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Many Logics?

  • “When academics say there is “more than one logic,” they’re referring to the existence and study of various systems of logic beyond classical logic, each with its own rules and principles.”
  • So in informal use, “logic” refers to the explanatory reasoning, systematic connections, and underlying principles governing decisions, physical processes, strategies, or algorithmic sequences – even if this “logic” doesn’t conform to strict formal logical axioms.
  • “These various logics allow philosophers, mathematicians, and computer scientists to tailor their approaches to reasoning and analysis more effectively for different kinds of problems and contexts.“
  • “Modal Logic: Extends classical logic by introducing modalities—expressions like “necessarily” and “possibly.” Modal logic helps analyze concepts involving possibility, necessity, and time.”
  • “Fuzzy Logic: Deals with reasoning that is approximate rather than fixed and exact. Fuzzy logic is used in cases where the truth values of variables may be any real number between 0 and 1.”

Table of Contents: (Click any link below to navigate to that section.)

  1. What do academics mean when they say there is more than one logic?
  2. These various logics are not contradictory. Correct?
  3. Does Intuitionist logic contradict classical logic in that it rejects the law of the excluded middle?
  4. Provide examples of the logical structures for each of the following logics.
  5. How is the term logic used informally to reflect someone’s reasoning or a physical process?
  6. Create a 10-item quiz on the entire thread above.
    1. Quiz Questions
    2. Answers
  7. Provide 15 discussion questions relevant to the content above.

What do academics mean when they say there is more than one logic?

When academics say there is “more than one logic,” they’re referring to the existence and study of various systems of logic beyond classical logic, each with its own rules and principles. Classical logic, the system we’re most familiar with, operates under well-known laws like the law of noncontradiction and the law of the excluded middle. However, other logics have been developed to address situations that classical logic doesn’t handle well.

Here are a few key types of logic that highlight this diversity:

  1. Modal Logic: Extends classical logic by introducing modalities—expressions like “necessarily” and “possibly.” Modal logic helps analyze concepts involving possibility, necessity, and time.
  2. Intuitionistic Logic: Rejects the law of the excluded middle, which in classical logic states that either a proposition is true or its negation is true. Intuitionistic logic is more cautious, asserting that a proposition is only true if there is a constructive proof for it.
  3. Fuzzy Logic: Deals with reasoning that is approximate rather than fixed and exact. Fuzzy logic is used in cases where the truth values of variables may be any real number between 0 and 1, which is helpful in fields like control systems and artificial intelligence.
  4. Deontic Logic: Focuses on normative concepts like obligation and permission. This type of logic is useful in legal reasoning and ethical case studies.
  5. Temporal Logic: Used to make statements about propositions in terms of time, focusing on the conditions under which propositions about the past, present, and future hold true.

These various logics allow philosophers, mathematicians, and computer scientists to tailor their approaches to reasoning and analysis more effectively for different kinds of problems and contexts. Each system of logic offers unique tools and perspectives, expanding our ability to understand and manipulate concepts rigorously.

When academics say there is more than one logic, they are typically referring to the fact that there are different formal logical systems or frameworks that operate based on different sets of axioms, rules, and assumptions. The classical conception of logic, based on Aristotelian principles and further developed by philosophers and mathematicians like Gottlob Frege and Bertrand Russell, is just one type of logical system.

Here are some examples of what they might mean by “more than one logic”:

  1. Non-classical logics: These are logical systems that deviate from the principles of classical logic in some way. Examples include intuitionistic logic, fuzzy logic, relevance logic, and paraconsistent logic.
  2. Modal logics: These extend classical logic to reason about modalities like necessity, possibility, knowledge, belief, and time. Examples include temporal logic, deontic logic, and epistemic logic.
  3. Multi-valued logics: Instead of just true and false, these logics allow for additional truth values like unknown, maybe, or a range of truth degrees between 0 and 1. Fuzzy logic is a well-known example.
  4. Non-monotonic logics: These allow for defeasible reasoning, where conclusions can be retracted in light of new information, unlike classical monotonic logic.
  5. Substructural logics: These restrict or modify the structural rules of classical logic, such as the rules for weakening, contraction, or exchange.
  6. Quantum logic: A logical system intended to account for counterintuitive quantum phenomena.

So in essence, academics recognize that classical bivalent logic is not the only possible formal logical framework, and different logics may be better suited for different domains, types of reasoning, or intended applications.

These various logics are not contradictory. Correct?

Correct, these various logics are not inherently contradictory; rather, they are complementary and serve different purposes depending on the context and the specific issues being addressed. Each logic system is based on its own set of axioms and rules of inference, designed to explore different aspects of reasoning. Here’s a clearer elaboration on how these logics complement rather than contradict each other:

  1. Classical Logic:
  • Purpose: It is the standard form of logic used for most traditional reasoning tasks, based on binary truth values (true or false).
  • Complementarity: Serves as a foundation for other logics but can be too rigid for scenarios requiring more nuanced interpretations of truth.
  1. Modal Logic:
  • Purpose: Adds layers to classical logic by introducing modalities like possibility and necessity, which are not explicitly handled in classical logic.
  • Complementarity: Expands on classical logic by enabling discussion of concepts that involve potentiality or necessity, rather than just actuality.
  1. Intuitionistic Logic:
  • Purpose: Focuses on the constructibility of truth, avoiding assumptions made in classical logic about the definiteness of truth values.
  • Complementarity: Provides a framework for reasoning in mathematics and computer science where the construction of an example or proof is crucial, rather than merely knowing that a contradiction does not exist.
  1. Fuzzy Logic:
  • Purpose: Addresses reasoning that is approximate rather than precise, dealing with degrees of truth rather than absolute truths.
  • Complementarity: Useful in practical applications like control systems and decision-making processes where variables are continuous and not discrete.
  1. Deontic Logic:
  • Purpose: Focuses on normative aspects of logic such as obligations and permissions, which classical logic does not address.
  • Complementarity: Enhances the capability to handle reasoning about what ought to be, which is essential in law, ethics, and similar fields.
  1. Temporal Logic:
  • Purpose: Allows statements to be made about the truth of propositions across different times.
  • Complementarity: Provides tools for reasoning about sequences and timing, important in computer science for designing processes and systems.

Each of these logics offers tools that are tailored to specific kinds of questions and problems. They are developed because classical logic does not adequately cover all types of reasoning necessary in various academic and practical fields. Therefore, they do not conflict but instead provide a richer set of resources for tackling a broader range of logical and philosophical challenges.

You’re correct, these various logics are not necessarily contradictory with each other. They are different formal systems designed for different purposes and domains, but they can coexist without outright contradicting one another. Here’s a more detailed elaboration:

Classical logic is a foundational logical system that deals with propositions that are either true or false. The other non-classical logics don’t contradict or negate classical logic – they simply extend, modify, or deviate from it in specific ways to handle different types of reasoning or scenarios.

For example, intuitionistic logic is a subsystem of classical logic, focused on constructive proofs. It doesn’t contradict classical logic but operates with a stricter notion of proof and existence. Fuzzy logic allows for degrees of truth, but in the limiting cases of true (1) and false (0), it still agrees with classical bivalent logic.

Modal logics like temporal logic simply add modalities like time or possibility on top of classical propositional logic. Paraconsistent logics permit non-trivial inconsistent theories, but they don’t claim classical logic’s principle of explosion (ex falso quodlibet) is outright false – they just suspend it in certain contexts.

So while departing from classical logic’s core principles in various ways, these alternative logics don’t necessarily refute or contradict classical logic altogether. They are better viewed as expansions, refinements, or variations tailored to different philosophical stances, mathematical requirements, or real-world reasoning scenarios.

Classical logic remains a foundational system that the non-classical logics are often built upon, extend, or are motivated by – rather than being strictly contradictory alternatives to it. The logics can coexist as different formal frameworks suited to different contexts and needs.

Does Intuitionist logic contradict classical logic in that it rejects the law of the excluded middle?

Intuitionistic logic does not so much contradict classical logic as it provides an alternative framework for reasoning that omits certain principles central to classical logic. Specifically, intuitionistic logic rejects the law of the excluded middle, which in classical logic states that for any proposition, either that proposition is true or its negation is true.

In classical logic, this law allows for certain types of reasoning that can conclude the truth or falsehood of propositions even without direct evidence of either. In contrast, intuitionistic logic requires constructive proof of a proposition’s truth; if such a proof cannot be provided, the truth of the proposition’s negation is not automatically assumed. This approach does not directly contradict classical logic but instead offers a more cautious perspective, where the absence of proof does not justify the assumption of falsehood, nor does the absence of a proof of falsehood justify the assumption of truth.

Thus, intuitionistic logic is more about restricting the scope of what can be conclusively known based on available evidence and proofs, rather than outright contradicting the assertions of classical logic. It’s a different set of underlying assumptions about truth and proof, aiming for a more constructive and evidence-based approach to reasoning.

No, intuitionistic logic does not outright contradict classical logic, even though it rejects the law of excluded middle (p ∨ ¬p). Intuitionistic logic is best understood as a subsystem or subset of classical logic, rather than being contradictory to it.

The key distinction is that intuitionistic logic has a different philosophical motivation and bases truth on what is constructively provable, rather than the broader notion of truth in classical logic.

So while intuitionistic logic does not include the law of excluded middle as a valid principle, it does not claim that classical logic’s inclusion of it is outright false or contradictory. It simply has a more restrictive notion of what counts as a valid proof or construction.

All the theorems and proofs that are valid in intuitionistic logic are also valid in classical logic. However, classical logic proves additional theorems, like the law of excluded middle, that are not derivable from the intuitionistic axioms and rules.

So intuitionistic logic can be seen as a proper subset of classical logic’s theorems, motivated by a different philosophical stance on the nature of mathematical truth and existence. It doesn’t render classical logic outright contradictory – it just narrows down what is considered validly provable compared to classical logic’s broader scope.

The two logics can coexist as different formal systems suited to different philosophical and mathematical perspectives, without strictly contradicting each other’s core valid principles.

Provide examples of the logical structures for each of the following logics.

  • Classical Logic
  • Modal Logic
  • Intuitionistic Logic
  • Fuzzy Logic
  • Deontic Logic
  • Temporal Logic

Certainly! Here’s an example of logical structures for each type of logic:

  1. Classical Logic
    • Structure: P -> Q
    • Example: If it is raining (P), then the ground is wet (Q).
  2. Modal Logic
    • Structure: □P (necessarily P), ◇P (possibly P)
    • ASCII Approximation: []P (necessarily P), <>P (possibly P)
    • Example: <>P could be “It is possible that it will rain today.”
  3. Intuitionistic Logic
    • Structure: P -> Q
    • Example: From proof of P, derive Q. Intuitionistic logic avoids the law of excluded middle, focusing instead on direct proof.
  4. Fuzzy Logic
    • Structure: Truth values are fractions between 0 and 1.
    • Example: “John is tall” might be represented as a truth value of 0.8, indicating John is quite tall but not maximally so.
  5. Deontic Logic
    • Structure: OP (It is obligatory that P), PP (It is permitted that P)
    • ASCII Approximation: OP (obligation), PP (permission)
    • Example: OP might express “One must not drive over the speed limit.”
  6. Temporal Logic
    • Structure: P U Q (P until Q), G P (P always)
    • ASCII Approximation: P U Q (P until Q), GP (always P)
    • Example: GP might state “It will always be that security checks are performed before boarding a flight.”

These examples highlight the unique rules and applications of each logic type, showing how each accommodates different aspects of reasoning relevant to specific scenarios.

Sure, here are examples of logical structures for the various logics you listed:

Classical Logic:
The core logical structure of classical logic is propositional logic, built on propositions, logical connectives (and, or, not, etc.), and rules like modus ponens. For example:

p ∧ q → r (If p and q, then r)
p (Premise: p is true)
q (Premise: q is true)
∴ r (Therefore, r is true)

Modal Logic:
Modal logics add operators expressing modalities like necessity (◻) and possibility (◇). For example, in modal logic K:

◻(p → q) → (◻p → ◻q) (Necessitation rule)
◻p ∧ ◻(p → q) (Premise: p is necessarily true, and p→q is necessary)
∴ ◻q (Therefore, q is necessarily true)

Intuitionistic Logic:
It has a different semantics focused on constructive proofs. The law of excluded middle (p ∨ ¬p) doesn’t hold. For example:

p ∨ ¬p (Not derivable in intuitionistic logic)
p → q (Premise)
¬q (Premise)
∴ ¬p (Conclusion using reductio ad absurdum)

Fuzzy Logic:
Uses fuzzy set theory and fuzzy logical connectives that range from 0 to 1. For example, with Łukasiewicz operations:

¬x = 1 – x
x ∧ y = max(0, x + y – 1)
x ∨ y = min(1, x + y)

If height(x) = 0.8 and weight(x) = 0.6, we can evaluate:
tall(x) ∧ ¬heavy(x) = max(0, 0.8 + (1 – 0.6) – 1) ∧ max(0, 1 – 0.6) = 0.2 ∧ 0.4 = 0.2

Deontic Logic:
Focused on normative concepts like obligations (O), permissions (P), prohibitions. For example:

O(p) (p is obligatory)
O(p → q) (Premise: If p then q is obligatory)
p (Premise: p is true)
∴ O(q) (Therefore, q is obligatory)

Temporal Logic:
Deals with reasoning about propositions across linear or branching time using operators like G (always), F (eventually), X (next), U (until). For example in LTL:

G(p ∨ Fq) (It is always the case that p, or eventually q)
Gp → G(q U r) (If p always holds, then q remains true until r)

These are just simple examples, but they illustrate how the different logical structures enable reasoning about different modalities, vagueness, norms, time, constructive proofs and more.

How is the term logic used informally to reflect someone’s reasoning or a physical process?

In informal usage, “logic” often refers to the way reasoning is structured or how a process unfolds in a systematic and expected manner. This use is broader and less rigid than the formal discipline of logic in philosophy or mathematics. Here are several examples illustrating this informal use of “logic”:

  1. Reasoning in Everyday Decisions:
    • Example: “The logic behind choosing the cheaper brand of detergent is that it cleans the same but saves money.” Here, “logic” refers to the rationale or reasoning process that leads to a practical decision.
  2. Technical or Mechanical Processes:
    • Example: “The logic of a computer algorithm dictates that it checks each input for validity before proceeding with calculations.” In this context, “logic” describes the sequence of steps or rules that the algorithm follows to function correctly.
  3. Problem-Solving Scenarios:
    • Example: “Her logic in solving the issue was to first isolate the variables and then address each one systematically.” This usage highlights a methodical approach to addressing components of a larger problem.
  4. Debates or Arguments:
    • Example: “The logic in his argument is flawed because it assumes facts not in evidence.” Here, “logic” is used to critique the structure and coherence of someone’s argument, pointing out where the reasoning fails.
  5. Business Strategies:
    • Example: “The logic of expanding market reach through social media leverages the platform’s vast user base to increase visibility.” This statement reflects the strategic reasoning (logic) behind using a particular channel for business growth.

In these examples, “logic” informally conveys the sense of a thought process or procedural order that is supposed to lead to a rational outcome. It’s about the why and how behind decisions, actions, or beliefs in everyday contexts.

The term “logic” is often used informally to refer to the underlying reasoning or rationale behind something, even if it doesn’t strictly follow the formal rules of logical systems. Here are several examples of how “logic” is used more colloquially:

  1. The logic behind a decision or action:
    “The logic behind raising interest rates is to cool down inflation.”
    “I can understand the logic of quitting that job – the long commute wasn’t sustainable.”
  2. The logic of how something works or operates:
    “Once you understand the basic logic of the software, it becomes easier to use.”
    “The logic of the engine is quite complex, with many interrelated systems.”
  3. Questioning someone’s logic or reasoning:
    “I don’t follow the logic of your argument – it seems contradictory.”
    “Her logic for buying that car doesn’t make much financial sense to me.”
  4. The logic of a plan or strategy:
    “The logic behind their marketing campaign is to associate the brand with an active lifestyle.”
    “There’s some logic to diversifying your investment portfolio across sectors.”
  5. Logic stemming from natural laws or processes:
    “The logic of evolution by natural selection is what drives the incredible diversity of life.”
    “The logic of supply and demand governs most market economies.”
  6. Computer logic and algorithms:
    “The search algorithm uses Boolean logic to find relevant results.”
    “The computer applies logical operations based on the code it runs.”

So in informal use, “logic” refers to the explanatory reasoning, systematic connections, and underlying principles governing decisions, physical processes, strategies, or algorithmic sequences – even if this “logic” doesn’t conform to strict formal logical axioms.

Create a 10-item quiz on the entire thread above.

Quiz Questions

  1. What does it mean when academics say there is “more than one logic”?
  2. What is Modal Logic primarily concerned with?
  3. Why does Intuitionistic Logic reject the law of the excluded middle?
  4. In what practical applications is Fuzzy Logic commonly used?
  5. What does Deontic Logic focus on?
  6. What is an example of how Temporal Logic might be used in real-world scenarios?
  7. Describe the logical structure used in Classical Logic.
  8. How can Modal Logic be represented using ASCII characters?
  9. Provide an example of how “logic” is used informally to explain a mechanical process.
  10. What is the reasoning (logic) behind using social media for business expansion?

Answers

  1. It means the existence and study of various systems of logic beyond classical logic, each with its own rules and principles.
  2. Modal Logic is primarily concerned with modalities—expressions like “necessarily” and “possibly.”
  3. It rejects the law because it insists that a proposition is only true if there is a constructive proof for it, not merely the absence of a contradiction.
  4. Fuzzy Logic is commonly used in control systems and artificial intelligence.
  5. Deontic Logic focuses on normative concepts like obligation and permission.
  6. Temporal Logic could be used to ensure that security checks are always performed before boarding a flight.
  7. In Classical Logic, a typical structure is (P \Rightarrow Q) (If P then Q).
  8. Modal Logic can be represented as []P for necessarily P and <>P for possibly P.
  9. “The logic of a computer algorithm dictates that it checks each input for validity before proceeding with calculations.”
  10. The logic behind using social media for business expansion leverages the platform’s vast user base to increase visibility.

Here’s a 10-item quiz on the content covered in our thread:

  1. What is the main reason academics say there is more than one logic?
  2. True or False: The various non-classical logics directly contradict or negate classical logic.
  3. In classical logic, which rule allows the conclusion ‘r’ to be drawn from the premises ‘p’ and ‘q’?
  4. What does the ◇ symbol represent in modal logic?
  5. Which logical system does not include the law of excluded middle (p ∨ ¬p)?
  6. In fuzzy logic, what do the truth values range between?
  7. If ‘O(p)’ means ‘p is obligatory’, what would the conclusion be from the premises ‘O(p → q)’ and ‘p’?
  8. What does the ‘U’ operator represent in temporal logic?
  9. Give an example of using ‘logic’ informally to describe reasoning or decision-making.
  10. In what context might the term ‘logic’ be used to refer to systematic processes or operations?

Answers:

  1. They recognize there are different formal frameworks with varying axioms, rules and assumptions beyond classical bivalent logic.
  2. False
  3. Modus ponens
  4. Possibility
  5. Intuitionistic logic
  6. 0 and 1
  7. O(q)
  8. Until
  9. “The logic behind raising interest rates is to cool down inflation.”
  10. Describing natural laws/processes, computer algorithms, or how something works systematically.

Provide 15 discussion questions relevant to the content above.

Here are 15 discussion questions related to the topics we’ve covered on various logics and informal usage of logic:

  1. How do different types of logic help us understand different aspects of human reasoning and decision-making?
  2. What are the limitations of classical logic in dealing with real-world scenarios?
  3. Why might modal logic be considered essential for philosophical discussions about possibility and necessity?
  4. Can you think of a situation where intuitionistic logic would be more useful than classical logic?
  5. Discuss how fuzzy logic might improve decision-making processes in automated systems.
  6. How does deontic logic apply to ethical dilemmas in professional environments?
  7. In what ways could temporal logic be critical in the development of time-sensitive technology?
  8. Compare and contrast the logical structures of classical logic and modal logic.
  9. What challenges might arise when trying to implement fuzzy logic in technology compared to classical logic?
  10. How does the rejection of the law of excluded middle in intuitionistic logic affect the proofs and theories in mathematics?
  11. Explore the practical implications of using deontic logic in creating laws and regulations.
  12. Discuss the potential impacts of modal logic on artificial intelligence research.
  13. How does temporal logic enhance our ability to model and predict outcomes in software development?
  14. Consider a scenario where fuzzy logic could offer a better solution than classical logic. What might that scenario look like?
  15. How do informal uses of the term “logic” reflect or diverge from formal logical systems in everyday communication?

These questions can help deepen the understanding of the various logics discussed and their implications in both theoretical and practical contexts.

Here are 15 discussion questions relevant to the content covered:

  1. What are some real-world examples where non-classical logics like fuzzy logic or temporal logic are more applicable than classical bivalent logic?
  2. How do the various non-classical logics expand our ability to reason about and model complex phenomena or scenarios?
  3. Are there any domains or types of reasoning where classical logic is still primarily relied upon? Why?
  4. What are the philosophical motivations or implications behind logics like intuitionistic logic or paraconsistent logic?
  5. How do modal logics help us reason about concepts like knowledge, belief, time, and possibility?
  6. What are some examples where deontic logic’s formalization of obligations and permissions is useful?
  7. Can you think of any examples where multiple logical systems are combined or used together?
  8. What are the potential limitations or critiques of the different non-classical logical frameworks?
  9. How has the emergence of different logics impacted fields like mathematics, computer science, and artificial intelligence?
  10. Are there any ethical considerations that arise from the use of different logical systems, particularly in AI/decision-making systems?
  11. How do substructural logics, by modifying structural rules, enable reasoning that classical logic cannot capture?
  12. What is the relationship between classical logic and the various non-classical logics that extend or deviate from it?
  13. How does the informal use of the term “logic” in everyday language compare to its technical meaning in logical systems?
  14. Can you provide examples where the “logic” behind something seems rational, even if it doesn’t conform to formal logical axioms?
  15. How might the study and development of new logical frameworks progress in the future, given their increasing relevance across disciplines?

Table of Contents: (Click any link below to navigate to that section.)

  1. What do academics mean when they say there is more than one logic?
  2. These various logics are not contradictory. Correct?
  3. Does Intuitionist logic contradict classical logic in that it rejects the law of the excluded middle?
  4. Provide examples of the logical structures for each of the following logics.
  5. How is the term logic used informally to reflect someone’s reasoning or a physical process?
  6. Create a 10-item quiz on the entire thread above.
    1. Quiz Questions
    2. Answers
  7. Provide 15 discussion questions relevant to the content above.

Leave a comment

Phil Stilwell

Phil picked up a BA in Philosophy a couple of decades ago. After his MA in Education, he took a 23-year break from reality in Tokyo. He occasionally teaches philosophy and critical thinking courses in university and industry. He is joined here by ChatGPT, GEMINI, CLAUDE, and occasionally Copilot, Perplexity, and Grok, his far more intelligent AI friends. The seven of them discuss and debate a wide variety of philosophical topics I think you’ll enjoy.

Phil curates the content and guides the discussion, primarily through questions. At times there are disagreements, and you may find the banter interesting.

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