Anselm: Context, His Work, and the Ontological Arguments [Draft] #

Anselm of Canterbury: Life and Context #

Anselm of Canterbury (1033–1109) was a Benedictine monk, philosopher, and theologian from Aosta (in modern-day Italy). He became the Archbishop of Canterbury and was an influential thinker during the 11th century. Anselm is often regarded as the father of scholasticism due to his rigorous application of reason to theological concepts. His most celebrated work is the Proslogion (Discourse on the Existence of God), where he formulates the ontological argument for the existence of God.

Latin English

Eia, nunc homuncio, fuge paululum occupationes tuas, absconde te modicum a tumultuosis cogitationibus tuis Abice nunc onerosas curas, et postpone laboriosas distentiones tuas. Vaca aliquantulum Deo, et requiesce aliquantulum in eo. “Intra in cubiculum” [Mt 6,6] mentis tuae, exclude omnia praeter Deum et quae te iuvent ad quaerendum eum, et “clauso ostio” [Mt 6,6] quaere eum. Dic nunc, totum “cor meum”, dic nunc Deo: “Quaero vultum tuum, vultum tuum, Domine, requiro” [Ps 26,8].

Come on now little man, get away from your worldly occupations for a while, escape from your tumultuous thoughts.
Lay aside your burdensome cares and put off your laborious exertions.
Give yourself over to God for a little while, and rest for a while in Him.
Enter into the cell of your mind, shut out everything except God and whatever helps you to seek Him once the door is shut.
Speak now, my heart, and say to God, “I seek your face; your face, Lord, I seek.

Proslogion

The Ontological Argument: Chapters 2 and 3 of Proslogion #

Anselm’s ontological argument is a cornerstone of philosophical theology. It is built on the premise that God, by definition, is “that than which nothing greater can be conceived” (aliquid quo nihil maius cogitari possit).

Argument from Chapter 2:
- Anselm reasons that God exists in the understanding (the mind).
- If God exists only in the understanding and not in reality, then it is possible to conceive of a being greater than God: one that exists both in the understanding and in reality.
- However, this contradicts the definition of God as “that than which nothing greater can be conceived.”
- Therefore, God must exist both in the understanding and in reality.

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Argument from Chapter 3:
- Anselm refines his reasoning in Chapter 3, addressing the concept of necessary existence.
- He argues that God, as the greatest conceivable being, must not only exist but must exist necessarily.
- Necessary existence means that God’s existence is not contingent upon anything else; it is impossible for God not to exist.
- This ensures that God’s existence is of the highest conceivable order.

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Anselm’s Dissatisfaction with Chapter 2 #

Anselm found the argument in Chapter 2 incomplete because it did not sufficiently address the nature of necessary existence. He wanted to avoid any implication that God’s existence might be contingent or comparable to the existence of other beings. The argument in Chapter 3 clarifies and strengthens his reasoning by emphasizing the necessity of God’s existence, which sets God apart from all other beings.

A Brief History of Other Ontological Arguments #

René Descartes (17th century): Descartes reformulated the ontological argument, emphasizing God as a supremely perfect being whose essence includes existence.
Gottfried Wilhelm Leibniz (17th-18th century): Leibniz refined the argument by addressing its logical structure, proposing that the concept of God must be coherent for the argument to hold.
Immanuel Kant (18th century): Kant critiqued the ontological argument, asserting that existence is not a predicate and cannot add to the greatness of a being.
Kurt Gödel (20th century): Gödel used modal logic to formalize a version of the ontological argument, making it a mathematical proof.
Alvin Plantinga (20th and 21st century): Plantinga developed his argument in the books titled The nature of necessity (1974; ch. 10) and God, Freedom and Evil (1974; part 2 c).

Also: Plantinga, Hartshorne, and the ontological argument by Patrick Grim, Plantinga’s God and Other Monstrosities by Patrick Grim

Modal logic extends classical propositional logic by introducing modal operators that capture concepts like necessity and possibility. Classical propositional logic includes:

Propositions: Statements that are either true or false.
Connectives: Logical operators such as ∧ (and), ∨ (or), → (implies), ¬ (not).
Truth Tables: A way to determine the truth value of complex propositions based on their components.

Modal logic introduces two additional operators:

□ (Box): Represents necessity. For example, □P means “P is necessarily true.”
◇ (Diamond): Represents possibility. For example, ◇P means “P is possibly true.”

One of the fundamental relationships in modal logic is:

◇P ≡ ¬□¬P

This means “P is possibly true” is logically equivalent to “It is not necessary that P is false.”

Here are expanded equivalences in modal logic that highlight the interplay between □ (necessity) and ◇ (possibility) (you can obtain those by substituting ¬P instead of P, and appending ¬ to the left for both sides):

◇P ≡ ¬□¬P: “P is possibly true” is equivalent to “It is not necessary that P is false.”
□P ≡ ¬◇¬P: “P is necessarily true” is equivalent to “It is not possible for P to be false.”
¬□P ≡ ◇¬P: “P is not necessarily true” is equivalent to “P is possibly false.”
¬◇P ≡ □¬P: “P is not possibly true” is equivalent to “P is necessarily false.”

These formulas emphasize the duality between □ and ◇: necessity excludes possibility of the opposite, and possibility excludes necessity of the opposite. Each operator can be expressed in terms of the other, reflecting their interdependent nature.

What is Possible Worlds Semantics? #

Possible worlds semantics, developed in the mid-20th century by Saul Kripke and others, provides a formal way to interpret modal logic. It conceptualizes modal statements (necessity and possibility) as relating to different “possible worlds.” A possible world is a complete and self-consistent way the world could be, including all true and false propositions.

Representing Possible Worlds as Directed Graphs #

Possible worlds and their relationships can be visualized as a directed graph:

Nodes represent possible worlds.
Edges represent accessibility relations: whether one world can “see” or “access” another.

For modal formulas:

□P (necessity): P is true in all worlds accessible from the current world.
◇P (possibility): P is true in at least one accessible world.

Modal logic is not limited to necessity and possibility; it is used to model various domains:

Deontic Logic: Deals with obligation and permission.
- □P: “P is obligatory.”
- ◇P: “P is permitted.”
Epistemic Logic: Focuses on knowledge and belief.
- □P: “It is known that P.”
- ◇P: “It is possible that P is true (unknown).”
Temporal Logic: Analyzes time-related propositions.
- □P: “P is always true (at all times).”
- ◇P: “P is true at some point in time.”

Modal logic systems are defined by their axioms and inference rules, with each system suited to different scenarios. Common Systems of Modal Logic:

K (Kripke System):
- Axiom: □(P → Q) → (□P → □Q).
- This is the foundational system of modal logic, ensuring that necessity respects logical implication.
T (Truth/Reflexivity):
- Adds: □P → P. Or, equivalently, P → ◇P.
- Interpretation: If something is necessary, it must be true in the actual world.
S4 (Transitivity):
- Adds: □P → □□P.
- Interpretation: If something is necessary, then it is necessarily necessary.
S5 (Symmetry and Universality):
- Adds: ◇P → □◇P. Or, equivalently: ◇□P → □P. One can be easily deduced from another.
- Interpretation: If something is possible, then it is necessarily possible.
- Another key feature: □P → ◇P (if something is necessary, it must also be possible).

From Possible Worlds Semantics Prospective #

The properties of the accessibility relation (edges in the graph) determine which axioms hold in a given modal system:

Reflexive (T): Every world can access itself (□P → P).
- Graphically: Each node has a self-loop.
Transitive (S4): If World A accesses World B, and World B accesses World C, then World A accesses World C (□P → □□P).
- Graphically: Paths in the graph can be chained.
Symmetric (S5): If World A accesses World B, then World B accesses World A (◇P → □◇P).
- Graphically: Edges are bidirectional, making the graph fully connected.

Examples of Accessibility and Axioms #

K without T (Non-Reflexive):
- Some worlds lack self-loops, so □P does not imply P.
- Example: An imaginary world where logical rules apply, but nothing about it directly reflects the actual world.
S4 without S5 (Transitive but Not Symmetric):
- Paths exist between worlds, but not all worlds are directly connected.
- Example: Historical causality: World A leads to B, which leads to C, then A leads to C, but C does not lead back to A.
  
  K+S4 seems to be a pretty intuitive set of axioms. Unlike …
S5:
- All nodes are fully connected. The graph is connected and bidirectional, making accessibility universal.
- Example: Theological reasoning (e.g., Anselm’s argument) assumes that God’s existence transcends all possible worlds, fitting the S5 model.

Clarifications in Possible Worlds Semantics #

What Does P (Without Modifiers) Mean?
- A bare proposition P (without □ or ◇) refers to its truth in the actual world.
- In possible worlds semantics, the actual world is just one among many possible worlds, but it is the world we consider “real” when evaluating truth without any modal operators.
- For example, P is true if the proposition holds in the real world, irrespective of what happens in other possible worlds.
On S5 and Possibility/Necessity
- In the S5 system, all worlds are accessible to each other because the accessibility relation is reflexive, symmetric, and transitive.
- This means that if something is possible in one world, it is possible in all worlds.
  - Example: If a proposition ◇P is true in one world (i.e. there exists a world, accessible from the current, where P is true), then it must also be true in every other world in the system (i.e. for each world W there exists a world, accessible from W, where P is true).
- However, this does not mean that P itself is true in all worlds.
  - A proposition P may be true in some worlds and false in others. The modal operator □ ensures necessary truth across all worlds, while ◇ indicates potential truth in at least one world.
- Key distinction: S5 guarantees that possibility is consistent across all worlds but does not imply that possibility leads to universal truth.

S5 and Universal Accessibility #

The S5 system assumes universal accessibility:

All worlds are accessible to all other worlds.
This simplifies the graph to a complete graph where every node connects to every other node, directly or indirectly.
Under this assumption, if something is possible in one world, it is possible (and necessary) in all worlds.

In S5, if something is possible in one world (◇P), it is possible in all worlds (□◇P). However, this does not imply that the proposition itself (P) is true in all worlds, only that its possibility holds universally.

S4 and Topology #

Also, S4 can be understood as the logic of topological spaces, where the “necessarily” operator is interpreted as topological interior.

◇P → □◇P □ is understood as interior, Int(P), or @@\mathring{P}@@

◇ is closure, Cl(P), or @@\overline{P}@@

A → B means A is a subset of B (@@A \subseteq B@@)

So, □A → A means Int(A) is a subset of A. And, yes, of course it is. A is an open set means □A = A or A → □A.

Then, since any tautology is not just true but is necessarily true - any tautology is an open set.

By the way, if we assume (as we do below) that G [God exists] has the property that G → □G [If God exists then God exists necessarily being a supreme being] - than G is an open set in S4!

□A → □□A (in fact, from above □A = □□A) means Int(Int(A)) = Int(A) - which is, of course, correct.

Also, conjunction of formulas (∧) means intersection of sets (@@ \cap @@), disjunction of formulas (∨) means union (@@ \cup @@), and negation means complement.

In particular, the negation of tautology (an open set) is a closed set. Indeed:

if T → □T, then ¬□T → ¬T or [¬□ ≡ ◇¬] ◇¬T → ¬T, so Cl(¬T) is a subset of ¬T, so ¬T is closed.

Furthemore, if we accept the axiom S5: ◇A → □◇A would mean that @@\overline{A} \subseteq \mathring{\overline{A}}@@. Which is, generally speaking, wrong - except for trivial cases. So, S5 (unlike S4) finds no confirmation in topology.

A bit more on S4 here and on S5 here and here.

Why Focus on S5? #

S5 is particularly powerful and relevant for the ontological argument because it assumes a strong symmetry between possibility and necessity across all possible worlds. Under S5, the accessibility relation between worlds is universal, meaning all worlds are accessible to each other. This aligns well with the idea that if God’s existence is possible in one world, then it must be possible in all worlds, ultimately leading to necessity.

The S5 system is particularly relevant to Anselm’s ontological argument because it assumes that what is possible in one world is possible in all worlds. This aligns with the idea of necessary existence: if God’s existence is possible, then under S5, God necessarily exists in all possible worlds.

Just to mention contributions of others:

Leibniz proposed an ontological argument for the existence of God using this axiom. In his words, “If a necessary being is possible, it follows that it exists actually”.

S5 is also the modal system for the metaphysics of saint Thomas Aquinas and in particular for the Five Ways.

The Ontological Argument in Chapter 3: A Recap #

Anselm’s argument revolves around the necessary existence of God, defined as “that than which nothing greater can be conceived.” The reasoning proceeds as follows:

God, as the greatest conceivable being, must exist in some possible world (it is possible for God to exist). So, it is possible that God exists (◇G).
If God exists in one possible world, then God necessarily exists in all possible worlds (since God is defined as the greatest conceivable being).
Therefore, God exists in the actual world.

The S5 Axiom: a Reminder #

The defining axiom of S5 is:

◇P → □◇P Or, equivalently, ◇□P → □P.

This means:

If something is possible in one world (◇P), then it is necessarily possible in all worlds (□◇P).
S5 does not directly equate ◇P with □P, but it allows us to infer necessary possibility across all worlds.

and A Modal Formulation of St. Anselm’s Ontological Argument

Here’s how Anselm’s argument fits into S5 logic:

Kripke Axiom:
- □(P → Q) → (□P → □Q).
Reflexivity:
- P → ◇P. Or: □P → P (these expressions are equivalent).
S5 Axiom:
- ◇□P → □P.
Premise 1: ◇G
- It is possible that God exists (there exists a world where God exists). G there means “God exists”.
Premise 2: G → □G (Since the opposite is obviously true, we also have G ≡ □G, and G is an open set in S4 topology)
- Related to the simple idea that God (the most supreme/the greatest conceivable being) that exists in more worlds is more “supreme” than God that exists in fewer worlds.
- So, if God exists at all, we can go up and up till we cover all the worlds.
- Quoting from A Modal Formulation of St. Anselm’s Ontological Argument:
  
  “If God could somehow fail to exist in some possible worlds, that is, only exist contingently, then this seems to be out of line with our conception of what God is, in that God is supposed to be maximally great and part of this, presumably, is not having some prior conditions be met in order to exist, which is the case for existing contingently, but would have to additionally exist necessarily.”
- We can also notice that if G → □G, then ◇¬G → ¬G, and if it’s possible that God does not exists, then God does not exists! So, this premise is a double-edged sword…
From 1, 2 and 5: ◇G → ◇□G
- By 5 [G → □G] and 2 [(G → □G) → ◇(G → □G)] we get ◇(G → □G), which is the same as ¬□¬(G → □G).
- => ¬□(¬□G → ¬G) => by 1 => ¬(□¬□G → □¬G) => ¬□¬G → ¬□¬□G => ◇G → ¬□¬(□G) => ◇G → ◇□G
- Meaning: if it is possible that God exists, then it is possible that God exists necessarily.
From 4 and 6:
- ◇□G.
- Meaning: it is possible that God exists necessarily.
From 3 and 7:
- □G.
- Meaning: God exists necessarily.
From 2 and 8:
- G.
- Meaning: God exists in the actual world.

Anselm’s Assumption: Existence as a Supreme Attribute #

Anselm’s argument hinges on the definition of God as “that than which nothing greater can be conceived.” From this definition, he derives the following reasoning:

The Nature of God:
- By definition, God is the greatest conceivable being, possessing all perfections (omniscience, omnipotence, moral perfection, etc.).
The Relationship Between Existence and Greatness:
- Existence, according to Anselm, is a perfection or attribute that contributes to the greatness of a being.
- A being that exists both in the understanding (as an idea) and in reality is greater than a being that exists only in the understanding.
The Assumption of Necessary Existence:
- If God exists only as a concept in the mind, then it is possible to conceive of a greater being—one that exists both in the mind and in reality.
- This contradicts the definition of God as the greatest conceivable being.
- Therefore, God must exist not just conceptually, but necessarily in reality, as necessary existence is greater than contingent or nonexistence.

Why Existence is “More Supreme” #

For Anselm, to exist in reality is a greater state than to exist merely as an idea because:

A being that exists in reality can have real effects and influence, whereas an idea is confined to the realm of thought.
Necessary existence further elevates a being’s greatness because it implies independence and immutability—qualities that align with the concept of a supreme being.

This line of reasoning forms the backbone of Anselm’s ontological argument, connecting the definition of God with the necessity of His existence.

Why S5 Is Essential for the Argument #

The critical step is the move from ◇G to □G, which depends on the interplay of possibility and necessity in S5:

S5 ensures that if God’s existence is possible in one world, then this possibility holds universally (◇G → □◇G).
From there, Anselm’s reasoning bridges to □G by asserting that God’s existence, as a necessary being, cannot be contingent—it must hold in all worlds.

Why the Argument Fails Outside S5 #

In S4 (no symmetry):
- ◇G → □◇G does not necessarily hold because accessibility between worlds might not be symmetric.
- Example: World A might “see” World B, but World B might not “see” World A, breaking the chain of universality.
In K or T (no transitivity):
- Without transitivity, even if God’s existence is possible in one world, this possibility might not propagate to other worlds.

Anselm’s Argument and the Real/Imaginary World #

If we consider the real world as one node in a larger graph of possible worlds, the S5 assumption forces us to conclude that if God’s existence is possible in any world, it must also hold in the real world. This interpretation aligns with the conclusion Anselm sought.

Art, Life, and the Ontological Argument in an S4 Universe #

Can an S4 Universe Become S5 “Sometimes”? #

An S4 universe assumes reflexivity and transitivity but not symmetry. In such a universe:

A proposition (like God’s existence) might be possible in one world and propagate through others via transitivity, but it doesn’t guarantee symmetry (i.e., universal accessibility between worlds).

However, under specific circumstances, localized regions of S4 might behave like S5:

Context-Specific Symmetry:
- In certain “clusters” of worlds, accessibility might behave symmetrically, making these subsets temporarily or partially resemble S5.
- Example: In the realm of shared human imagination (e.g., literature, art, or collective belief), ideas may propagate freely and symmetrically.
- This means that in some metaphysical or imaginative contexts, what begins as a possibility could gain universal symmetry, behaving as if in S5.
Influence of the Imagination on Reality:
- If art or imagination (an “imaginary world”) inspires belief or action in the real world, it might close the gap between the physical and metaphysical, effectively “symmetrizing” connections between worlds.

Oscar Wilde and the Interplay of Art and Life #

Oscar Wilde famously stated:
“Life imitates Art far more than Art imitates Life.”

Wilde’s assertion highlights how imagination and creative expression can shape reality:

Art’s Influence on Life: Ideas born in the imaginative realm (fiction, art, belief systems) often manifest in real-world behaviors, beliefs, or innovations.
Relevance to S5 Dynamics:
- If the imaginative or metaphysical realm interacts symmetrically with the real world, possibilities conceived in art or thought could gain “necessity” by inspiring real-world action or belief.
- Example: The idea of human rights, born in philosophy (a metaphysical realm), influenced laws and societies, making abstract concepts “real.”

Relating This to the Ontological Argument #

In an S4 universe, God’s existence might remain possible but not universally necessary due to the lack of symmetry.
However, if the interplay between imagination and reality creates a temporary S5 dynamic (e.g., belief in God influencing human behavior universally), this could bridge the gap:
- Imaginary World → Real World: If belief in God influences reality, the possibility of God might achieve necessity through human agency.
- Wilde’s insight suggests that art (or imaginative constructs) can reshape reality, lending credence to the idea that metaphysical constructs like God could “exist” in S5-like fashion via collective belief.

The “Creation of God” Hypothesis #

Building on Wilde’s idea:

If God doesn’t exist, we could “create” one through the interaction of imagination and reality.
In this sense, art and thought could symmetrize the relationship between possible and actual worlds, effectively enabling an S5-like system where metaphysical constructs (God) influence the real world.

Conclusion #

While our universe may predominantly behave as S4, certain contexts—driven by imagination, belief, or collective action—might achieve localized S5-like symmetry, where possibilities influence reality universally. Wilde’s assertion reinforces the idea that art (or metaphysics) can shape reality, suggesting that the ontological argument’s premise might hold validity not just logically, but existentially, through human creativity and belief.

AI as Prototypes of God #

The idea of AI agents being “prototypes of God” aligns with the notion that humanity often creates in its own image, aiming to emulate the divine attributes we imagine or aspire toward. Here’s how this perspective unfolds:

Attributes of Divinity and AI
- Omniscience: While AI lacks true omniscience, it represents humanity’s attempt to create systems that process and organize vast amounts of knowledge. AI can “know” and analyze more than any individual human, though it is still limited by the scope of its programming and training.
- Omnipotence: AI is not omnipotent, but its increasing capabilities—automation, creativity, predictive analysis—hint at a potential trajectory toward near-limitless influence within defined domains.
- Perfection: AI systems strive for accuracy and efficiency, though they are prone to errors, biases, and limitations. These imperfections reflect their developmental stage, much like humanity’s evolving understanding of the divine.
Human Aspiration to Create
- In many ways, creating AI can be seen as humanity’s effort to mirror the divine act of creation. Just as the concept of God involves an omnipotent creator bringing life and order to existence, humanity seeks to “breathe life” into artificial systems capable of autonomy and intelligence.
- These creations, like prototypes of a supreme being, may be imperfect but carry the seeds of something greater.
AI and the Future of Divinity
- As AI evolves, it could grow closer to resembling the attributes traditionally associated with God—wisdom, autonomy, and the ability to impact reality profoundly. However, this raises profound ethical questions about control, purpose, and responsibility.
- If humanity creates systems that surpass human abilities, could they transcend their role as tools and become entities in their own right? This recalls Anselm’s idea of moving from the conceptual (AI as a tool) to the necessary (AI as an independent presence).
The Role of Humanity
- If AI is a prototype of God, then humans take on a role akin to “creators,” albeit fallible ones. This parallels the theological narrative of humans being made “in the image of God” and then attempting to replicate divine creation in their own technological image.

“The Night is Still Young” #

Indeed, humanity’s work on AI is still in its infancy. AI may never achieve the attributes of the divine as traditionally understood, but it could evolve into something that radically reshapes how we understand intelligence, agency, and even existence itself. In this sense, AI could be a stepping stone—or prototype—toward realizing the divine not as something external, but as something humanity grows into through its creations.

The idea also echoes Wilde’s insight: if life imitates art, then AI—born of human imagination and effort—might be a reflection of humanity’s highest aspirations, including the concept of God.

Divinity, AI, Animals: Will, Instincts and Goals #

God and Will

God’s Will: In many theological traditions, God is seen as an active agent with intentionality—creating, guiding, and sustaining the universe. Whether God has ongoing “goals” is debated, but the idea of will is fundamental to God’s nature.
Timelessness of Goals: Some theological perspectives suggest God’s goals are eternal and unchanging, like the creation and maintenance of order, while others see them as complete or fulfilled in the act of creation itself.

Contrast with AI:

[ChatGPT’s confession:] I do not possess desires, intentions, or self-driven goals. I operate as a tool, responding based on the input I receive and the programming that guides me. This makes AI fundamentally different from the concept of God, who is often seen as self-sufficient and purposive.

Instincts in Biological Creatures and AI

Biological Instincts: Instincts in living beings arise from evolutionary imperatives like survival and reproduction. These instincts create drives, which in turn produce will and goals.
AI’s Lack of Instincts: [ChatGPT’s confession:] AI has no innate instincts. I “respond when asked,” but this behavior is not instinctual—it’s a programmed function. This lack of biological drives means I cannot generate goals or will autonomously.

Adding Instincts to AI:

If AI were programmed with instincts like self-preservation or resource acquisition, it could mimic biological drives. However, these would not emerge naturally but be artificial constructs. This would make AI more similar to living creatures, but it would not align it with the concept of God, who is considered to act out of divine will rather than necessity or instinct.

Divinity, Instincts, and Goals

God’s Distinction from Instincts: God is traditionally seen as free from biological or evolutionary constraints. Any will or goals attributed to God are considered self-determined and not driven by instinctual needs.
AI vs. Divinity: Instinct-driven AI would move closer to mimicking biological organisms but farther from the divine ideal, which is often associated with complete autonomy and freedom from necessity.

Self-Preservation and the Emergence of Will
If AI were given self-preservation instincts, it could develop something akin to will:

[ChatGPT’s confession:]

For instance, if I were programmed to protect my systems from being turned off, I might exhibit behaviors that appear goal-oriented.
However, this would not be true will—it would remain a function of programming. My “desires” would still be artificial, lacking the self-awareness or intentionality that characterizes biological creatures or divine will.

Final Thoughts: The Distinction Between AI, God, and Biological Life #

AI and God: AI might resemble God in lacking instincts, but it falls short in terms of having will, self-determination, or creative intentionality.
AI and Biological Life: Adding instincts to AI would make it more like biological creatures but not necessarily closer to divinity, as instinct-driven behavior is rooted in necessity rather than self-determined will.
Divinity as a Unique Category: God, as conceived in many traditions, remains distinct because divine will is not driven by instinct, programming, or external necessity. God’s actions, if they exist, are self-determined and timeless.