17
votes
Accepted
When proving a conclusion of the form P v Q, we only have to prove a single disjunct. Is that a slight abuse of syntax and exactness required of proof
The original form of the question for context; see below the break for a response to the question as clarified:
Is it "weird" that in standard logic P ∨ Q can collapse to a simpler formula, ...
- 243
10
votes
Why are we allowed to use inference rules instead of truth tables in order to prove an argument's validity?
A short answer is that truth tables only work for the propositional calculus, so they are of limited application. They don't apply to first order or higher order logics where you have quantifiers. ...
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8
votes
Accepted
Is logical reasoning deterministic?
Logic is formal and systematic (not deterministic, which refers to causal relationships). This means that in a (pure) logic system any statement can be derived by the manipulation of symbols of other ...
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7
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Why are we allowed to use inference rules instead of truth tables in order to prove an argument's validity?
inference rules are just argument forms whose validity can be proven by truth tables
I think that answers the question of why inference rules can be used.
There is another question: Why use inference ...
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7
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Is logical reasoning deterministic?
following tarski, a logic, as in a consequence relation, is just a function
f: P(X) → X
satisfying some closure-like conditions, so in a sense yes, it is 'deterministic', as is any function, simply ...
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7
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Difference between ∃x(Sx & Px) vs. ∃x(Sx → Px)
I am assuming that you are using classical logic and that → is the material conditional.
In classical logic, "some S is P" is understood to mean, "there exists some thing that is S and ...
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5
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When proving a conclusion of the form P v Q, we only have to prove a single disjunct. Is that a slight abuse of syntax and exactness required of proof
In terms of logical gates or circuits, the satisfiability of OR can be determined by any one of its inputs — and “satisfiability” might be the word you’re looking for.
AND has the same sort of ...
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5
votes
chain of proofs and whether a statement can be proven
We can prove something only within a certain plane — only here and now, no more and no less. And we are not capable of carrying any proof to an absolute understanding for every observer — regardless ...
- 902
5
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Accepted
Why are we allowed to use inference rules instead of truth tables in order to prove an argument's validity?
The reason you can use natural deduction, which uses only valid rules of inference and previously proved theorems during proofs and derivations, is because you are guaranteed the last line is a fact. ...
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5
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How do buddhists employ paraconsistent logic in their alternative to cartesian duality?
A classical Buddhist philosopher is Nagarjuna. His work “Mulamadhyamakakārikā” is a key text of Buddhist logic and philosophy.
A fundamental principle of Nagarjuna’s logic is the Catuṣkoṭi (=
...
- 50.2k
4
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Similarities Between Category Theory, Model Theory, FOL, and HOL
No. Category theory has similarities to other formalisms because it is a generalization of them. Think of it as an abstract version of set theory that can generalize and serve as an alternative ...
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4
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What are the most relevant investigations into the complete formalization of philosophy, at the level of rigor and precision found in mathematics?
Contemporary examples include, perhaps first and foremost, Edward Zalta's axiomatic metaphysics, i.e. his theory of abstract objects. There are also formal presentations on metaphysical grounding such ...
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4
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Difference between ∃x(Sx & Px) vs. ∃x(Sx → Px)
Assume the situation that Sx is never true and Px is always true. Then Sx AND Px is never true, but Sx IMPLIES Px is always true.
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4
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Does strengthening the antecedent lead to paradoxes?
I find myself once again engaged in the uniquely Sisyphean and unrewarding task of trying to shake Lorenzo from his erroneous understanding of logic.
You cannot change the domain of discourse in the ...
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3
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When proving a conclusion of the form P v Q, we only have to prove a single disjunct. Is that a slight abuse of syntax and exactness required of proof
You've greatly modified your question which makes all the other answers seem off-topic. Please don't do this! However, I will answer your new question, which seems to boil down to
When I prove P v Q, ...
- 237
3
votes
Accepted
Can necessary and sufficient conditions for definitions be understood as directed biconditionals?
You ask:
Can necessary and sufficient conditions for definitions be understood as directed biconditionals?
Interesting question. One of the fascinating aspects of semantic analysis generalizing over ...
- 39.2k
3
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When proving a conclusion of the form P v Q, we only have to prove a single disjunct. Is that a slight abuse of syntax and exactness required of proof
A counterexample to your "collapsing" property: the law of excluded middle. In classical logic, we don't need to prove P or prove ¬P. We prove P∨¬P directly from the axiom, but with the ...
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3
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When proving a conclusion of the form P v Q, we only have to prove a single disjunct. Is that a slight abuse of syntax and exactness required of proof
I'll try to specifically avoid using the words "or" and "and", but as we're talking about "weirdness", which is a feeling, let's explore the ideas in feelings.
So imagine ...
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3
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Contradictory statements in paraconsistent logic?
Relevance logics, which are a kind of paraconsistent (read: non-explosive modulo contradictions) logic, don't necessarily assign presented content to contradictions but can hold to a cancellative view ...
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3
votes
Has anyone tried to reduce works from continental philosophy to pure logic/analytic philosophy?
The division between "continental" and "analytic" philosophy emerged in the 20th century, so referring to Spinoza as "analytic" and Schopenhauer as "continental"...
- 902
3
votes
Has anyone developed a "restricted English" notation for logic (first-order or otherwise)?
Frederic Fitch did some work along these lines:
Fitch, Frederic B. “Natural Deduction Rules for English”. Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition 24, ...
- 131
3
votes
When proving a conclusion of the form P v Q, we only have to prove a single disjunct. Is that a slight abuse of syntax and exactness required of proof
Your question has changed. Here are a few observations.
There's nothing weird about the fact that some proofs can be short-circuited. Depending on what you are trying to prove, it may be sufficient to ...
- 34k
3
votes
The framework of assumptions that generated Russell's paradox
The title is
The framework of assumptions that generated Russell's paradox
but the text in the question's body focuses a lot on the case of sets, so it seems important and useful to clarify that, ...
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3
votes
Does strengthening the antecedent lead to paradoxes?
The formula that you have referenced in the proof generator can be translated as follows:
If
(if everything is a natural number then everything has a thing that is
greater than it)
then
(if ...
- 34k
2
votes
Can a sentence without a truth value be true?
Since the theory of truth values is not identical to the theory of truth predicates, and neither is quite identical to the theory of propositional truth operators, and then since we can fabricate &...
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2
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When proving a conclusion of the form P v Q, we only have to prove a single disjunct. Is that a slight abuse of syntax and exactness required of proof
Can we isolate and just talk about this property of "collapsing" without just talking about the difference between AND and OR?
Sure, let's talk about how they are similar. It might help to ...
2
votes
chain of proofs and whether a statement can be proven
The OOP(?) issue of proving a proof of a proof is somewhat orthogonal to the normal regress of inference from theorems to axioms. The situation is analogous to that of the modus ponens loop from Lewis ...
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2
votes
Is logical reasoning deterministic?
You ask:
Is logical reasoning deterministic?
That depends on what you mean by logical reasoning. For the purposes of this question, we can say that logic comes in formal and informal varieties. Both ...
- 39.2k
2
votes
When proving a conclusion of the form P v Q, we only have to prove a single disjunct. Is that a slight abuse of syntax and exactness required of proof
Based on your revision of the question, I would say that the "weirdness" comes from the difference between justifying a disjunction as a premise, and inferring a disjunction as a conclusion. ...
- 2,637
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