PHILUN3601 Metaphysics

Quine - Two Dogmas of Empiricism
Metaphysics Midterm Preparation
Metaphysics is the study of the relationship between what appears on the surface and the underlying mechanisms, theories, etc.
- Without clarifying this relationship, the natural sciences, which conduct experiments and construct theories, cannot be established.
Logical positivists believe that if two theories have the same empirical consequences, they are the same, even if they appear different.
Quine, on the other hand, argues that theories do not have empirical consequences in the first place.
Everything is called into question when we observe anything.
- This includes everything, even logic, which is based on empirical evidence.

Symbolic logic:

Aristotle
- Modus ponens is not included in the 14 valid syllogisms, but it is used in meta-logical proofs of assertions.
- What is the difference from boolean logic?
  - Is it simply a matter of incompleteness or errors?
Boolean logic
- Assigns 1 or 0 to each atomic sentence.
- Well-formed formula refers to all correct logical expressions.

Logic that deals with predicates P(x) that have some variables.
- Quantifiers forall and exists can be used with x.
In second-order logic and beyond, quantifiers like forall/exists can also be used with predicates.

Frege:

Sentences, for Frege, are built of names (a, b, c…), variables (x, y, z…), function symbols (f, g, h…), predicates (P, Q, R…), and quantifiers (∃x and ∀x).
- Quantifiers specify variables (for example, for all integers).

This leads to the topic of semantics.

Alfred Tarski later developed the semantics of Frege’s symbol system. Tarski provided a method for assigning “truth values” to propositions and structures in a language. The interpretation (or model) of first-order logic involves specifying a particular “domain,” assigning names and variables to objects within that domain, and assigning predicates to sets of objects (or sets of k-tuples of objects) within that domain.

How to give meaning to Frege’s logical expressions
- For example, if the domain is natural numbers, forall x means “for all natural numbers.”
- The domain can be anything, such as apples or humans. (blu3mo)
The truth value depends on the domain.
- Conversely, values that are not affected by the domain are called logical truths.

From there, the concepts of intension and extension emerge.

Comprehension and extension, I understand these.
- The concept of defining sets.
- https://www.nihongo-appliedlinguistics.net/wp/archives/8451
  - I see.
Quine’s claim:
- Quine held that ordinary language is an unreliable guide to metaphysics. Science, broadly construed, should be extensional and first-order.
  - It means that intensional things/second-order or higher are unreliable, so let’s use minimal logic, right?
- Quine advocates for an “extensional” approach to science and philosophy, he is emphasizing a focus on the actual objects and sets involved, rather than the nuanced and potentially ambiguous ways those objects might be described or characterized.
  - I see.
Psychology from an Empirical Standpoint is the assignment.
The concept of modal logic
- intensional extension of first-order logic called modal logic.
- An extension of first-order logic called modal logic, which is an intensional logic, right?
https://en.wikipedia.org/wiki/Extensional_context
- In extensional context/logic:
  - If the truth values or specific results remain the same even if the terms change, it can be said that they are the same sentence.
- In intensional context:
  - Since the concept of “specific results” does not exist, changing the term makes it a different sentence.
  - When there is a concept like belief, it becomes intensional.
    - Ah, I understand now. (blu3mo)(blu3mo)
- The use of the terms intentional context and extensional context in this sentence.
Generally, is mathematics dealt with in an extensional manner?
- Meaning doesn’t matter.
Prominent locutions are not extensional. ‘Hyperintensional’ distinctions, like grounding or constitution, are examples. But so are counterfactual, causal, and modal distinctions.
- Most locutions are intensional.
Modal logic is also considered intensional.
- Modal logic
- https://www.is.nagoya-u.ac.jp/dep-ss/phil/kukita/seminars/Introduction_to_Modal_Logic.pdf
  - For example, “If this coin lands heads, then Tokyo is the capital of Japan” would be considered a true conditional statement in classical logic.
  - It is indeed a good example of discomfort.
- forall x (□ F(x)) and □ forall x (F(x)) are different.
  - The former:
    - This necessity is applied to each individual separately, meaning that in some possible worlds, some individuals might not have the property F(x) while others do.- I see. The latter is not much different from saying “forall x (F(x))” in classical logic.
Quine notes that modal contexts like []Fx (‘x is necessarily F’) and <>Fx & <>~Fx (‘x is contingently F’), are intensional.
- Intensional means that changing the term changes the sentence.
- That’s true. When considering multiple possible worlds, we can’t say that the truth value remains the same if the specific elements are the same even if the terms are changed.
  - “Necessarily 8>7” and “Necessarily the number of celestial bodies > 7” are extensional if we think about it.
- This is what Quine criticizes as intensional in modal logic.
- I’m starting to understand, but I still need to understand more.
  - I want to think about other specific examples.
  - Leibniz’s law is about confirming extensionality.
  - Modal logic is explained in more detail by [sno2wman], but there seem to be few specific examples.
From there, it leads to a discussion of intentionality.
- Intensionality is a semantic property, which refers to the failure of substitutivity of co-referring terms.
- Intentionality is a property of mental states, referring to being about things.
- Believing, desiring, assuming, imagining, and fearing are examples of intentional states.
- The connection between the two is that attributions of intentional states are typically intensional.
- It’s like creating a copy of real things in our minds and referring to them.
- Quine has a different attitude from Brentano. Quine sees the Brentano thesis as either showing the indispensability and importance of an autonomous science of intention or showing the baselessness and emptiness of a science of intention, and Quine’s attitude is the latter.

20230919

This question should impress you. Just by having thoughts, we can, it seems, relate ourselves to: the planet Pluto, the size of the continuum, all events outside our lightcones, and whatever was on Lincoln’s mind when he died.

TARSKI’S THEORY OF TRUTH

https://plato.stanford.edu/entries/tarski-truth/
- Tarski described several conditions that a satisfactory definition of truth should meet.
- The definition was to be in terms of syntax, set theory, and the notions expressible in L, but not semantic notions like ‘denote’ or ‘mean’.
Field criticized the mistaken interpretation of Tarski’s theory as “truth is reducible to a syntactic notion and not a semantic notion”.
Field discusses what is involved in breaking down truth, whether it is about meaning or form.
Tarski’s work changed the perception that semantic concepts such as truth and denotation cannot be dealt with in science.
Tarski’s work should make the term ‘true’ acceptable even to someone who is initially suspicious of semantic terms.
However, Field argues that Tarski did not actually exclude semantic elements.
Tarski merely reduced truth to other semantic notions.
“A is F” in a language is true if and only if A is F in the metalanguage.
However, Tarski’s theory of truth cannot explain reference.
In the statement “the apple is red,” there is reference to sets of apple and red (extensional context).
Physical facts fix everything, including chemistry and biology, that are built on top of them.
Q: reference
Bonus: metaphysics vs epistemology
- Epistemology asks questions that metaphysics does not.
- Assumptions like “we are not in the matrix or VR” or “the world exists” are made in metaphysics.
Kripke
- Physicalism, causality
What is reduction?
- It’s difficult, there is no conclusion for now.
Skipped about 4 classes on Quantum, need to review.

Handout 7 Individuals

Preview
- Classical logic does not fully represent reality.
- Intuitionistic Logic 直観主義論理
  - The law of excluded middle does not hold.
  - It doesn’t mean that (P and not P) is absolutely true, but it means that both t and f are possible.
  - It was easier to understand when interpreted with the double negation elimination.
    - First, let’s consider $\neg P$ as an abbreviation for $P \to ⊥$ .
      - If $P$ is true, it leads to a contradiction, so $P$ cannot hold.
    - Then $\neg\neg P$ can be interpreted as “Even if $P$ were true, it would not lead to a contradiction.”
    - Intuitionistic logic sees a leap between this and “P is true”.
      - Isn’t it too much to claim that we can construct a proof of $P$ just because even if $P$ were true, it would not lead to a contradiction?
        
        It’s not like we actually constructed $P$ .
      - It is easier to understand if we replace $P$ with $\exists x . Q (x)$ .
The formula $\neg\neg\exists x . Q (x)$ simply states that “even if there exists an $x$ that satisfies $Q$ , it does not lead to a contradiction.”
On the other hand, $\exists x . Q (x)$ means “we can specifically demonstrate an $x$ that satisfies $Q$ .”
- Without double negation elimination, the only way to prove $\exists x . Q (x)$ is to find a specific $x$ that satisfies $Q”.
However, just because we know that something does not lead to a contradiction does not mean we can specifically demonstrate it.
Classical logic does not distinguish between the fact that something does not lead to a contradiction and the fact that it is true.
Reference: /sno2wman/Brouwer-Heyting-Kolmogorov解釈
- One interpretation of intuitionistic logic.
- The explanation here is concise and easy to understand.
https://mathlog.info/articles/2890
However, in the semantics based on truth values used in the previous example, the law of excluded middle and double negation elimination become valid. Therefore, it contradicts the purpose of criticizing classical logic. Therefore, it is necessary to criticize the semantics based on truth values (more generally, the semantics based on predicate logic). This raises the level of the discussion by one layer. It means that the resolution of the debate on inference rules is determined by the debate on semantics.
Quantum Logic 量子論理
- The distributive law does not hold.
Both logics have fewer constraints than classical logic, so they are more general.
Is it possible to consider a logic that does not assume “individuals”?
- Individuals: An Essay in Revisionary Metaphysics
Validity vs actually true
- Validity means being logically true.
- Even if it lacks validity, it is possible to be true.
If we think based on a language that assumes “things,” our thinking is constrained by it.
- Quine and Dasgupta argue that it is possible to consider a language without subjects.
- They argue that a thing-less language is more correct for doing science.
  - Such as physics.
Language G

Shutaro Aoyama

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