are half-separated: although knowledge implies belief, the latter does not imply the former.

In order to go further, I will establish some formal properties of K and B. The following diagram (D) is instrumental in this respect:[5]

This diagram is a generalization of the well-known logical square for categorical and modal propositions. Let symbol ■ denote K or B. Understanding α as ■A, β as ¬A, γ as ♦A and δ as ♦¬A, we have the following facts, among others:

(6) ¬(α ∧ β) (α and β are contrary)

(7) (α ⇒ γ) (α entails γ; γ is subordinated to α)

(8) (β ⇒ δ) (β entails δ; δ is subordinated to β)

(9) (α ⇔ ¬δ) (α and δ are contradictory)

(10) (β ⇔ ¬γ) (β and γ are contradictory)

(11) (γ ∨ δ) (γ and δ are complementary)

(12) (■A ⇔ ¬♦¬A) (■ is definable as ¬♦¬A)

(13) (♦A ⇔ ¬■¬A) (♦ is definable as ¬■¬)

(14) (β ⇔ ■¬A) (β is definable as ■¬A)

(15) ν ⇔ (α ∨ β) (ν ⇔ (■A ∨ ■B)

(16) µ ⇔ (γ ∧ δ) (16) µ ⇔ (♦A ∧ ♦¬A)

(17) ¬(ν ⇔ µ) (ν and µ are contradictory)

(18) α ∨ β∨ µ (α, β, µ are mutually exclusive and jointly exhaustive)

(19) κ ⇔ A

(20) λ ⇔ ¬A)

(21) ¬(α ⇒ κ) (¬(■A ⇒ A))

(22) ¬(κ ⇒ α) (¬(A ⇒ ■A))

(23) ¬(λ ⇒ β) (¬(■¬A ⇒ ¬A))

(24) ¬(β ⇒ λ) (¬(■¬A ⇒ ¬A)).

These dependencies (in particular, (21)-(24)) normalize the operators K and B as functioning in non-normal modal logic. It is important to observe that knowledge that A does not implies that A is true, but this fact is not at odds with (a)-(c) (see above). The main reason is that ‘to know’ and ‘to belief’ is a strongly intensional word. A well-known criterion of the sentence I∃xPx implies neither ∃xPx nor ¬∃xPx, if the letter I stands for a strong intensional operator. By analogy, neither KA nor BA implies A. Note also that (5) has no justification in (D). However, we can say that accepting KA ∧ ¬BA would not be rational, because if someone knows that A he or she should (is cognitively rational to) believe that A as well. Of course, there are always problematic cases of assertions like ‘I know that A, but I do not believe that A’ (this example puzzled Moore and Carnap), but if we distinguish consistency and rationality, (5) is defensible via an appeal to the latter. Some special views concerning the plausibility of BA ⇒ KA and it its strengthening to the equivalence KA ⇔ BA will be discussed later.

If we accept (5) as a conceptual truth or an additional axiom, we have the following consistent possibilities:

(25) KA ∧ BA

(26) ¬KA ∧ BA

(27) ¬KA ∧ ¬BA.

Thus, I can know that A and believe that A or not know that A and believe that A or not know that A and not believe that A. Further, not knowing that A is consistent with believing that A as well as not believing that A, but not with both simultaneously. Consequently, not knowing that A does not imply anything definite about believing. Some additional possibilities appear when we consider:

(28) K¬A ⇒ B¬A.

This generates the following coherent cases:

(29) K¬A ∧ B¬A

(30) ¬K¬A ∧ B¬A

(31) ¬K¬A ∧ ¬B¬A.

By combining, (25)-(27) and (29)-(31), we obtain:

(32) ¬K¬A ∧ ¬KA ∧ BA

(33) ¬K¬A ∧ ¬KA ∧ B¬A

(34) ¬KA ∧ ¬BA ∧ ¬B¬A

(35) ¬K¬A ∧ ¬BA ∧ ¬B¬A

(36) ¬KA ∧ ¬K¬A ∧ ¬BA ∧ ¬B¬A.

Now, if we skip the formulas which are proper parts of others or follow from others, there remain (25), (29), (32), (33) and (36).

If we apply the above considerations to (*), we obtain:

(37)  I know that God exists and I believe that God exists

(38)  I know that God does not exist and I believe that God does not exist

(39)  I do not know that God exists and I do not know that God does not exist and I believe that God exists

(40)  I do not know that God exists and I do not know that God does not exist and I believe that God does not exist

(41)  I do not know that God exists and I do not know that God does not exist and I do not believe that God exists and I do not believe that God does not exist.

Combination (37) expresses the standpoint of the Catholic Church on which knowledge and faith (religious belief) are complementary and mutually irreducible; the famous formula fides quaerens intellectum displays this view. (38) is characteristic for atheism popular in the French Enlightenment or in Sartre. (39) is the credo of fideism. (40) defines agnosticism and (41) describes the radical epistemological-theological skepticism (I will skip this view, because I consider it not very interesting). Theism and fideism contain ‘I believe that God exists’ as their common part and this statement can be taken as expressing theism (or fideism) in a wider sense. Similarly, atheism, agnosticism and radical skepticism (as given by (41)) agree with respect to ‘I do not believe that God exists’ and this assertion characterizes atheism in a broader sense; this picture is historically faithful, because all three views were and are pointed out as atheist. Atheism and agnosticism accept the statement ‘I do not know that God exists and I believe that God does not exist’. I propose to count this attitude as indirect atheism. If we take the set {theism, fideism, atheism, agnosticism}, all of the standpoints listed in it are mutually independent and contrary, but not all are mutually contradictory. Theism in the broader sense and indirect atheism are also contraries, but the former is inconsistent with atheism in the broader sense.

The above analysis is based on a purely formal criterion. This does not preclude that particular views differ in their account of how knowledge and faith are related. Now, I can supplement remarks (see above) about various motives of religious belief. Let me start with fideism. The fideist will probably say that he or she accepts (5), but providing that knowledge as applied to religion is of a special kind and, therefore, ‘I know’ in the religious discourse has a specific meaning, different from that in the statement ‘I know that 2 is the first prime number’. Thus, when the fideist announces that we have no rational knowledge of God’s existence, that is, based on intersubjectively accessible empirical evidence, this does not exclude another kind of experience, for example, mystical experience. A more ordinary understanding of fideism is perhaps given by:

(42)  I believe that God exists, although I have no rational support for my belief and I claim that there are no rational reasons for denying God’s existence.

Yet I think that (39) gives a better account of fideism, at least for its logical analysis and provided that ‘I know’ refers to rational knowledge. Logical reasons also dictate that I prefer ‘I know (do not know) that God exists’ than ‘I know (I do not know) whether God exists’. The theist of the Thomistic brand (or more generally, recommended by the Catholic Church) accepts the equivalence (i) KA ⇔ BA and justifies this by the above quoted formula fides quaerens intellectum, at least so far as the matter concerns the existence of God. Pragmatically speaking, faith (religious belief) is the starting point and its equivalence with knowledge is dictated by ratio recta. Since (i) holds for ¬KA and ¬BA as well, the theist has to accept it, too. According to standard theism, (ii) ¬KA ⇔ ¬BA describes unbelievers. They do not believe if and only if they do not know. Their lack of knowledge and belief may be unconscious and non-culpable, but if it is deliberate, they are deprived of ratio recta. Paradoxically, the atheist also shares (i) and (ii), but the motivation is different here. First of all, the atheist strengthens (ii) to (iii) K¬A ⇔ B¬A, because atheism consists in a positive view about God’s non-existence, not only in expressing the lack of knowledge and faith concerning His non-existence. Thus, the atheist asserts (iv) K¬A ⇔ ¬KA, (v) B¬A ⇔ ¬BA, and (vi) K¬A ⇔ B¬A, and argues that the theist abuses the concept of knowledge. In other words, the atheist maintains that belief and knowledge as attitudes concerning (***) should be entirely rational. This position is best expressed by (vi), but (ii) holds only because KA and BA are false. Finally, the atheist can agree that KA and BA are true, but knowledge, belief and truth are understood differently than in rational discourse. Since (i)-(vi) do not follow from (D), their acceptance or rejection has their source in decisions motivated by various factors.

The agnostic proceeds similarly to the fideist in the matter of the relation between knowledge and faith, namely he or she derives some consequences from the lack of knowledge. However, agnosticism insists that knowledge should be taken as rational, but belief as rational if it is grounded by knowledge, otherwise it is irrational. Thus, there are obvious affinities between atheism and agnosticism. The skeptic, according to his or her general epistemological position, is completely neutral about the above issues. Skepticism is not particularly interesting, because it consists of merely negative assertions. The remaining views propose at least some positive beliefs, but theism and atheism additionally insist that their adherents know something about God’s existence or non-existence. A positive element in belief or knowledge seems to be of the utmost importance for world-view matters, because negative declarations do not suffice in this area. Although, on the one hand, every view (I omit skepticism) in the set {theism, fideism, atheism, agnosticism} is contrary to the rest, on the other hand, theism and atheism operate on a different epistemological level than fideism and agnosticism. In particular, the latter assume epistemological skepticism concerning the knowledge of God’s existence, but reject theological skepticism in this question. This positive content, directly related to theology in the case of fideism and indirectly in the case of agnosticism, is contained in statements ‘I believe that God exists’ and ‘I do not believe that God exists’, respectively. Similarly as in the case of theism and atheism, the agnostic or fideist approach to (i)-(vi) exceeds the logical space. If we start with ‘I do not know whether God exists’, we have two choices concerning God’ existence, namely we can either assert or deny it; fideism chooses the first possibility, agnosticism opts for the second one.

Saying that decisions in world-view matters are practical and motivated by many factors, I do not deny that defenders of attitudes towards religion support their own positions and criticize those of their opponents. Yet, due to the variety of evidence involved in the proposed arguments, we have an actual mosaic of issues and their epistemological qualifications. Let me appeal to historical examples. The theism of the Aquinate and similar thinkers looks for deductive arguments for God’s existence, eventually supplemented by data from physics and metaphysics. Anselm of Canterbury and other theological rationalists formulated ontological proofs, which are deductive and entirely conceptual. Circumstantial proofs or arguments for God’s existence invoke very different circumstances. For example, the defenders of the idea of Intelligent Design frequently say that the mathematical nature of physical reality cannot be explained without an appeal to God. Other defenders of religion point out miracles, historical facts (for instance, Jesus’ life), morality or mystical states. It is difficult to say something more about this variety than point out that its elements have a very different argumentative force. The situation is additionally complicated by the fact confirmed by almost every world-view discussion, that the use of arguments pro and contra (*) is very deeply determined by the positions of people involved in such debates, much more than in scientific or even philosophical controversies.

Nevertheless, the difficulties in assessing and collecting possible arguments for God’s existence or against this belief do not preclude some general observations about the strategies of theism, fideism, atheism and agnosticism; I exclude skepticism once again. In particular, we can say something about onus probandi in related proofs or disproofs. The onus probandi (the burden of proof) indicates which part in a discussion should prove or justify its thesis. In general, the onus probandi belongs to the party which asserts something positively. Clearly, the issue primarily concerns (***) (this form is the most suitable for a logical discussion), because proving this sentence motivates the theist to take his or her position captured by (37). Hence, the atheist is not obliged to prove the negation of (***). Thus, the demand, frequently directed to the atheists, that they should prove God’s non-existence, is simply misguided for logical reasons. This does not mean that the atheist has nothing to do with justifying his or her own standpoint. In general, the atheist asserts that the set {God} is empty. One argument consists in objecting to given proofs that this set is non-empty. This way is of limited force, just because the theist can always say that other proofs are possible. Other atheist arguments try to show that no empirical data suggest that the predicate ‘is God’ refers to a non-empty set. For example, many atheists claim that no empirical support can be given for assertions about beings devoid of empirically graspable arguments. Further analysis leads to a substantial discussion concerning reasons invoked by defenders of theism and atheism. Since the fideist does not appeal to knowledge (e.g. scientific knowledge) of God or His existence, he or she limits his or her arguments to personal ones. Thus, fideists consider the onus probandi as related to ‘I believe’, not to ‘I know’. Similarly, the agnostic stresses ‘I believe that God does not exist’ and claims that there are no sufficient reasons for believing that God exists. Fideists and agnostics, contrary to theist and atheists, are more interested in believing than in knowing, both related to God. They agree that knowledge (in the rational or scientific sense) does not apply to God, but they can disagree whether there is another kind of knowledge or what the word ‘knowledge’ means. Historically speaking, atheism and agnosticism usually point out the lack of scientific knowledge of God and insist that other types of knowledge are merely personal and subjective cognitive states, and thereby cannot pretend to objectivity. This last observation shows well how the whole issue is complicated and what the consequences are of the fact that theism and atheism operate on different epistemological levels, although, on the other hand, fideism is closer to theism and agnosticism is closer to atheism, in particular, if we take into account the practical consequences of discussed views.

Demonstratio appeared in mathematical texts in the 17th century as regards speaking about deductive proofs (in our sense). The letters QED (quod erat demonstrandum, which was to be proved) remind us of this terminological custom until now. It is interesting that the word deductio had a different meaning at that time. Its employment in medieval Latin was related to law and applied to deriving a claim or right before a court. The process of such a derivation was divided into two parts, namely factual and legal demonstration. Assume that a person Y borrowed money from a person X and did not return the loan. If X put the case before the court, he had to prove that the loan actually took place and that there is a legal rule, which generated the claim in question. In other words, one problem concerned quid facti?, and the second one, quid juris? Kant used this sense of deductio in explaining his idea of the transcendental deduction, pointing out that every case of this reasoning has a quid facti? part and a quid juris? part. When Kant spoke about deduction in our sense, he used the term Ableitung (derivation by logically admissible steps). Of course, Ableitung is analytic, apodictic and, thereby, a priori. For Kant, the transcendental deduction has the same nature, except that it is analytic. Thus, Kant argued that some, in fact, the most important pattern of reasoning, is apodictic and a priori, but is not reducible to purely logical steps. A less sophisticated philosophically usage occurred in Newton, who also employed deductio in a very loose sense as deriving hypotheses on the basis of some evidence, but who proved the theorems in the Principia Philosophiae Naturalis by rigorous demonstratio.

The development of logic and meta-mathematics in the 19th and 20th centuries resulted in, among other things, establishing the meaning of ‘deductive proof’ in the sense of Kant’s Anleitung in the following manner. The deductive proof of a sentence (more generally, a formula; but we can speak about sentences in the present context without any loss of generality) A on the basis of a set X of sentences (notation: Pr(A, X)) is a finite sequence of sentences S = A₁, ..., An-1, An such that An = A and for any 1 ≤ i ≤ n, Ai is either a logical axiom (Ai is provable from the empty set, that is, Pr(Ai, ∅)) or An ∈ X or An is provable (derivable) from the earlier items of S by the applying merely logical rules of inference. By a logical rule of inference we mean a rule which preserves truth, that is, never leads to false conclusions from true premises. Accordingly, the concept of deductive proof essentially uses the notion of logical consequence or logical entailment. Thus, we can say that A is a logical consequence of X or X logically entails A instead of saying that A is provable from X. If X is an axiomatic system, we speak about axiomatic proofs, that is, proofs based on some sentences that are adopted as axioms. Traditionally, one says that axioms are accepted without proof. However, if one writes axioms, one automatically constructs one-term sequences and any such sequence can be considered a proof for an axiom which constituted this sequence. In the case of axiomatic proof (as in geometry), every member of S is either an axiom of a consequence of axioms obtained by the logical rules of inference. By conditional proofs or proofs from assumptions, we understand derivations in which S consists of assumptions, not necessarily considered to be axioms. If A is an axiom, we think of it as a true sentence, unless it is demonstrated incorrect. This presumption does not concern conditional proofs, because these assumptions are only accepted, but not accepted as true. For example, we can derive deductive conclusions from falsehoods or sentences temporally or hypothetically accepted. Since the general concept of proof does not require that all elements of S are true, one traditionally distinguishes formal and material correctness of proofs. A proof Pr(A, X) is formally correct if A is logically entailed by X, but it is materially correct if it is formally correct and its premises are true. It is taken as granted that axiomatic proofs are materially correct. This allows us to say that proving A consists in proving that A is true, but this is an oversimplification. Anyway, the concepts of proof and deductive proof can be considered as equivalent under the above analysis.

Not all arguments are deductive. Consequently, if we accept the equivalence of proofs and deductive proofs, not all arguments fall under the form Pr(A, X). This looks trivial, but it is not, because we should distinguish between worthless non-deductive arguments and good non-deductive arguments. If I say that tomorrow it will rain because one week ago it was Friday, my argument fails entirely, but if I say that tomorrow it will snow because today it is cold, this reasoning has some credibility. Now, there are well-known problems with good non-deductive arguments consisting in difficulties in defining non-deductive goodness, soundness, correctness, etc. Let us assume, without attempting to state how the qualities of non-deductive arguments might be measured, that we have intuitive criteria in this respect. The tradition of using such words as apoidexis, demonstratio or ‘proof’ suggests broadening the concept of truth in order to apply it to good non-deductive arguments. In general, such arguments consist in acquiring empirical evidence to support the proposed statements or hypotheses. Legal arguments, related to the question quid facti? and empirical reasoning in science are paradigmatic examples of proofs in the presently considered sense. Let us observe that the word ‘proof’ has a definite rhetorical flavour. One could propose that we should use the generic term ‘justification’ and divide justifications into deductive (proofs) and inductive (in the broad sense), and eventually the latter into good and worthless. Under such a convention, every proof is a justification, but not reversely. However, it is difficult to expect that this proposal would be accepted, perhaps except by philosophers of science. For example, judges will say ‘We do not merely justify our statements concerning that someone is guilty, we must prove them’. Similarly, when Newton said hypotheses non fingo, he had in mind that he proposed something proved, not merely justified. The reason for such an attitude seems to be related to an evaluative or normative aspect of the meaning of ‘proof’, suggesting that if something is proven, it is beyond a reasonable doubt, at least subjectively. This observation decides that apodeixis is understood as apodictically convincing (the same concerns de-monstratio), although etymology does not suggest this. Anyway, according to common opinion it is much better to prove than to merely justify.

Deductive proofs and proofs from various empirical circumstances are closely related. As it has been observed many times not everything is deductively provable. Even if we agree that logical tautologies are derivable from the empty set and proofs of axioms are one-term sequences, it remains to justify that these, but not other axioms, are adopted. The issue is particularly important when the postulates of empirical theories are proposed, but also mathematical axioms cannot spring from nothing. I will illustrate this issue by an example relevant for the main topic of my paper.[6] The proofs of the existence of God ex motu and ex causa efficiendi fall under the following scheme:

(****) if ∀x∃yP(x, y), then ∃y∀xP(x, y).

The following arguments are instances of (****): (a) if every moving thing is moved by its mover, then there exists a mover of all things; (b) if everything is an effect of a cause, then there exists a cause of all things. Clearly, (****) is invalid. As counter-example we can take a the fallacious argument saying that if for every natural number n, there is a number greater than n, then there is the natural number greater than all natural numbers. Russell in his famous conversation with Father F. Copleston used another example, namely that the assertion that there exists the mother of all people does not follow from the assumption that every man has a mother, because the former is false and the latter true.

Thomas Aquinas supplemented (****) in his proofs by reasoning to the effect that infinite sequences of movers or efficient causes are impossible. I do not evaluate whether his demonstrations are sound or not. He appealed to Aristotelian physics, while his followers are rather inclined to rely on metaphysical necessity. Anyway, Aquinas intended to prove the existence of God. He used the label demonstratio, applied syllogistic forms and proceeded by reductio ad absurdum. Although his reasoning also appeals to empirical circumstance, the Five Ways are deductive and have the structure of syllogisms. Thus, interpretations of Aquinas that point out[7] that he did not offer proofs, but only suggestions for unbelievers or defenders of the faith, are historically unfaithful. There is still another reason to think that Thomas Aquinas considered the Five Ways to be proofs. His idea of religious belief regarded every act of faith as veridical, if it comes from right reason (ratio recta). This means that ratio recta produces knowledge, independently of whether it concerns religion or not. There can be a problem with revealed truths that transcend human cognitive capacities (for example, the Trinity Dogma), but (***). This statement is subjected to a normal demonstratio by syllogisms and reductio ad absurdum employing factual information. The Aquinate, a very faithful follower of Aristotle, could not accept knowledge without its being proven. Anyway, there is no obstacle to consider Aquinas’ and other proofs to be arguments for theological models in the sense exhibited at the beginning of this paper.

The role of considering theological argument as concerning some model is particularly impressive in the case of ontological proofs of God’s existence. I shall comment upon a proposal made by the late Jerzy Perzanowski.[8] Perzanowski offers a reconstruction of the reasoning proposed by Charles Harthshorne. It is based on two axioms:

(a)   if a being is the most perfect, its existence is necessary

(b)   the existence of the most perfect being is possible (Leibniz lemma).

Using modal system S5, we can obtain:

(c)   the most perfect being exists.

Perzanowski simplified Harthshorne’s argument by proving (b) in a stronger modal logic equating truth and necessary truth. Moreover, this logic proves (this is another version of the Leibniz lemma):

(d)   if the most perfect being is possible (its existence is possible), it is necessary (its existence is necessary).

Hence, via modus ponens, we have:

(e)   the most perfect being exists as necessary (necessary existence implies existence).

The last step ends Perzanowski’s reconstruction.

If the matter concerns (a), one might observe that it is a conditional assertion having an existential antecedent. Hence, any further application of this axiom essentially depends on the truth of the sentence ‘the most perfect being exists’. If (b) is applied in proving that the most perfect being exists, the entire argument is burdened by petitio principii. This is quite evident if we consider the equivalence:

(f)   the most perfect being exists if and only if the most perfect being exists and it is possible.

However, (f) is trivial because the sentence ‘a exists’ entails ‘it is possible that a exists’. As far as the issue concerns Perzanowski’s simplification, the success of his main move based on (d) requires a very strong modal logic (the logic of strong rationalism). Although the scope of the term ‘logic’ is conventional to some extent, one can express some serious doubts as to whether logic should lead to such strong existential consequences. Personally, I would prefer to say that the proof of the Leibniz lemma and further steps of the entire argument proceed in the context of some formal theory involving modal concepts and relations between them and that this theory is not purely logical. Nevertheless, the fact that Anselm’s argument can be presented via a precise formal machinery shows the power of formalization applied to concepts of rational theology or theological ontology.[9]

Since, according to Leibniz, possibility is logically equivalent to consistency, (d) can be rendered as:

(g)   if the most perfect being is consistent, its existence is necessary.

Let the letter B denote the predicate ‘is the most perfect being’. Assume that B is introduced by some consistent set K of sentences. By the Gödel-Malcev completeness theorem (a set X of sentences has a model if and only if this set is consistent), K has a model. This assumption meta-mathematically guarantees that K is non-empty. One can be even tempted to say:

(h)   if a set of sentences has a model, it possesses it necessarily (it is impossible that the set in question has no model).

However, this last assertion requires additional constraints (see below). The way in which K is given as defining the item conceived as greater than any other being, is very important. First of all, the phrase ‘the conceived item’ means ‘the item described by a set of sentences’. Secondly, according to the common opinion, the most perfect being (MPB, for brevity) is introduced by the maximalization procedure. Meta-logically speaking, it consists in identification of MPB with a collection of perfections (omniscience, omnipotence, the greatest goodness, immutability, infinity, etc.); existence belongs to perfections. Assume that K⁰ is an initial consistent collection of perfection. It might be maintained that application of the Lindenbaum maximalization theorem (every consistent set of sentences has a maximally consistent extension) provides an argument for generating the adequate set K. However, the Lindenbaum extensions are not unique. More specifically, if X is a consistent set of sentences, it has more than one (in fact, there are infinitely many) maximally consistent extensions. Every such extension E has a model (a possible world) in which elements of E are true. Moreover, since these sentences cannot be false in this model, they must be true in it as well. Note that the relativisation to a specific model is crucial.

Let us apply these observations to the set K⁰ and K. Clearly, the former set has several different maximally consistent extensions, including K as their part; in fact, K itself is not maximal, but it is a minor point. Meta-mathematical observations about K do not suffice for a demonstration that this set is true in every possible world. Note that we could consider K as a body of absolute necessities (necessary truths), that is, sentences true in every model. It is obvious now that necessity of truth, with respect to a specific model is not absolute, but just relative, because a sentence true in one model can be false in other possible worlds. Moreover, there is no a priori reason to refer to one and only one initial set K⁰ of perfections. One could ask, for instance, why the greatest goodness or immutability, but not changeability or goodness directed to people deserving it to some degree, should be regarded as prima facie perfections. Typically, listed perfections are recommended by Christian (or other similar) theology, but it is only a religious argument, but not a logical one. Thus, contemporary versions of Anselm’s ontological proof do not show that B is not empty in virtue of logical necessity. Furthermore, one cannot demonstrate independently the assumptions of a given monotheistic theology that the denotation of B is unique.

If we apply a similar technique to any consistent collection of anti-perfections (that is, pointing out the lack of perfections in the ordinary or theological sense), it is easily provable by the dual logic and meta-mathematics (it takes falsity as the distinguished value) that there is a being such that nothing lesser can be conceived. We can interpret this being as the Absolute Evil (AE, for brevity). If we entirely omit the ordinary or religious meaning of perfection, AB is a maximal being, because it is constructed by a similar maximalization strategy as employed in the case of the denotation of B. There is no reason, at least logical, in order to maintain that AE is less real than MPB or Christian God. This leads to a Manichean theology with its perennial battle between forces of Goodness and Evil or even a radicalized Marcionic heresy in which the world presents itself as a emanation of the personified Evil, in particular, deceiving people in order to make them suffer more. The theist could presumably answer that the meta-mathematically phrased ontological proof is enough for demonstrating the existence of MPB, let us say a god of philosophers which has properties which are not entirely coherent with Christian theology. This being is necessary and thereby exists. However, this standpoint is not correct, even if we agree some perfections actually occur in our world. Lindenbaum’s maximalization procedure does not imply that there exist maximal perfections in the sense of Anselm and his followers. Although people know something, can do something or behave properly, these facts do not entail that there is a being (subject) which realizes these properties in the most maximal manner. Thus, there are models in which no MPB occurs. In fact, some maximally consistent extensions of K⁰ contain the sentence ‘there exists the MPB, according to a given specification of perfections’, but this sentence is false in other extensions. Since we do not know which model represents our universe, we can only say that the gods of philosophers (and theologians as well) exist in some possible models.[10]

We can also make some general observations about the strategies of theism, fideism, atheism and agnosticism. In particular, we can say something about onus probandi in related proofs or disproofs. The onus probandi (the burden of proof) indicates which part in a discussion should prove or justify its thesis. In general, the onus probandi belongs to the party which asserts something positively. Clearly, the issue primarily concerns (***), because proving this sentence motivates the theist to take his or her position captured by (37). Hence, the atheist is not obliged to prove the negation of (***). Thus, the demand, frequently directed at atheists that they should prove God’s non-existence, is simply misguided for logical reasons. There is also a psychological argument in a similar direction. It applies to atheists and agnostics grouped together, let us call them disbelievers for a while. A religious attitude, independently of whether it is understood as knowledge or merely faith, has grounds in positive mental states. Of course, one can say that disbelievers lack such states and that the absence of a mental state is also a mental event. This is true, but there are serious doubts as to whether negative state of affairs can be causally efficient. However, the practice of religious disbelievers suggest that, if their atheism or agnosticism is not caused by growing in a non-religious environment, they always look for the weak points of theological arguments.

Although I am skeptical about the scientific knowledge that God does not exist, I do not suggest that the atheist has no arguments for justifying his or her own standpoint. In general, the atheist asserts that the set {God} is empty. One argument consists in objecting to given proofs that this set is non-empty. This way is of limited force, just because the theist can always say that other proofs are possible. Other atheist arguments try to show that no empirical data suggest that the predicate ‘is God’ refers to a non-empty set. For example, many atheists claim that no empirical support can be given for assertions about beings devoid of empirically graspable arguments. Further analysis leads to a substantial discussion concerning reasons invoked by defenders of theism and atheism. Since the fideist does not appeal to knowledge (e.g. scientific knowledge) of God or His existence, he or she limits his or her arguments to personal ones. Thus, fideists consider the onus probandi as related to ‘I believe’, not to ‘I know’. Similarly, the agnostic stresses ‘I believe that God does not exist’ and claims that there are no sufficient reasons for believing that God exists. Fideists and agnostics, contrary to theist and atheists, are more interested in believing than in knowing, both related to God. They agree that knowledge as a cognitively rational enterprise does not apply to God, but they can disagree whether there is another kind of knowledge or what the word ‘knowledge’ means. Historically speaking, atheism and agnosticism usually point out the lack of scientific knowledge of God and insist that other types of knowledge are merely personal and subjective cognitive states and thereby cannot pretend to objectivity. Agnosticism is my own position. It is motivated by logical difficulties of all known proofs of God’s existence. They allow me to reject (***) and, consequently, to accept the statement ‘I do not believe that God exists’. Additionally, I argue that the language of theology and the language of science are semiotically incommensurable. Since any proof (deductive or empirical) assumes that the language in which it is embedded possesses semantic uniformity, prospects of proofs from the world to transcendence or in the reverse direction are not very great. On the other hand, logic helps in understanding the nature of the theological enterprise and its quality.

Zapraszamy do zakupu pełnej wersji książki

Footnotes

CHAPTER 1
Jan Woleński. Theology and Logic

* This contribution was made possible through the support of a grant “The Limits of Scientific Explanation” from the John Templeton Foundation.

[1] See Th. Flint, M. Rea, The Oxford Handbook of Philosophical Theoology, Clarendon Press, Oxford 2009 for a comprehensive survey.

[2] For details of this movement, see J. Woleński, Polish Attempts to Modernize Thomism by Logic (Bocheński and Salamucha), “Studies in East European Thought” 2003, vol. 55, p. 299-313.

[3] See Th. Flint, M. Rea, The Oxford Handbook, op. cit., p. 5.

[4] I follow J. Woleński, Theism, Fideism, Atheism, Agnosticism, [in:] L. Johannson (ed.), Logic, Ethics, and All That Jazz. Essays in Honour of Jordan Howard Sobel, Department of Philosophy, Upsala University, Uppsala 2009; some fragments presented are taken verbatim from this paper.

[5] See J. Woleński, Logical Squares: Generalizations and Interpretations, “Logica” 1995, Proceedings of the 9th International Symposium, T. Childers, P. Kolár, and O. Majer (eds), Filosofia, Praha 1996, p. 67-75; Applications of Squares of Oppositions and Their Generalizations In Philosophical Analysis, “Logica Universalis” 2008, 2 (1), p. 13-29; reprinted in J. Woleński, Essays..., op. cit., p. 255-269.

[6] See H. Sobel, Logic and Theism. Arguments for and against Beliefs in God, Cambridge University Press, Cambridge 2004 for the comprehensive survey of proofs of God’s existence.

[7] See for example J. Schmidt, Philosophische Theology, Kohlhammer Verlag, Stuttgart 2003 in the introduction to Part I.

[8] See J. Perzanowsaki, O wskazanych przez Hartshorne’a modalnych krokach w dowodzie ontologicznym Anzelma (1), unpublished.

[9] See also M. Tkaczyk, Is the Ontological Proof of God’s Existence an Ontological Proof of God’s Existence?, “Logic and Logical Philosophy” 2007, no. 16, p. 289-309 for embedding Anselm’s proof into first-order logic without modalities.

[10] See also J. Perzanowski, Reasons of Monodeism, [in:] J. Perzanowski, Art of Philosophy. A Selection of Jerzy Perzanowski’s Works, Ontos Verlag, Frankfurt am Main 2012, p. 41-44, where one can find a good example of model-theoretical thinking in theology; Perzanowski gives a set of definitions for monotheistic philosophical theology.