After some thought I realized this is a denying the antecedent fallacy. Put another way we have
If the person is a teenager then they are bad
If the person is a non teenager then they are not bad
To add to @Jon's answer, denying the antecedent often comes up due to confusion with the valid argument form Modus Tollens
((p → q) ∧ ¬q) → ¬p)
or ((if p then q) and not q) then not p)
Which is equivalent to
(if p then q. Therefore, if not q then not p), i.e.
If a person is a teenager, then they are bad
If a person is not ...
"Positive" is what Leibniz and other proponents of the ontological argument called qualities that make something "better" than it is without them (Anselm spoke of "good" as in summum bonum). In particular, Leibniz defined "perfections" ascribed to God as "simple, positive qualities in the highest degree", see What makes Leibniz's definition of perfection ...
Gödel’s Incompleteness Theorem is a result about formal systems.
Its proof requires certain assumptions about the properties of specific formal system F: basically, about its "expressive capabilities".
In a sense that can be specified rigorously, system F must have the capabilities to manufacture the provability predicate for F, i.e. a suitable formula PrF(...
Hegel's logic has already been formalized by physicist and mathematician Urs Schreiber. However, there are likely only a few dozen people on earth who can understand it due to the formalization being done with cutting edge mathematical logic (such as homotopy type theory) and with a deep familiarity with the Science of Logic: https://ncatlab.org/nlab/show/...
(1) is a special case of the general principle that if you accept a statement, then you accept that the statement is true. If you believe snow is white, then you believe "Snow is white" is true. It remains handwavy until you give a precise explanation of what you mean by "X is true." And that turns out to be a tricky business.
Yes - the key term is "generalized quantifiers." They are studied in the contexts of both natural language and in mathematical logic. I'll focus on the logic side, about which I know more.
A name which crops up in both contexts is Jon Barwise, and this article of Vaanaanen describes much of Barwise's work on generalized quantifiers; this paper of Barwise on ...
why do we stick with the material implication?
First of all, we have to consider that the propositional connectives are a (very simple) mathematical model of natural language, suited for modelling formal arguments.
In the context of classical logic, their definition is through truth-table; having defined them, we may check how they are "proxing" natural ...
Suárez discusses the applicability to the Persons of the Trinity of this form of the principle of non-contradiction,
A = C
B = C
∴ A = B
in On the Various Kinds of Distinctions p. 59 (Disputationes Metaphysicæ, Disputatio VII, De Variis Distinctionum Generibus),
§2 The Signs or Norms for Discerning Various Grades of Distinction in ...
There is a philosophical branch which is all about limits of understanding. It is called Hermeneutics and Gadamer's Truth and Method may be a start.
Basically, you are talking about what he calls understanding The Other, ie. real understanding involves being able to conceptually grasp what something is able to "tell" you about its history of effect (...
While it’s not quite a perfect parallel, a related concern about decision problems had been phrased some years before Gödel’s work: see https://en.m.wikipedia.org/wiki/Entscheidungsproblem . David Hilbert was pretty convinced as to the decidability of arithmetic prior to the Incompleteness/Incomputability proofs, but was aware that the problem remained open.
Deductive proofs in first-order logic are essentially transformations of one statement of the language into another. You start with some statement (or several or none at all) and then produce from it an ordered sequence of new statements derived by successive applications of the established rules of the system. You can end this sequence at any time and any ...
"⊨A is a tautology" is not true.
You have assumed f(A)=1, but this does not mean that A is a tautology.
The same for the second case : "When ⊨A → B, B must be true since ⊨A is a tautology" is wrong.
A → B is a tautology: this means that, whenever A is true also B is, and this is enough to conclude with A⊨B.
Perhaps we can here begin to show, without specifically answering the question posted on this page, that a regular use of two-valued Aristotelian logic, useful though it is in its own right and realm of specific operation, fails to apply to the powerful doctrine of the Most Blessed Trinity.
The intrinsic relationality of the One divine essence is self-...
Short answer: (1) amongst our 16 possible binary operators, we have an operator at hand that never gives as output "Truth" when the input is ( Truth, Falsity) (2) we use it because this operator is the best tool we have to model the notion of logical implication (3) this is the reason we also call it implication with the lower status of " material ...
Suarez has commented on Aquinas's Summa Prima Pars, and he also wrote a theological summa called De Deo Uno et Trino (cf. Suarez in Latin online).
The image below shows that
(1) Pater est Deus
(2) Filius est Deus
(3) Pater non est Filius
However, from (1) and (2) ( with the symmetry and the transitivity property of equality) one can infer that
Long comment (too long for a comment) regarding my preceding comment's link to
I think that might be analogous to what you're trying to think about in the following way, which is analogous in some ways, but very different in other ways...
You introduce C as a "cause" with mutually ...
If C is a cause in a non-deterministic system leading to mutually exclusive and jointly exhaustive outcomes Y and Z, and at time T, C causes Y, to answer the question why Y and not Z will inevitably be a function of the context. That is to say, your example of a causal system is an abstraction, and the only answer in the abstract sense is ...
First of all, note that nothing prevents us from using multiple different kinds of implication at once; indeed, nobody (I think!) would quibble with the claim that most of the time when we use conditionals in natural language we are not invoking the material conditional. Even in mathematical contexts we can consider non-material implications.
That said, ...
One attractive feature of the material conditional is that, in conjunction with the universal quantifier, it offers a natural way to translate classical logical statements of the form "all A are B" into propositions in first-order logic, like "for all x, A(x) -> B(x)". Consider for example the classic example "all men are mortal", translated as "for all x, ...
there is no alternative for truth functional classic logic. It's the only thing you can build out of the truth tables for propositional logic that even remotely resembles an indicative conditional, and it works in most contexts.
I recommend you look at the truth tables yourself and try to find another way to define it.
Allow me to start with something that will sound like a quibble: there is a difference between 'evidence' and 'data.' The world is filled with data: sense data, analytical facts and figures, mathematical axioms, etc. But data by itself is meaningless, substance-less. For data to have meaning and substance, it must be interpreted so it fits within a ...
Yes, there is a logical fallacy to identity politics. The fallacy has two parts.
Identity politics begins with a great big ad hominem argument. The site LogicallyFallacious describes the ad hominem technique as
Attacking the person making the argument, rather than the argument
itself, when the attack on the person is completely irrelevant to the
John of St. Thomas’ following remark, introducing his comparative treatment of Suárez and St. Thomas, captures a distinctly Catholic principle:
“Non Contrariatur Principiis Luminis Naturalis Hoc Mysterium.”
We may also say of the doctrine of the Most Blessed Trinity—inspired by the same principle:
Supra luminis naturalis rationis sed non adversus eam.
if you'd like to teach yourself through reading, I'd sincerely recommend the following books. the prerequisite is familiarity with propositional logic and first-order logic. and I suppose you've met this requirement as a math student.
First-Order Modal Logic By M. Fitting & Richard L. Mendelsohn.
I learnt modal logic by reading this book. its exposition ...
I'd like to add an addition to A.K.'s otherwise very good answer to address the topic of progressing to the "state of the art." A.K.'s reading list will get you the basics. That will be very doable on your own. By contrast, it's very difficult to progress to the state of the art without a mentor or PhD advisor, simply because you won't have any orientation ...
Statement "X" must be true because it is clearly laid out as such in
mutually reliable source "Y".
That sounds like an appeal to authority, spiced with some "weasel words" ("mutually reliable source").
Joe, the statement "X" seems outlandish and therefore it ought to be
rejected in its own right.
It's hard to know if that's a logical statement ...
Positivity is a property of unary predicates that Gödel defines implicitly, i.e. he writes axioms for positive properties. These include the one you cite and consequences of positive axioms being positive, so if any positive property were unsatisfiable all properties would be positive, a contradiction.
Similarly, Sobel showed Gödel's axioms imply there are ...
I think there are different kinds of necessity at play in your argument that should be kept separate. (I may be repeating some of what you said, and some of what you and Conifold discussed in chat.)
If a proposition is true at all moments in time, let's call that, let's call that proposition "eternally true".
If a proposition is true in some possible world,...