Belief and Doxastic Logic

Knowledge is factive.

Belief is not.

A person can believe something false.

That difference is enough to require a different logic.

Doxastic logic studies formal reasoning about belief.

Belief Operator

A common notation is:

[ B_a P ]

meaning:

Agent (a) believes that P.

For example:

[ B_{Alice}(Raining) ]

means:

Alice believes it is raining.

The proposition may be true or false.

Belief vs Knowledge

Knowledge usually satisfies:

[ K_a P \rightarrow P ]

Belief does not.

We do not assume:

[ B_a P \rightarrow P ]

People are mistaken.

That simple difference is philosophically enormous.

Possible-World Semantics

Doxastic logic often uses possible-world semantics.

A world is doxastically accessible if it is compatible with what the agent believes.

Then:

[ B_a P ]

means:

P is true in all worlds the agent considers possible according to their belief state.

False Beliefs

Suppose the actual world has no rain.

Alice believes it is raining.

Her belief-accessible worlds may all contain rain.

The actual world need not be among them.

This is how false belief differs from knowledge.

Consistent Belief

Many formal systems idealize agents as having consistent beliefs.

If an agent believes:

[ P ]

and:

[ \neg P ]

their belief set is inconsistent.

Classical idealizations often try to exclude such states.

Humans do not.

Human Beliefs Are Messy

People can simultaneously believe things that do not fit together.

They may:

• compartmentalize,
• forget,
• fail to notice contradiction.

Doxastic logic often models idealized belief rather than psychological reality.

KD45

A standard doxastic system is often called KD45.

Very roughly, it encodes assumptions like:

• beliefs are consistent,
• agents know what they believe,
• agents know what they do not believe.

These are strong idealizations.

Consistency Axiom

One common principle is:

[ B_a P \rightarrow \neg B_a \neg P ]

If the agent believes P, they do not also believe not-P.

This rules out direct contradictory belief.

Positive Introspection

Another principle is:

[ B_a P \rightarrow B_a B_a P ]

If the agent believes P, they believe that they believe P.

Real people may fail this.

Formal agents are cleaner.

Negative Introspection

A further principle is:

[ \neg B_a P \rightarrow B_a \neg B_a P ]

If the agent does not believe P, they believe that they do not believe P.

Again, this is idealized self-knowledge.

Belief Without Certainty

Belief need not mean absolute conviction.

Ordinary language uses “believe” across degrees.

Doxastic logic is often binary:

believes P

or does not.

This can be too coarse.

Graded Belief

Probability theory offers a richer model.

Instead of:

[ B_a P ]

we may represent:

[ P_a(P)=0.8 ]

The agent assigns 80% credence to P.

This connects belief logic to Bayesian reasoning.

Credence

A credence is a degree of belief.

Credences can express uncertainty much better than binary belief.

An agent may be:

• nearly certain,
• moderately confident,
• unsure.

Probability turns belief into graded structure.

Belief Revision

What happens when evidence contradicts belief?

A rational agent should update.

But updating is not trivial.

Which beliefs should be abandoned?

Which should be preserved?

This is the problem of belief revision.

AGM Theory

A famous formal framework for belief revision is associated with Alchourrón, Gärdenfors, and Makinson.

AGM theory studies operations such as:

• expansion,
• contraction,
• revision.

Belief change becomes formal.

Expansion

Expansion adds a new belief without removing old ones.

This can create inconsistency.

It is the simplest update.

Contraction

Contraction removes commitment to a belief.

The problem is deciding what else must change with it.

Beliefs form networks.

Removing one proposition can affect many implications.

Revision

Revision adds new information while restoring consistency.

If new evidence conflicts with old belief, something must give.

Rational revision tries to minimize unnecessary change.

Confirmation Bias

Humans often fail ideal belief revision.

We may prefer evidence that preserves existing commitments.

This is confirmation bias.

Doxastic rationality and actual psychology diverge.

Entrenchment

Some beliefs are more resistant to revision than others.

Basic assumptions may be deeply entrenched.

Peripheral beliefs may change easily.

Formal belief revision can represent degrees of entrenchment.

Belief and Evidence

A belief can be:

• true,
• false,
• justified,
• unjustified.

These dimensions are separate.

Doxastic logic tracks belief structure.

Epistemology asks whether beliefs deserve acceptance.

Belief About Belief

We can nest belief operators:

[ B_a B_b P ]

Alice believes that Bob believes P.

This matters in:

• games,
• negotiations,
• deception.

Strategic reasoning often depends on beliefs about others’ beliefs.

False Belief Tasks

Psychology studies whether children understand that another person can hold a false belief.

This ability is central to theory of mind.

To predict another person’s behavior, we must model their beliefs, not just reality.

Deception

Deception deliberately manipulates another agent’s belief state.

A liar wants:

[ B_{other} P ]

while knowing:

[ \neg P ]

Social intelligence is partly doxastic engineering.

Belief and Common Belief

Groups can share beliefs.

But group belief is complex.

Everyone believing P does not automatically imply:

everyone believes that everyone believes P.

Higher-order structure matters just as in knowledge.

Rumors

Rumors propagate beliefs without guaranteed truth.

A belief network can amplify false claims.

Doxastic logic separates:

• transmission,
• truth.

This is essential for understanding misinformation.

Belief and Action

Beliefs interact with desires.

If an agent believes:

the restaurant is closed,

they may not go.

Action depends not only on what is true but on what the agent thinks is true.

Decision theory therefore needs belief states.

Bayesian Agents

A Bayesian agent represents uncertainty probabilistically.

New evidence updates prior beliefs using Bayes’ theorem.

This is one formal ideal of rational belief change.

But real agents have limited data and computation.

Logical Omniscience Returns

Standard doxastic systems can imply that agents believe all logical consequences of their beliefs.

Humans clearly do not.

We may believe premises without noticing what follows.

Resource-bounded belief models try to address this.

Belief and Self-Deception

Can a person believe P while also, at some level, knowing not-P?

Self-deception challenges clean logical models.

Human belief is layered.

Different cognitive systems may encode incompatible commitments.

The Philosophical Lesson

Belief is weaker than knowledge because belief can be false.

Doxastic logic gives us tools for reasoning about:

• mistaken agents,
• nested beliefs,
• revision,
• uncertainty.

It formalizes not the world itself, but how agents take the world to be.

The Next Question

Agents do not only believe.

They also act under rules.

Some actions are:

• required,
• permitted,
• forbidden.

How can logic represent obligation?

That leads to:

deontic logic.