Saturday, March 1, 2025

Materna Semantics

In the context, I gave ChatGPT the text of Is Strong Correspondence (Theory of Truth) Possible?, by Pavel Materna, which is omitted. I am sharing this conversation because ChatGPT kept offering to do more different examples to explain it to me, which I found interesting.

formally how would Materna use possible worlds semantics to describe truth

Materna's use of Possible Worlds Semantics (PWS) to describe truth relies on the idea that truth is not an inherent property of sentences alone, but a function of how sentences relate to facts across possible worlds. His formal approach builds upon Transparent Intensional Logic (TIL) as developed by Pavel Tichý. Here’s how he might formally describe truth using PWS:

1. Basic Framework: Types and Functions

In PWS, entities are categorized into different types:

  • \( o \): Truth-values (\( T, F \))
  • \( \iota \): Individuals (entities in the world)
  • \( \tau \): Real numbers or time points
  • \( \omega \): Possible worlds
  • \( (\alpha \beta_1 ... \beta_m) \): Functions, where \(\alpha\) is the range type and \(\beta_1 ... \beta_m\) are argument types.

A proposition is defined as a function from possible worlds and time points to truth-values: \[ p: \omega \times \tau \to \{T, F\} \] This means that a proposition is true in a given world \( w \) and time \( t \) iff: \[ p(w,t) = T \]

2. Truth in Possible Worlds Semantics

Materna follows a representational semantics approach, where:

  • A sentence \( S \) denotes a propositional function \( p_S \).
  • This function assigns truth-values across possible worlds and times.

Thus, the truth of \( S \) in the actual world \( w_0 \) at time \( t_0 \) is defined as: \[ S \text{ is true in } w_0, t_0 \text{ iff } p_S(w_0, t_0) = T. \]

This means truth is not an absolute property of a sentence but depends on which world and time we evaluate it in.

3. Truth and Correspondence

For a sentence to be true, it must correspond to a fact, which can be defined in two ways:

A-Construal (Fact as a True Proposition)

A fact is simply a true proposition in the actual world: \[ \text{Fact} = \{p \mid p(w_0, t_0) = T\} \]

  • This means equivalent sentences (e.g., "Mt. Everest is taller than Mt. Blanc" vs. "Mt. Blanc is shorter than Mt. Everest") correspond to the same fact.
  • This supports weak correspondence: different expressions can point to the same truth conditions.

B-Construal (Fact as a Construction)

A fact is a structured construction leading to a true proposition: \[ \text{Fact} = \lambda w \lambda t [^0 A_{wt} [^0 H_{wt} \ ^0 M_{wt}]] \]

  • This means different constructions correspond to different facts, even if they are logically equivalent.
  • This supports strong correspondence, where truth-bearers must match structured facts.

4. Defining Truth Formally

A sentence \( S \) is true iff its meaning (a construction) constructs a fact in the actual world: \[ S \text{ is true} \iff M(S) \text{ constructs a fact in } w_0, t_0. \] where:

  • \( M(S) \) is the meaning (construction) of \( S \).
  • A fact is a structured entity (B-construal) or a true proposition (A-construal).

5. The Role of Experience in Truth

Since we do not know which possible world is actual (we are not omniscient), we require experience to determine which sentences are true: \[ \text{Knowing whether } S \text{ is true} \Rightarrow \text{ empirical verification}. \] This aligns with Davidson’s intuition that truth depends on reality, not just on our definitions.

Conclusion

Materna’s Possible Worlds Semantics approach to truth formally expresses the idea that:

  1. Truth is a function from possible worlds to truth-values.
  2. Empirical truth requires correspondence to facts, either as true propositions (weak correspondence) or structured constructions (strong correspondence).
  3. Experience is necessary to determine truth in the actual world.
  4. Deflationary theories fail, because they ignore the necessary connection between truth, meaning, and facts in different worlds.

Would you like a worked-out example using PWS?

(Branching off from here, see the notation explanation.)

yeah sure

Let's work through an example using Possible Worlds Semantics (PWS) in the style of Materna’s approach.

Example: "Mount Everest is higher than Mount Blanc."

We analyze the truth of this sentence using Possible Worlds Semantics and Transparent Intensional Logic (TIL).

1. Define the Types and Functions

We need to formalize the entities involved:

  • Truth-values: \( o = \{T, F\} \)
  • Possible worlds: \( \omega \) (set of all possible worlds)
  • Time points: \( \tau \) (set of all times)
  • Individuals: \( \iota \) (Mount Everest, Mount Blanc, etc.)
  • Height function: \( H: \iota \times \omega \times \tau \to \mathbb{R} \)
    • This means \( H(x, w, t) \) returns the height of \( x \) in world \( w \) at time \( t \).

Now, the proposition expressed by the sentence: \[ p_{S} = \lambda w \lambda t \left[ H(Everest, w, t) > H(Blanc, w, t) \right] \]

  • This function takes a world \( w \) and a time \( t \) and returns true (T) or false (F) depending on whether Everest is taller than Blanc in that world and time.

2. Truth in the Actual World

The sentence "Mount Everest is higher than Mount Blanc" is true iff: \[ p_S(w_0, t_0) = T \] where \( w_0 \) is the actual world, and \( t_0 \) is the current time.

This means that in the actual world at this moment, Everest must be taller than Blanc.

3. Alternative Sentences and Correspondence

Consider a logically equivalent sentence:

  • "Mount Blanc is lower than Mount Everest."
  • It expresses the proposition: \[ p_{S'} = \lambda w \lambda t \left[ H(Blanc, w, t) < H(Everest, w, t) \right] \] Since both functions return the same truth-values in all worlds, they are L-equivalent.
A-Construal (Fact as a True Proposition)

Under the A-construal, facts are just true propositions: \[ \text{Fact} = \{ p \mid p(w_0, t_0) = T \} \]

  • Both sentences correspond to the same fact.
  • This supports weak correspondence, meaning different expressions can refer to the same fact.
B-Construal (Fact as a Structured Construction)

Under the B-construal, facts are structured constructions:

  • The sentence "Mount Everest is higher than Mount Blanc" has the construction: \[ \lambda w \lambda t [^0 > (^0 H_{wt} ^0 Everest) (^0 H_{wt} ^0 Blanc)] \]
  • The sentence "Mount Blanc is lower than Mount Everest" has a different structure: \[ \lambda w \lambda t [^0 < (^0 H_{wt} ^0 Blanc) (^0 H_{wt} ^0 Everest)] \]
  • Since these constructions are not identical, they correspond to different facts (even though they are equivalent).

This supports strong correspondence, where every sentence has a structurally distinct fact.

4. The Role of Experience

Since we cannot determine a priori which world is actual, we need empirical verification.

  • We measure the heights of Everest and Blanc.
  • If we confirm \( H(Everest, w_0, t_0) > H(Blanc, w_0, t_0) \), then: \[ p_S(w_0, t_0) = T \]
  • Thus, the sentence is true in the actual world.

Final Takeaways

  1. Possible Worlds Formalism: Truth is a function from worlds to truth-values.
  2. Weak vs. Strong Correspondence:
    • Weak correspondence (A-construal): Equivalent sentences correspond to the same fact.
    • Strong correspondence (B-construal): Equivalent sentences correspond to distinct facts.
  3. Empirical Verification is Required: Since we cannot know which world is actual, experience is needed to determine truth.

Would you like me to extend this with a modal example (e.g., "Everest could have been shorter") to show how PWS handles possibility?

sure!

Modal Example: "Mount Everest could have been shorter than Mount Blanc."

Now, let’s analyze a modal statement using Possible Worlds Semantics (PWS) in Materna’s framework.

1. Formalizing the Sentence

We need to express "Mount Everest could have been shorter than Mount Blanc", which introduces modal operators.

Entities and Functions

  • Possible worlds: \( \omega \) (the set of all possible worlds)
  • Time points: \( \tau \) (the set of all time points)
  • Individuals: \( \iota \) (Mount Everest, Mount Blanc, etc.)
  • Height function: \( H: \iota \times \omega \times \tau \to \mathbb{R} \)
    • \( H(x, w, t) \) returns the height of \( x \) in world \( w \) at time \( t \).

The proposition we analyze is: \[ p_S = \lambda w \lambda t [H(Everest, w, t) < H(Blanc, w, t)] \]

  • This function returns T (true) or F (false) based on whether Everest is shorter than Blanc in a given world \( w \).

2. Introducing Modality

The modal claim uses the possibility operator \( \Diamond \) (◊): \[ \Diamond p_S \] which means: \[ \exists w' \in W \quad p_S(w', t_0) = T \]

  • There exists at least one possible world \( w' \) where Everest is shorter than Blanc at time \( t_0 \).
  • This does not require Everest to be shorter in the actual world \( w_0 \), only in some alternative world.

3. Truth in the Actual World

The sentence "Mount Everest could have been shorter than Mount Blanc" is true in the actual world \( w_0 \) if and only if: \[ \exists w' \in W \quad p_S(w', t_0) = T. \] This means:

  • There is at least one possible world \( w' \) where Everest is shorter.
  • If such a world exists, the modal sentence is true in \( w_0 \), regardless of Everest’s actual height.

4. Possible Worlds Interpretation

Case 1: True Modal Statement

  • Suppose there is a physically possible world \( w' \) where geological changes resulted in Everest being shorter than Blanc.
  • Then: \[ \Diamond p_S(w_0, t_0) = T. \]
    • The modal claim is true in the actual world, even though Everest is not shorter in the actual world.

Case 2: False Modal Statement

  • If no physically possible world allows Everest to be shorter (e.g., if it is geologically impossible), then: \[ \forall w' \in W, \quad p_S(w', t_0) = F. \]
    • Then, the modal sentence is false in \( w_0 \).

5. Correspondence and Possible Worlds

Now, how does this relate to correspondence theory?

A-Construal (Fact as a True Proposition)

  • The sentence corresponds to a fact only if there exists a world where the proposition is true.
  • If such a world exists, the fact is the set of all worlds where Everest is shorter: \[ \text{Fact} = \{ w' \mid p_S(w', t_0) = T \} \]
  • Weak correspondence applies because the same fact could be described in different ways.

B-Construal (Fact as a Structured Construction)

  • The fact itself is a structured construction describing a possible world where Everest is shorter: \[ \lambda w' \lambda t [H(Everest, w', t) < H(Blanc, w', t)] \]
  • Strong correspondence requires a structured fact, which makes modal statements more complex since facts now depend on counterfactual worlds.

6. Deflationist Objection and PWS Defense

A deflationist might argue:

  • The sentence "Everest could have been shorter" is just a linguistic shortcut with no deep metaphysical implications.

Materna’s Possible Worlds Semantics rejects this, because:

  • Modal truth is determined by counterfactual facts, not just by linguistic conventions.
  • If there is no world where Everest is shorter, then the statement is false, not just meaningless.
  • Thus, modal truth must be grounded in facts across worlds.

Conclusion

  1. Modal truth requires Possible Worlds Semantics:
    • A sentence is true iff it holds in at least one accessible world.
  2. Weak correspondence (A-construal):
    • The sentence corresponds to the set of possible worlds where it is true.
  3. Strong correspondence (B-construal):
    • The sentence corresponds to a structured fact describing a possible world.
  4. Empirical verification is limited:
    • Unlike empirical sentences, modal sentences cannot be directly verified but require an analysis of possible worlds.

Would you like a more complex example, such as a counterfactual modal statement (e.g., "If Everest had eroded, it would be shorter")?

yes please

Counterfactual Example: "If Everest had eroded, it would be shorter than Mount Blanc."

Now, we analyze a counterfactual modal statement using Possible Worlds Semantics (PWS) and Materna’s approach.

1. Understanding the Statement

The sentence "If Everest had eroded, it would be shorter than Mount Blanc" expresses a counterfactual conditional: \[ \text{If } P, \text{ then } Q. \] where:

  • Antecedent \( P \): "Everest had eroded."
  • Consequent \( Q \): "Everest would be shorter than Mount Blanc."

This suggests that in some hypothetical world where Everest eroded, it must also be shorter than Mount Blanc.

2. Formalization Using Possible Worlds

We use David Lewis’s counterfactual logic and PWS to represent this formally.

  1. Define the relevant functions:

    • Truth-values: \( o = \{T, F\} \)
    • Possible worlds: \( \omega \) (the set of all possible worlds)
    • Time points: \( \tau \) (set of all time points)
    • Individuals: \( \iota \) (Mount Everest, Mount Blanc, etc.)
    • Height function: \( H: \iota \times \omega \times \tau \to \mathbb{R} \), where \( H(x, w, t) \) returns the height of \( x \) in world \( w \) at time \( t \).

  2. Propositions for the antecedent and consequent:

    • Let \( P \) be: "Everest has eroded significantly": \[ p_P = \lambda w \lambda t [Eroded(Everest, w, t)] \]
    • Let \( Q \) be: "Everest is shorter than Mount Blanc": \[ p_Q = \lambda w \lambda t [H(Everest, w, t) < H(Blanc, w, t)] \]

  3. Counterfactual Conditional Representation:

    • The statement "If Everest had eroded, it would be shorter than Mount Blanc" is true in the actual world \( w_0 \) if and only if: \[ w_0 \models (P \Rightarrow Q) \text{ iff } Q \text{ is true in the closest world } w' \text{ where } P \text{ is true}. \]
    • This means we evaluate the truth of \( Q \) in the nearest possible world \( w' \) where \( P \) holds.

3. Evaluating Counterfactual Truth

According to Lewis-Stalnaker counterfactual analysis, the truth of the statement depends on:

  • Whether there exists a "nearest" possible world \( w' \) where Everest has eroded.
  • Whether in that world, Everest is indeed shorter than Mount Blanc.

Case 1: True Counterfactual

  • Suppose in the closest possible world \( w' \) where Everest eroded, its height is reduced.
  • If Everest’s height in \( w' \) satisfies: \[ H(Everest, w', t_0) < H(Blanc, w', t_0) \] then \( Q \) is true in \( w' \), and thus: \[ w_0 \models (P \Rightarrow Q). \]
    • The counterfactual statement is true in the actual world.

Case 2: False Counterfactual

  • Suppose in the closest possible world \( w' \), Everest has eroded, but due to other geological factors, it is still taller than Blanc.
  • That is, in \( w' \): \[ H(Everest, w', t_0) > H(Blanc, w', t_0). \]
    • Then \( Q \) is false in \( w' \), meaning: \[ w_0 \not\models (P \Rightarrow Q). \]
    • The counterfactual statement is false in the actual world.

4. Truth and Correspondence in Counterfactuals

A-Construal (Fact as a True Proposition)

  • Under A-construal, facts are just true propositions.
  • The fact that supports the counterfactual is: \[ \text{Fact} = \{ w' \mid p_Q(w', t_0) = T \text{ and } w' \text{ is closest to } w_0 \text{ where } p_P(w', t_0) = T \}. \]
  • The counterfactual is true in \( w_0 \) if such a world \( w' \) exists.
  • Weak correspondence holds, as different sentences might express the same proposition.

B-Construal (Fact as a Structured Construction)

  • Under B-construal, facts are structured constructions that explicitly describe the logical steps leading to the truth of the counterfactual.
  • The fact corresponding to the counterfactual is the construction: \[ \lambda w' \lambda t [ \text{If } ^0 Eroded(Everest, w', t), \text{ then } (^0 H(Everest, w', t) < ^0 H(Blanc, w', t))]. \]
  • Since counterfactuals involve reasoning about alternate worlds, the structured construction explicitly encodes this.
  • Strong correspondence holds because the fact itself includes its logical dependencies.

5. Why PWS is Essential for Counterfactuals

  1. Counterfactual truth requires evaluating multiple worlds:
    • Unlike simple empirical truth, counterfactuals require checking which worlds are "closest" to the actual world.
  2. Truth depends on possible world accessibility:
    • If no world where \( P \) is true also satisfies \( Q \), then the counterfactual is false.
  3. Different sentences can express the same fact (weak correspondence):
    • "If Everest had eroded, it would be shorter" and "Had Everest suffered major erosion, it would be lower than Blanc" may describe the same truth conditions.
  4. Structured constructions provide strong correspondence:
    • Explicit constructions show the dependency between erosion and height reduction.

Conclusion

  1. Counterfactual truth is evaluated in "nearest" possible worlds:
    • The statement "If Everest had eroded, it would be shorter" is true if in the closest world where Everest eroded, it is indeed shorter.
  2. Weak correspondence (A-construal):
    • The statement corresponds to the set of all closest worlds where Everest eroded and became shorter.
  3. Strong correspondence (B-construal):
    • The statement corresponds to a structured logical construction linking erosion and height reduction.
  4. Deflationism fails:
    • Counterfactual truth cannot be a mere linguistic shortcut—it requires a formal method of evaluation.
  5. PWS is necessary for counterfactuals:
    • Unlike standard logic, PWS allows us to evaluate truth across different possible worlds.

Would you like a more complex nested modal example, such as:
"If Everest had eroded, it could have still been taller than Blanc"?

yes, go ahead

Nested Modal Example: "If Everest had eroded, it could have still been taller than Mount Blanc."

Now, we analyze a nested counterfactual modal statement using Possible Worlds Semantics (PWS) and Materna’s framework.

1. Understanding the Sentence

The sentence "If Everest had eroded, it could have still been taller than Mount Blanc" contains both:

  • A counterfactual conditional ("If Everest had eroded, ...")
  • A possibility modal (\(\Diamond\)) ("... it could have still been taller than Blanc.")

This requires analyzing two levels of modal evaluation:

  1. First modal level (Counterfactual condition):
    • We consider a world where Everest has eroded.
  2. Second modal level (Possibility claim):
    • Within that counterfactual world, we check whether there exists another possible world where Everest remains taller.

2. Formalization Using Possible Worlds Semantics

Defining the Functions

We define the relevant functions:

  • Truth-values: \( o = \{T, F\} \)
  • Possible worlds: \( \omega \) (set of all possible worlds)
  • Time points: \( \tau \) (set of all time points)
  • Individuals: \( \iota \) (Mount Everest, Mount Blanc, etc.)
  • Height function:
    \[ H: \iota \times \omega \times \tau \to \mathbb{R} \] where \( H(x, w, t) \) returns the height of \( x \) in world \( w \) at time \( t \).

We express the propositions:

  • \( P \): "Everest has eroded significantly." \[ p_P = \lambda w \lambda t [Eroded(Everest, w, t)] \]
  • \( Q \): "Everest is still taller than Mount Blanc." \[ p_Q = \lambda w \lambda t [H(Everest, w, t) > H(Blanc, w, t)] \]

Now, we incorporate the modal operators:

  1. Counterfactual conditional (\(\Rightarrow\)):
    • "If Everest had eroded (\( P \)), then..."
  2. Possibility modal (\(\Diamond\)):
    • "... it could still be taller (\(\Diamond Q\))."

Thus, the formal representation of the sentence is: \[ w_0 \models (P \Rightarrow \Diamond Q) \] which means: \[ w_0 \models \text{"In the closest world \( w' \) where Everest has eroded, there exists a world \( w'' \) where Everest is still taller."} \]

3. Evaluating Truth in Possible Worlds

To evaluate whether \( w_0 \models (P \Rightarrow \Diamond Q) \), we apply Lewis-Stalnaker counterfactual logic:

  1. Find the closest world \( w' \) where Everest has eroded.

    • This is the world where \( p_P(w', t_0) = T \).

  2. Within \( w' \), check if there exists a world \( w'' \) where Everest is still taller.

    • We must find \( w'' \) such that: \[ p_Q(w'', t_0) = T \]
    • This means Everest is still taller than Blanc in some world accessible from \( w' \).

4. Cases for Evaluating Truth

Case 1: The Sentence is True

The statement is true in \( w_0 \) if:

  • In the closest eroded world \( w' \), Everest is still close to its original height.
  • There exists another nearby world \( w'' \) where slight variations in erosion keep Everest taller than Blanc.

Mathematically: \[ \exists w'' \in W, w'' \sim w' \quad H(Everest, w'', t_0) > H(Blanc, w'', t_0). \]

  • If such a \( w'' \) exists, the nested modal sentence is true in \( w_0 \).

Case 2: The Sentence is False

The sentence is false if:

  • In all eroded worlds \( w' \), Everest’s height drops below Blanc.
  • No world \( w'' \) accessible from \( w' \) allows Everest to remain taller.

Mathematically: \[ \forall w'' \in W, w'' \sim w' \quad H(Everest, w'', t_0) < H(Blanc, w'', t_0). \]

  • If this holds, the nested modal sentence is false in \( w_0 \).

5. Correspondence and Possible Worlds

A-Construal (Fact as a True Proposition)

  • The fact corresponding to the statement is the set of all worlds where \( P \) holds and at least one world \( w'' \) makes \( Q \) true: \[ \text{Fact} = \{ w' \mid p_P(w', t_0) = T, \quad \exists w'' \sim w' \quad p_Q(w'', t_0) = T \}. \]
  • This supports weak correspondence, where different sentences might map to the same fact.

B-Construal (Fact as a Structured Construction)

  • The fact is explicitly structured: \[ \lambda w' \lambda t \left[ \text{If } ^0 Eroded(Everest, w', t), \text{ then } (\Diamond (^0 H(Everest, w', t) > ^0 H(Blanc, w', t))) \right]. \]
  • Since this includes logical steps, it supports strong correspondence.

6. Why PWS is Essential for Nested Modalities

  1. Counterfactual truth must be evaluated across two levels of possible worlds.

    • The first step requires finding the closest world where Everest eroded.
    • The second step requires checking whether another world allows Everest to stay taller.

  2. Model-theoretic semantics (Tarski) fails to capture nested modal relationships.

    • Model-theoretic approaches only define truth within a single world.
    • PWS allows us to track truth conditions across multiple interconnected worlds.

  3. Weak vs. Strong Correspondence is resolved in modal contexts.

    • Weak correspondence groups sentences into truth-equivalent sets of possible worlds.
    • Strong correspondence requires structured mappings between sentences and logical constructions.

Final Takeaways

  1. Nested counterfactuals require multi-layered world evaluations.

    • First, we assume the antecedent (Everest eroded).
    • Then, we check if Everest’s height remains taller in some nearby world.

  2. Weak correspondence (A-construal) holds if equivalent statements describe the same set of possible worlds.

  3. Strong correspondence (B-construal) requires explicit construction modeling of how modal and counterfactual dependencies interact.

  4. Deflationism fails to account for nested modality.

    • Simply "disquoting" truth fails when multiple modal layers affect evaluation.

  5. PWS is essential for formalizing complex modal statements.

    • Without PWS, there is no rigorous way to evaluate "counterfactual possibility" as used in everyday language.

Would you like to see a diagram illustrating this multi-world structure?

sure, you can try to render one with code

Here is a diagram illustrating the nested counterfactual structure:

  • \( w_0 \): The actual world, where we evaluate the truth of the statement.
  • \( w' \): The closest world where Everest has eroded.
  • \( w''_1 \): A possible world accessible from \( w' \), where Everest is still taller than Mount Blanc.
  • \( w''_2 \): Another possible world accessible from \( w' \), where Everest is not taller than Mount Blanc.

How to Interpret the Diagram:

  • The first modal step moves us from \( w_0 \) → \( w' \) (the erosion counterfactual).
  • The second modal step checks if \( w' \) has a world \( w'' \) where Everest is still taller.
  • If \( w''_1 \) exists, the statement is true in \( w_0 \).
  • If only \( w''_2 \) exists, the statement is false in \( w_0 \).

Would you like further refinements or additional annotations on this visualization? 😊

I did not continue from here. This last output took two tries (the diagram was broken on the first try); no other outputs were tried more than once.

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