This is an example of the confusion inherent in switching between a natural language like English, and a formal language of logic.
X only if Y
is rare in spoken English, but perfectly grammatical, and it typically has a logical meaning equivalent to
If X then Y
Both statements are saying you can't ever have X without Y. However, ...
Consider the sentence:
If I am in America then I am in New York.
One could make the antecedent, "I am in America", true by being in Chicago. But then the consequent, "I am in New York", would be false. So this conditional would be false unless we are given other information, such as travel plans, in addition to knowing that I am in America.
"Some" does not exclude "all", but you cannot deduce "all" from "some".
Having said that, the above argument is not valid.
From premises 1 and 2 we can derive :
Some reasonable people are criminal
that is equivalent to : Not all reasonable people are not criminal.
Having said that, from "Some reasonable people are criminal" we cannot conclude by logic ...
Gödel's theism is discussed by Franzen in Gödel’s Theorem: An Incomplete Guideto Its Use and Abuse. He penned a version of the ontological argument, and in 1961 ranked the worldviews “according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion)... Skepticism, materialism, and positivism stand on ...
Is the argument valid?
"I will not need a loan if I do not buy a house" is the same as "If I do not buy a house, then I will not need a loan".
This is not implied by "If I buy a house, I will need a loan".
See Denying the antecedent.
"A only if B" and "if A, then B" mean the same.
The truth-condition for "if A, then B" excludes the case when A is True and B is False.
"A only if B" means that we cannot have A without B.
The two are equivalent.
See necessary and sufficient.
The contrapositive of both statements is :
If I am not in America, then I cannot be in New York.
A conditional statement is logically equivalent to its contrapositive. It means both your statements are equivalent since they have the same contrapositive.
We are asked to imagine the following conversation:
Alice: I believe that X.
Bob: Do you also believe Y? (Alice says yes) But that means you believe in Z.
Alice: That's true, but I still don't believe in Z.
At this point Alice looks like she is inconsistent in her beliefs. She believes in X. She believes in Y. She believes there is a warrant ...
I see two interpretations of the sentence here. They mean logically different things. In both cases "only" is interpreted as "must be true and cannot be false".
I am in New York (only if I am in America).
If I am in New York, it can only be true that I am in America.
New York => America.
This is the interpretation everyone else is responding to....
Wikipedia describes validity as follows:
In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
The argument we want to test for validity is the following:
If interest rates go down, then I will buy a house. If I buy a house, I will need a loan. ...
When is a connective truth functional?
Short answer : when it is defined by a truth table.
Classical propositional logic is a truth-functional logic in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a ...
HINT: I'll sketch the derivation. Since the theorem is a conditional, try using conditional proof/conditional-introduction by assuming P(a) and trying to derive ∀x(P(x) ∨ ¬(x = a)) from it. Here, to derive it, I would try an indirect proof by assuming the negation ¬∀x(P(x) ∨ ¬(x = a)) and trying to derive a contradiction. Use quantifier equivalence rules to ...
The author gives an example (page 15) of an argument that is NTP but not valid:
The glass on the table contains water.
∴ The glass on the table contains H2O.
He then says (page 17):
In the case of (7), to see that the premise cannot be true while the conclusion
is false, we need specific scientific knowledge: we need to know that the ...
It's an excellent question.
Heisenberg thought that QM forced us to modify the tertium non datur rule. So do many scientists. They are wrong, and here's why.
The principle of bivalence is not the issue here since it is unnecessary in dialectical logic that all statements are true or false, only that the statements we subject to our logical processes are. ...
This statement is provable. You can turn it into an equivalent implication: ¬□(□A → B) → □(□B → A)
Then make the hypothesis that the first term is true and show that the second follows.
The first term is equivalent to
◇¬(□A → B)
◇(□A ∧ ¬B)
From which follows that
□A ∧ ◇¬B
Then in any possible world, we have
From which we can trivially show ...
Unless you're seeking an understanding of the history and philosophy of logic, you really should not need to know anything about term logic to study propositional or predicate logic itself. Certainly the way propositional and predicate logic are standardly taught is self-contained. You may begin with some intuitive syllogisms, but ultimately you don't need ...
In a natural deduction system (if that is what you are using) to prove a conditional, such as is P → Q, you must use a Conditional
This takes the form of assuming the antecedent (that is P) aiming to derive the consequent (that is Q) through valid inferences (also using the premises; that is ￢P ∨ Q). Then discharging the assumption allow the ...
These claims have distinctly different connotations. From a pure formal-logic perspective, the "X only if Y" is equivalent to "Y or not X" which is the same as "X implies Y", which is the same as "if X then Y". However, natural language carries more information than its simple-minded reduction to predicate logic.
The second formulation "If I am in NY then I ...
The last statement suggests that buying a house is the only reason you would need a loan. Not buying a house does not rule out other reasons for needing a loan. Therefore it's logically false.
If it were explicitly stated that you would only ever need a loan when buying a house, it would be logically correct, even though it would be potentially false in ...
Some newspaper readers are criminal
This means that at least one newspaper reader is a criminal. It can be more, it can even be all of them. We do not know. But at least one is.
All newspaper readers are reasonable people
All of the newspaper readers are reasonable people. Because at least one of the readers was a criminal, we must have one reasonable ...
We can clarify a little bit the relationship between some fundamental logical concepts using their definitions.
In classical logic we have that A entails B (or : B is a logical consequence of A) iff A → B is valid (in propositional logic : is a tautology).
But a formula is valid iff its negation is unsatisfiable.
Thus, A entails B iff ¬ (A → B) is ...
What Smith means by the syntax of propositional logic are
the basic symbols of the language and
how those symbols can be combined to make sentences of the language.
These sentences and only these sentences are considered well formed formulas (wff).
On the top half of page 41 he defines the syntax.
The basic symbols of propositional logic (PL) are
You are quite right to say that in general conditionals "if A then B" are not truth functions and so cannot be defined by a truth table. The only exception is the classical material implication (or material conditional) which is defined by equivalence to "(not A) or B", or to "not(A and not B)". Material implication is the simplest of all the conditionals ...
All the upvoted arguments are valid. Here's just another way of phrasing the answer.
You start with this:
(Lower interests) IMPLIES (purchase house)
(Purchase house) IMPLIES (take loan)
You can drop the first one entirely.
Now you're asking : "Logically, are the following two statements equivalent?"
(Purchase house) IMPLIES (take loan)
(NOT purchase ...
Rewrite the phrases in a more formal-like manner as
1. For all x, N(x) implies R(x)
2. There exists x, N(x) and C(x)
And notice these do imply there are reasonable criminals, ie,
There exists x, R(x) and C(x)
Now, "Not all reasonable people are criminal" would be
Not for all x, R(x) implies C(x)
which is (classically) equivalent to
There exists x, R(...
One can always topologise:
As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the ...
Your question is a deep one about the very nature of logic. If logical truths are true, what exactly are they true of? And how do we know? There are many possible answers to these questions, and no general agreement among philosophers. Since the literature is pretty huge, I can only briefly summarise the main positions.
Simple logical realism would have it ...
I don't believe that there is wide agreement among contemporary philosophers on the answer to your question. The layperson's position you describe above sounds like logical realism. Logical realism states that there are facts of logic, and these facts are completely independent of us (our minds and our language). If humans had never existed, the law of non-...
Edit You can only use conjunctive, disjunctive, and negation intro/elim and only uses TautCon for DeMorgans.
You also seem to be using rules called "contradiction introduction", and "contradiction elimination". So I suspect what you call "negation elimination" is what is more usually called "double negation elimination".
Anyway, you have the first five ...
Proof created with proofs.openlogicproject.org:
The proof assumes that you can use the double-negation rule and the disjuction-elimination rule. (When asking for help with logic problems, it's a good idea to say which rules you're allowed to use since different texts/programs allow different rules that others may not.) Also, lines 6-8 can be simplified by ...