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22

Here is part of the question: My only idea is v must be introduced, but how would I use subproofs to show one of A/\C or B/\D is never false if A v B? It might be best to think of using disjunction elimination initially although disjunction introduction may be needed later. The OP notes the following: Obviously since A → C and B → D then if A v B one ...


15

They're not equivalent, but they do seem very close together in most contexts when you assume a bivalent (two truth valued) logic. But they pull apart when it comes to several controversial decisions we have to make in formal semantics and the philosophy of language. Lets consider two prominent examples. Supervaluationism: This is one solution to the ...


9

Non sequitur I'll go off of the example in the comments, namely “One dollar” = “money” : “Nickel” = “money.” Therefore, “one dollar” = “nickel.” This is non sequitur - there's no logical reason to assume that Therefore. Or, alternatively, this could be ambiguity fallacy as this seems to be caused by (intentional?) misapplication of the symbol "=" with ...


7

When you say, NOT(A and NOT A) = NOT A or A you are actually doing a few things at once. First, you are using de Morgan's Laws to state NOT(A and NOT A) = NOT A or NOT NOT A Then, you use double negation elimination to state NOT A or NOT NOT A = NOT A or A i.e. NOT NOT A = A. This makes your reasoning circular. Assuming that double negation ...


7

You can use proof by contradiction: p1: A v B p2: A -> C p3: B -> D assume ~(C v D) ~C & ~D (from 1) ~C (from 2) ~D (from 2) ~A v C (from p2) ~A (from 3, 5) B (from 6, p1) ~B v D (from p3) D (from 7, 8) D & ~D (4, 9) Since D & ~D is a contradiction, our assumption must be false. Therefore C v D.


6

~(A & ~A) The Law of Non-contradiction is that a statement can not be both true and false. It does not prohibit the assignment of some other value, but restricts how many values may be assigned. (A v ~A) The Law of Excluded Middle is that a statement must true or false. It does not prohibit statements from having multiple values, but restricts what ...


6

This is a question in philosophy that deals with the metaphysics of identity. A classic problem in philosophy is the Ship of Theseus and goes back to the pre-Socratics, particularly Heraclitus and his proposition that one cannot stand in the same river twice. In logic, one often draws a distinction between a name (symbol) and the thing it represents (...


6

To approach this from a slightly different angle, this concept is important in computer programming. In a lot of languages, the programmer can decide what attributes make an object "equal to" another object. For example, if you have two "People" objects represented by "first name", "last name" and "address"; you could choose to say that if the first and ...


5

The short answer is: no, there's no generally accepted solution of this 'problem' (if it is one). Some simply reject Bradley's argument(s) since they reject some of its (their) assumptions, for instance, that particulars are bundels of qualities or that qualities are tropes (e.g. Russell). Those that accept Bradley's assumptions have responded in various ...


5

"Neither true nor false” means that the statement has no definite truth valued : it lives in a sort of limbo, a truth value-gap between true and false. “Either true or false” means that the statement has (exactly) one of the two truth values. To say that "Lexical definitions are either true or false" means that a Lexical definition : also known as ...


4

No. If and only if [time passes] then [Uranium238 decays]. Time does not cause the decay, but it can't happen unless time passes. Another example is quantum entanglement. Given two entangled photons A and B, then: If and only If [A changes phase 45°] then [B changes phase 45°]. Because these changes happen at the exact same instant then by definition ...


4

I do believe you've missed the point of 'duplicate' here. 'Sameness' in this context is a fairly loose and utilitarian construct. Consider: if the temple priestess says she needs a statue of Zeus for entryway, and everyone in the village steps up to sculpt a statue of Zeus, well... the priestess still only needs (and will only use) one of those statues. The ...


4

Hint You have to apply OR-elimination to first premise and used 2nd and 3rd premises to derive "C OR D" under both cases. Then the conclusion follows. See also Proof by cases.


4

Yes-ish: it takes some work to formalize it, but it can be done. Specifically, the proof of the relevant model checking theorem gives a general method for proving, for an appropriate sentence p, a sentence of the form "If q_i implies p for each i < n, then p is true" where n is the appropriate bound and {q_i: i < n} are sentences characterizing each ...


4

This is perhaps one of the earliest surviving derivations of the law of explosion, ex falso sequitur quodlibet, contradiction implies anything. It seems very counterintuitive, but it is hard to pinpoint where things go wrong. "Conditional exchange" is the definition of the material conditional, which is controversial. Indeed, it means that "2 is not odd or ...


3

This answer is offered as a supplement to Daniel Prendergast's answer. I hope to address the following question: Both laws basically mean the same thing. One is a restatement of the other. Am I wrong in any way? Wikipedia offers a way to look at the law of the excluded middle differently from the law of non-contradiction if one thinks of A and not A ...


3

In formal logic, identity is a two-place predicate. It could be written Identical(x,y). It has the value true when x and y are numerically identical and false otherwise. As a matter of syntactic sugar, it is usually written as x=y, but it is still a two-place predicate. In first-order logic, at least, there is no need for an additional relation of equality, ...


3

Different sources (contexts) may approach these concepts differently. The authors of the introductory logic text forallx uses the equality symbol (=) to stand for a two place predicate and calls it identity: (page 222) This does not mean merely that the objects in question are in-distinguishable, or that all of the same things are true of them.Rather, it ...


3

Regarding the Law of Identity, Leibniz formulation "A is A" (of a very old idea) encapsulates a basic idea that something is itself and seems trivial and incapable of being contradicted. It is a tautology of proposition rather than a tautology of inference. In that regard there are no exceptions. I think the general purpose of it's formulation, however, is ...


3

I depends on what you want your logic to model. You might want a logic that applies to some everyday process, like the laws, and not only to 'perfect' situations. One way to look at this is in terms of types of modality. Modalities of obligation or of preference do not enforce non-contradiction. If you create a contradiction of obligations, there may be ...


3

Like almost all linguistic conventions, all forms of equality are relative. It is convenient to have numerous synonyms for the different kinds of equality. But they are really interchangeable at some level. What each means is determined by context. Within the domain of mathematics, one constantly contrives 'equivalence relations' which assign groups of ...


3

Edward Feser, speaking in the context of the philosophy of mind regarding arguments between dualism and materialism, claims that the "positivist" view that the OP presents is common and is indeed a misunderstanding of philosophical argumentation. (page 234) A related misunderstanding - and this time, one that even many philosophers are prone to - is to ...


3

It's a straw man argument: where you change an argument someone's made to make it weaker than it is, and then refute it as if it were the other person's argument: https://en.wikipedia.org/wiki/Straw_man


3

I don't think you can do what you want to do within classical propositional or quantificational logic. At least not elegantly, there's some solutions you can do, such as letting a sentence like "John fell down after hitting his head" be an atomic formula, represented by a variable like "P". It seems that no one uses propositional variables to represent ...


3

First of all, a relation by itself is not a sentence. A relation applied to one or more terms is a sentence. Similarly, a function by itself is not a term. A function applied to one or more terms is a term. The difference between functions and relations is that functions yield terms when applied to terms, while relations yield truth values. For instance, '...


3

Validity and truth Do you regard these as separate notions ? In a deductively valid argument, the conclusion cannot be false if the premises are true. However, a valid argument can have a false conclusion : (a) All cats are white; (b) X is a cat; therefore (c) X is white. This is a deductively valid argument. If (a) and (b) are true, then (c) must be true. ...


2

You are confusing two uses of the word argument. In one sense, an argument is an extended discourse with limited aims such as education or persuasion. In the second sense, argument is a synonym for the technical term inference which is the process by which a single proposition can be constructed from a collection of premises (sometimes unstated). So, in ...


2

"Correct" is a bit of a loaded term here. Arguably, the correct way is to say that the English is ill-formed for use in quantified logic or at best unclear as to whether it is making a universal claim. Probably, the simplest rendering will be seeing it as equivalent to "all roses are red." See "girls are humans." But it might be "many roses are red" or ...


2

See Dialetheism : Probably the master argument used by modern dialetheists invokes the logical paradoxes of self-reference : In its standard version, the Liar paradox arises by reasoning on the following sentence: (1): (1) is false where the number to the left is the name of the sentence to the right. As we can see, (1) refers to ...


2

If one thinks of the law of non-contradiction and the law of the excluded middle as two parts of a dichotomy, mutually exclusive and jointly exhaustive, it may be difficult to find examples that don't illustrate both at the same time. What one needs is a class of propositions that one expects to be either true or false and so passes the jointly exhaustive ...


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