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If philosophy is mathematics and mathematics is computation, can I conclude that philosophy is computation? Yes. So is philosophy merely computation? No because philosophy isn't mathematics and mathematics isn't computation. Can we axiomatize philosophy? If you want. Getting philosophers to agree on a set of axioms should be amusing. Can a ...


6

Firstly, what you have written there is a proposition, not an argument. An argument has premises and a conclusion. What you are asking is how do we interpret the quantifier scope in propositions that have more than one quantifier. The short answer is to read the proposition left-to-right and take the leftmost quantifier as having wider scope. So, (∀y)(∃x)Ryx ...


5

Formal philosophy is an approach to philosophical questions which attempts to answer them by developing formal systems. Formal systems are those in which ideas (terms, claims, etc) are formalized, meaning symbolized. In such systems, conclusions drawn are typically reached as a function of the form of premises alone, which is to say their logical structure. ...


4

Formal philosophy, as others pointed out, is an approach with formalized language. To ease things here, let us assume that the main idea is that by formalization it is possible to infer the thruth value and propositional content directly from grammar and syntax. The ultimate goal is a perfect scientific language where equivocations and misunderstandings are ...


4

Preamble Your fears are regrettably well-founded. Unlike mathematical logic, the philosophy of language is not so clear cut and remains to this day a deeply controversial topic. There are a dozen schools of thought on the philosophy of language, both classical and contemporary. It is a fundamental and unquestionably significant field of philosophical inquiry,...


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Your rephrasings of formulas in words are correct, but in this case moving the quantifier makes no logical difference (classically). You can verify this by converting formulas into equivalent form without implications using A → B = ¬A ∨ B, and then using the fact that quantifiers can be freely moved across conjunction and disjunction as long as the variables ...


3

You ask what the implications are for formal modal logic of there being differences among languages in what modal terms they have — or differences with respect to terms we might treat as modal, since it depends on how you define modality. There are no such implications, because the question mixes together two things which logicians keep separate. Terms like ...


3

The way you have chosen to express the rules implies you are assuming a non-monotonic form of reasoning. Rule #1 as stated has no exceptions, while rule #2 expresses an exception to rule #1. In a monotonic system of logic (which includes classical logic) this would lead to a contradiction: if Bob hits Charlie, rule #1 says Charlie may not hit Bob back, but ...


3

Firstly, I suggest that you make sure that we're talking about formal philosophy and not 'formal logic', 'formalism' or other areas of philosophy that might use the term in their title, as these likely have much more specific meanings. As far as I know, there is no real definition for what formal philosophy is. In my experience, most people and writers seem ...


3

You're correct that moving from the integers to the rationals does not fit, because the generalisation that inductive reasoning refers to is a generalisation of statements (or predicates) not of the objects themselves. For example, you could generalise from the statement "all the even numbers above 3 we ever tried can be written as the sum of two primes" ...


2

There is a particular school of philosophy, called "analytic philosophy" which aspires to formal rigor, and there is a subject area within philosophy, called "formal logic" which is very closely related to mathematics and is often taught in a mathematical manner. Researching either of those topics should lead you to a wealth of good sources for what you are ...


2

The arity of a relation R is the number of its "argument places". Thus, the relation "less than" is a binary relation: it has arity 2. We usually write: x < y, but we can write it "more formally" as: <(x,y). An extensional relation of arity n on D is a subset of D^n (i.e. D x D x ... x D: the set of all n-uples). The set of all extensional relations ...


2

The key question here is “Can a computer think for us?” This can be reformulated as “Will strong AI ever be realized?” Or, “Can human understanding be reduced to a program running on a Turing machine?” There are two answers: Yes or No. If Yes, then a program can produce human understanding. Since philosophy is the result of human understanding, ...


2

Mark Balaguer (Free Will, MIT, 2014 (FW)) claims that “we can distinguish many kinds of free will” (FW 50), that is, there are many different ways to define free will. He looks at two very different definitions: Hume-style free will used by compatibilists and not-predetermined free will which he wants to use within a materialist context like that of the OP’...


2

What you are looking for does not exist. First off, there is no one formal definition for freewill. Not everyone agrees on what free will is beyond generally agreeing that humans have it. There are myriad variants, each with their own little twists. One variant that will give you trouble is the metaphysical freewill approach, which explicitly states that ...


2

There is no need for any special mapping between a simulation in general and the underlying "hardware". Neither a temporal correlation of simulation time and real-time, nor between parts of the simulated world and parts of the hardware processing units. A simulation is a mere sequence of calculations, that can be done by humans on paper, given ...


2

Short Answer Broadly speaking, in order to have a "simulation", we must first have a physical computer of some kind... How can we rigorously (mathematically) describe the relationship between the physical computer and the formal system being simulated? If you are talking about a formal simulation on a computer, then you are talking about a ...


1

1: AND. The contrast denoted by "but" has no relevance because Mick's oldness does not negate Keith's. 2: exclusive or (denoted by XOR). The premise "she cannot be both" rules out the OR connector. 3: (airplane departed)AND(you will succeed OR world will implode). Removing the term "or both" would require usage of XOR instead of OR. Yes, #4 is equivalent ...


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The OP has this question: Which books should I read in order to understand these entries and this logic from scratch and be able to write reasoned arguments like this? To learn to use truth-functional logic and first order logic from a natural deduction perspective you might try forallx. The text is available on-line without cost and there exists a proof ...


1

We need two ingredients : high-order logic, because we have a property of properties. And, IMO, we need modalities, in order to express the necessary "connection" involved in the concept of essence. Compare with Gödel's ontological proof. I would suggest : Ess(F,x) ↔ ( F(x) & □∀y (~Fy → y ≠ x)).


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One form of determinism would be a Turing machine. Turing machines have a formal definition. Turing machines cannot decide every language. For example the Halting problem is undecided. However, if you generalize the notion of a Turing machine to have an infinite number of states, you can get around this and recognize any language: Consider a deterministic ...


1

First, I'm not sure exactly what you expect by a "formal" definition of free will. However, I will try to report about a conception of free will that "clearly differentiates it from determinism, randomness and any kind of determinism-randomness hybrid." This sort of conception of free will, which tends to be called "libertarian" free will, often means ...


1

The example, and the way it is phrased, is very confusing, I had to read it several times to parse it correctly. (It also originally had a small but decisive typo in your version of it in the question, I have edited to fix that.) The hypothesis is that (all) unobserved grasshoppers live south of Canada --which is to say, there are no unobserved ...


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Frank Hubeny's answer is quite good, but let me add to it re: Herbrand semantics. In particular, I'm going to push back against the argument pro Herbrand logic cited in that answer. It is true that Herbrand semantics permits only countable structures*, but that's not the full story: since first-order logic has the downwards Lowenheim-Skolem property, if ...


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The Stanford Introduction to Logic course offers the following difference between Relational Logic and Herbrand Logic in the Recap of Chapter 9: Herbrand Logic is an extended version of Relational Logic that includes functional expressions. Since functional expressions can be composed with each other in infinitely many ways, the Herbrand base for Herbrand ...


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If theses can be separated from their antitheses, then yes its 'unlikely'; but Hegel pointed out that this was an assumption that required justifying; and he held that no, it wasn't; theses, contained within themselves their counter-motion (ie negation); and hence to his theory of sublation; see Hegels Logic. It's worth looking at the beginning of paragraph ...


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I don't have an answer to your first question; but an observation that may help bring forward more helpful answers by focusing on a particular discipline where formal arguments are made as a kind of case study. Take physics, here simplicity is often put forward as a key criteria and thus also a kind of mantra to characterise useful ideas and theories and ...


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