Philosophy

• Mathematics, logic and truth are defined circularly

  • $\to$ truth’ effectively cannot be captured, the notion of ‘truth’ becomes relative and based upon a ‘shaky bottom.’ Thus, there is no such thing as an objective truth, but rather just interpretations.

  • An objective truth can be considered as the totality of all interpretations, which is something that cannot be obtained within this reality, any such ‘totality’ must exist outside of our reality. Thus, any view contained within our reality is subjective (we may have more ‘informed’ views which are based off of a larger database of information, although ultimately the information it is based off of is a relative notion). $\to$ We could interpret this through ‘horizontal’ structure (rather than a ‘vertical’ structure like a hierarchy) like possible worlds which each contain a fragment of objective truth (relative truth, other ‘dimensions’ for frameworks?)

  • There is a sense in truth can be (somewhat) unrelativized, specifically unrelativized truth about relativized objects. i.e., consider the claim ”$\varnothing$ is a set” w/in the context of $ZF$. This is an categorical/unrelativized truth about the object ${\varnothing }_{ZF}$ ($\varnothing$ in the context of $ZF$). Though such an unrelativized truth would be viewed as unrelativized by some (meta)-system which cannot capture its own truth (and thus, is also relative).

• Nested/meta Paradoxes (we can index this by meta-levels, paradox of level 2 ranges over the paradox of level 1)

  • $\to$ i.e., claiming the statement “truth is relative” as if it is an objective truth is paradoxical. if this statement is an absolute truth, then not all truths are relative, contradicting the initial statement. If the statement is relative, then this also results in contradictions (depending on the scenario, see On Truth). We can also consider statements such “it is absolutely true that all truths are relative” which will also result in contradiction. Possible Solution: “all truths are relative, except for statements about truth”
  • Consider a similar statement that “the notion of truth is paradoxical”, treating this as an objective statement results in a paradox, so we get a paradox of a paradox (this is if we interpret “truth is relative” as “truth is paradoxical”, see the above argument which results in a paradox if consider “truth is relative” as true).
  • Alternatively consider the statement “this statement is paradoxical” (adaptation of liar’s paradox). All of these paradoxes rely on self-reference
  • Paradoxicality of truth is evidence for the relativity of it.
  • Partial truths may be reflected internally, i.e., Tarski’s undefinability theorem, how we know that truth (internal to this reality) must be incomplete? $\to$ It is shown in the metatheory (or meta-layer) that base layer cannot define its own truth predicate

• The Question “What is the meaning of life” is undefined/circular/paradoxical

• There is no absolute meaning or definition

• Is ‘chaos’ just a really complex system (i.e., alternative logic systems, extra logical rules), or perhaps even an infinitely complex system? E.g., something like trivialism (anything can be derived from a valid logical procedure within the logic, this is in contrast to something like the liar’s paradox for a formal system which disobeys the laws of the logic, so a triviality of results isn’t allowed, but in this view of trivialism it doesn’t go against the laws of the logic as it follows a valid inference → we can emulate an absurd result)

  $\to$ This could be built up (like in a hierarchy, modifying rules at each level) where there are less and less statements which are logically invalid (violating the internal rules of the logic system, i.e., violating the law of non-contradiction in classical logic) and more that are logically valid, eventually reaching a fixed point where there are no more statements which could be logically invalid and so all are logically valid? (This is similar to Kripke’s hierarchy where truth gets more defined as you go up.)

• Logical Pluralism: The view that multiple logics which are equally valid, logic is a theory of validity, that is, a study of valid inference from some given rules (internal to the logic). According to this philosophical view, other logics could hold in certain instances/contexts or other ‘possible’ worlds (i.e., see extended modal realism below). This results in the notion of ‘truth’ to be relative to the logic you are working in (i.e., logical relativism).

  • Logical Monism: The view that there is only a single logic, which holds in all circumstances. Philosophers of this view would argue that classical logic is inherent to his world (i.e., there are no contradictory observation in reality) and other logics cannot exist in reality
  • Logical Nihilism: The view that there is no ‘logic’ in any form and there are no laws of logics as all fail universally. This appears to delve into trivialism (on the other hand, this could be seen as an extreme form of logical pluralism or nihilism where there are trivialist logic, as trivialism can be formulated as a logic, see Trivialism)

The Tarskian hierarchy framework supports the notion of logical pluralism via requiring different logics (metalanguages) to justify the semantics of a logic as well as being a flexible framework where we can choose different object languages and metalanguages to formalize. Hence, resulting in a variety of logics across different meta-levels. However, it does not say anything about multiple logics on a given meta-level (which logical pluralism may be re-interpreted as). This does not disregard logical monism as in a similar way this could still work via requiring at most one logic at different meta-levels or that there is unique correct ‘logic’ (this involves invoking a Platonic metaphysical notion of truth) and that other logics are just useful frameworks that ultimately do not fully capture ‘truth’. What can be said is that the Tarskian framework requires talking about different logics. $\to$ Other ideas: Multiplicity of logics may be reduced to a single one upon logical analysis

• Modal realism is the view that possible worlds as formalized in modal logic, actually exist and are equally ‘real’ as the ‘actual’ world (within this context our actual world as that is what possibility is being referred to relative to us).

  • These possible worlds are existentially (spatiotemporal) isolated and so you cannot interact or travel to another possible world (if you could then they would be the same possible world). Different from a multiverse or other cosmological structures.

• (Extended) Modal Realism is the view that impossible worlds also exist, which allow contradictions. These worlds would operate on paraconsistent (and even fully trivialist) logics. ‘Modality’ is treated as a dimension similar to time & space (i.e., an object is not just extended in space & time, but also through possible and impossible worlds; some parts of it exist in the actual world and other parts in non-actual worlds, although technically they are all actual worlds).

A being operating on trivialist logic would be omnipotent, (this is also a valid solution the omnipotence paradox)

Is logic discovered or invented? Logic is a set of inference rules and laws (i.e., the law of noncontradiction) that is abstracted off of both physical/material and immaterial observation, as well as human intuition. $\to$ This also similarly applies to mathematics

A Platonist would say that logic is the metaphysical framework reality operates in, and a formalist would respond to the question saying that logic is a syntactic construction (the above response is about logic being abstracted to the point of syntactic construction, although goes beyond symbol manipulation as it has been based off of intuition and observation).

Logicism: The view that mathematics is entirely reducible to logic. Gödel’s incompleteness theorem showed that there is a discrepancy between mathematical truth and logical provability as there exists unprovable truths (depending on the system). Meaning there are mathematical truths which cannot be derived using logic. This is why it is often stated that “logic is the intersection between philosophy and mathematics.”

$\to$ Can our reality be purely ‘logical’? (in a logical framework, not necessarily classical logic) Related: Can anything be made as a premise (questions are assigned the ‘undefined’ truth state)? Yes anything can be talked about in ‘logical’ syntax given that a logic is a system which yields only ‘valid’ results via a set of internal inference rules, especially trivialist systems (i.e., first-order logic would not be able infinitary statements).

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Although truth cannot be grounded by a formal system (as any attempt to do so devolves into paradoxicality), we find solace in the notion that we have a stratified framework (i.e., Tarskian hierarchies) to analyze truth (at different levels). Any attempt on the validity of this can be answered from some higher meta-level which is external to the initial Tarski hierarchy (from which we define the Tarskian hierarchy). $\to$ i.e., We infinitely defer the issue to the next meta-level, this avoids the issue although does not fully solve it (a full solution is impossible).

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Determinism: The view that all events (in particular, human actions like ‘choices’) are determined by past causes and certain laws of nature. Another example is that we cannot “will what we will”, that it, we have no control over what you desire (this idea is linked to Schopenhauer), it is easy to see that this will devolve into infinite descent where even if we could “will what we will”, then we could not “will what we will to will.”

Compatibilism: The view that free will exists and determinism are not mutually exclusive, and rather that they are compatible. $\to$ The idea is that we still have a ‘degree’ of free will (an illusion of choice?) in that we can choose between many choices, although all of which have been casually determined.

Consider the following though experiment:

You go to an ice cream store which offers two flavors of ice cream; vanilla and chocolate. You have always preferred chocolate and brought up in an environment where everyone prefers chocolate, so obviously you choose chocolate and pay the store person $6.Nowconsiderthatyouhaveatimemachineandtravelbacktotheverymomentbeforeyoupayforicecream(ignoringtimeparadoxesandthesuch)anddecidethistimetotryvanillaasitissomethingdifferentfromtheusual(thiswayyouareeffectivelypaying$6 for two ice creams). It can be seen that both decisions are already determined, the choice of chocolate from prior biases, and the choice of vanilla which was determined from your ‘prior’ choice of chocolate and the want to try something new. In this sense there is a fragment of free will preserved in that you can choose between options, but any chosen option has already been determined.

Note: Determinists that argue against compatibilism will often say that they are playing a semantics game and are redefining what it means to be ‘free’ (i.e., unbound from any external influence), in this sense determinism is correct. A compatibilist will then reply that the notion of free will which their discussing is a more intuitive one w/ what we actually mean when we say ‘free.’

Predeterminism: Future events have already been predetermined i.e., you could look into the future and know w/ 100% accuracy that this will occur. Disagrees completely w/ the notion of ‘free will’ in that not even a fragment or ‘illusion’ is preserved.

Are all definitions circular? $\to$ No because of ostensive definitions (i.e., person A: “what is an apple?”, person B: points at an apply and replies “that is an apple”). These definitions rely on a shared understanding of use. $\to$ One may ask the meta-question “what is a definition?” (i.e., in an attempt )

What does it mean for something to ‘real’? What is ‘real’ and what actually exists are different things. i.e., existence may be defined as the set of all things that can be ‘located’ w/in space and time. This notion becomes when we consider abstract concepts which are ‘real’ but do not exist physically w/in space and time (perhaps a platonic metaphysical realm depending on who you ask).

Laws cannot be defined w/out ‘loopholes’ Note: This is not because of the incompleteness theorem as laws and the legal system are not a formal language (so it would be improper to use it), Instead this naturally arises from ambiguity w/in language.

Apophatic theology responds to the ineffability paradox (i.e., saying “God is ineffable” is a description and hence is effable) by instead saying “God is not effable”, assigning a negative description to ‘God’ rather than a positive one. An apophatic theologist argues that ‘negative descriptions’ are not actually descriptions in that nothing exactly is being ascribed to God’s nature. Instead negative descriptions such as “God is not effable” are meta-linguistic statements about the limits of our language not being able to capture the notion of ‘God.‘

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Idealism

There may be differing perspectives or sense data (i.e., color, texture,…,etc.) of a physical object (the material) depending on the person. $\to$ i.e., the shape of a table may appear square or diamond shape or some other shape depending on the position of the observer relative to the table.

Sense data is caused from a object.

Idealism: The view that reality is composed of ideas and is mental in nature, thus reality is dependent on the mind. This form of idealism, often called ‘ontological idealism’, does deny the existence of objects independent of the mind. The view that reality (our perception of it) is constituted of ideas and knowledge is dependent of the mind.

Are all perspectives equally ‘real’? Berkeley’s Idealism: What defines an interpretation of something is if it is an idea in the mind of god. Descartes Idealism: What defines a ‘real’ interpretation of something is an idea in the collective consciousness Berkeley’s view has some issues in particular w/ collapsing the distinction between an idea and the act of ‘perceiving’ something, which is also treated as an idea (i.e., “of the mind”).

Transcendental Idealism: Transcendental Idealism (also called ‘epistemological idealism’), is the view that our perception of reality and knowledge of it are mind dependent. In this view we do not perceive reality directly but rather our minds structure interprets it. Transcendental idealists do not deny the existence of an external reality independent to us, but rather say as Kant has put it, that that the intrinsic nature of physical objects - the “thing in itself” (noumena) - are fundamentally unknowable. Kant’s particular form of transcendental idealism goes further asserting that space & time are a priori knowledge through which we structure experience.

Note: A transcendental idealist will argue that our sense data are an approximation of reality, but we can never fully gleam the total knowledge about an object (i.e., sense data may be just illusions produced by noumena).

There are issues w/ noumena being fundamentally ineffable or inaccessible to us as it could then not be stated logically.

A priori knowledge is knowledge that can be understood w/out experience, a subset of this is knowledge which can derived from purely logical processes. ‘Intuitive’ knowledge is our direct/immediate knowledge of truths (i.e., something asserted as a proposition). A lot of intuitive knowledge is a priori, although not all.

References

Tarski’s theory of truth Trivialism On Truth