10

Circular reasoning is generally used to refer to an argument (or part of one) where the conclusion is essentially one of the premises. In short, you could think of it as something like: A ⊃ B, B ⊃ A, ... , ∴ A. Naturally this of often more subtle that the above line makes it look but the idea is the same - you're using the conclusion in a premise to prove ...


7

As you seem to suspect, the phrase "vacuous tautology" is pleonastic. The modifier "vacuous" is not necessary. However, it is probably being used to rhetorically highlight the particular vacuity of the tautology. It is not a term of art, and has no specifically defined meaning as a sub-type of tautology.


4

The confusion here is that W, X and Y don't represent variables, they are premises with internal structure. It might make it more clear to call them P1, P2 and P3. What Baggini is saying is that any argument of the form P1, P2 and P3, therefore C can be rewritten as a tautology of the general form IF (P1 AND P2 AND P3) THEN C It won't look like a ...


4

The reason it is not a tautology is that there is no reason a priori that deterministic patterns should be so easy to find (inverse square law?! How easy is that!), nor that behaving creatures should be composed of parts for which such laws act at such an incredibly low level as to be nearly useless in understanding the whole organism. To put it another ...


4

Tautologies are logical truths in the context of propositional logic: φ is a tautology       =def   φ is assigned ⊤ by all rows of the truth-table for φ. Logical truths are something more general, and can be defined as follows: φ is a logical truth   =def   a true ...


4

Some tautologies are obvious; others are not. I assume that the writer was trying to convey that the tautology was obvious. A non-obvious tautology can still be illuminating, since you may approach a problem two different ways and not realize that two different claims actually work out to be the same thing.


3

See : 5.4733 Frege says: Every legitimately constructed proposition must have a sense. Thus, we may equate nonsense [unsinn] with an illegitimate grammatical combinations of words, something like an ill-formed formula of a formal language. Only well-formed combinations of words generate proper sentences, i.e. sentences that express a thought [3.2] or ...


3

There are usages in which 'tautology' and 'logical truth' are interchangeable. However, from the literature (Quine, Tarski et al.) I'd venture that 'logical truth' is a more theoretical concept, open to highly contestable interpretations. For instance, it would be commonly agreed (I think !) that 'not (p & not-p)' is a tautology, true by virtue of its ...


3

I think there's an even simpler reason: it's hard to preserve truth when all you start out with are falsehoods! I take it that you've been studying axiomatic systems of logic (I'll restrict my focus to propositional logic). Typically these systems will have a number of axioms (which will be tautologies) and an inference rule or two--- typically something ...


3

In his book "About Time" author Adam Frank explains a philosophical (opposed to the mathematical) aspect of work on the cosmological argument. Managing the troubling issue of a universe which evolved under "exactly the right conditions", scientists have used tautologies. Here are a few examples, "The universe and its laws must take a form consistent with ...


3

In logic, a tautology is defined as a logical truth of the propositional calculus. If your preferred semantics of logical truth is 'true in all possible worlds' then yes, a tautology is true in all possible worlds and hence necessarily true. Other semantics for logical truth include model theory, category theory and various kinds of substitutivity. In some ...


3

I think your exercise has a typo, or the point was to catch the error in the exercise. Quoting: In logic, a tautology (from the Greek word ταυτολογία) is a formula that is true in every possible interpretation. ... A formula of propositional logic is a tautology if the formula itself is always true regardless of which valuation is used for the ...


2

How is this a tautology? Isn't a tautology something that is necessarily true regardless of the truth values assigned to W,X and Y (in this case)? Yes, and the statement you quote is also correct. By the definition of validity, the conclusion must be true if all of the statements in the argument are true. In this case, we don't really care about the ...


2

Circular reasoning refers to certain arguments in which a single premise asserts or implies the intended conclusion. A tautology is a single proposition, not an argument, that is true due to its form alone (therefore true in any model).


2

'Identity' has several shades of meaning. 'X = X' does seem particularly vacuous. But if you translate identity to some perfectly ordinary contexts, it proves a useful concept. Diachronic identity - identity over time. If I am the same person now who defrauded a client a year ago, then I now am responsible for what I did a year ago. I am the same person, ...


1

I was quite confused when I read in Carnap's "Introduction to Symbolic Logic and its Applications" (in chapter A section 5) all tautologies are L-true but many L-true sentences are not tautologies. He referenced chapter A section 14, but does not there explicitly say "here is an L-true sentence, yet it is not a tautology". This discussion has clarified that ...


1

Draw a truth table, like this:


1

Generally, there are 2 main ways to demonstrate that a given formula is a tautology in propositional logic: Using truth tables (a given formula is a tautology if all the rows in the truth table come out as True), which is usually easier. Using natural deduction with no premises, which is usually harder. If you get a conclusion using no premises then it is a ...


1

I suppose a tautology may be meaningless. All four-horned unicorns have four horns. seems to be tautological, but since the set of four-horned unicorns is empty, you can also say All four-horned unicorns have three horns. And yes, all four-horned unicorns (zero out of zero) have three, or four, or five hundred and twenty seven, horns. And so the ...


1

It is possible that the exercise has a typo. However, there are logics that construe its claim as correct. Tautologies are logically true, by logic alone, it is a linguistic notion. Necessity, on the other hand, is a modal notion, and a priori is an epistemological notion. There is no conceptual reason why the three can not come apart. This said, contingent ...


1

Yes, because mathematics is the second of the three degrees of abstraction: Physics deals with that which is in motion and is material. Mathematics deals with that which is material and is not in motion [∵ mathematical objects do not move or change] Metaphysics deals with that which is not in motion nor is material. In other words: there is no mathematics ...


1

In the main line of the Western tradition of logic (which follows the Aristotle) contradictions are emphasised by being the limit of what can be said. They are not the centre of a logic or given attention for themselves. Having said that Chrysippus, the Stoic logician wrote 23 books (chapters) on the liars paradox, and in fact it was this paradox that ...


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