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Do the paradoxes of material implication in relevance logic, invalidate reductio ad absurdum?

One of the paradoxes of material implication in relevance logic is P → (Q → P) A proof of this statement in classical propositional logic is: P [OSC1] P ∨ ¬Q [1; addition] ¬Q ∨ P [2; commutativity of ...
lee pappas's user avatar
1 vote
1 answer
79 views

What is the reason behind the fourth axiom in Gödel's ontological proof?

In Gödel's ontological proof, axiom 4 goes like this: And I'm not sure about what it means. If that P(φ) is true, then isn't it necessarily true as well? There's some basic concept about modal logic ...
Elvis's user avatar
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0 answers
59 views

How can you derive Conjunction if your only underived rule of inference is modus ponens? [closed]

Suppose you are working with a logistic system for the propositional calculus that only has one underived rule of inference, namely Modus Ponens. How can you derive Conjunction in it? Rule 1: A, if A ...
lee pappas's user avatar
4 votes
1 answer
56 views

Can assumption in Hilbert style proof system be contradictory?

⊢(¬A→A)→A I don't know how to solve this proof with the Axiom, Theorem and Inference rule in Hilbert-style proof system so I ask my classmate and he show me his answer. After viewing his proof, I was ...
san zhang's user avatar
2 votes
0 answers
29 views

Is this a valid Disjunctive Syllogism

According to the website it is not, the alternative is to use a double negative, but double negation elimination, is only valid as an atomic expression, for context this is the full question.
Jacob Duffy's user avatar
2 votes
0 answers
129 views

Why not just give up on the idea of truth-functionality?

I understand that today only a minority of academics who are specialised in formal logic accept the horseshoe (aka "Classical Logic" or "First-Order Logic") as an accurate, or even ...
Speakpigeon's user avatar
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4 votes
9 answers
771 views

How can “a implies not a” be true?

Why does the capacity of the formula a -> ~a to be true seem so counterintuitive? Can you give me some ordinary language examples of this case?
Егор Галыкин's user avatar
9 votes
11 answers
2k views

Do statements have an intrinsic, unchanging truth value, even when currently unknown, or can they have different truth values at different times?

Example: I state that a coin will come up heads, then flip it. While the coin is flipping, does the statement 'the coin I just flipped will come up heads' have a truth value? Based on my ...
Anathemically's user avatar
5 votes
3 answers
635 views

Can an argument be valid even when its premise refutes the conclusion? (Trying to disprove my professor)

I want to ask about something that I saw in philosophy class today. Let's say that this is a valid argument, and let's call it argument A: Premise 1: P1 Premise 2: P2 Conclusion: C And there's ...
femboy420's user avatar
10 votes
9 answers
2k views

What framework or tool solves the Barber Paradox?

The Barber Paradox is usually phrased as follows: I know a barber whose policy is to shave everyone who doesn't shave himself. If a person shaves himself, the barber does not shave that person. If ...
pygosceles's user avatar
1 vote
2 answers
108 views

A priori argument fallacy

Considering the "A priori argument logical fallacy" (see https://utminers.utep.edu/omwilliamson/engl1311/fallacies.htm) Is there a way to formalize it (technical expression) using the ...
Vincent ISOZ's user avatar
1 vote
1 answer
40 views

Using the Universal Quantifier, Existential Quantifier, and logical connectives, write a description of a time when somthing bad for you was not bad

The following is an exercise in ethics, logic, the use of the universal quantifier and the the use of the existential quantifier. In the following context, the syntax for the universal quantifier is ...
Toothpick Anemone's user avatar
5 votes
3 answers
2k views

What's the difference between "iff" and "=df"?

Just a quick question I stumbled upon from my readings. When some philosophers write A ↔ B and others write A =df B, is there supposed to be a difference?
John Smith's user avatar
1 vote
4 answers
136 views

Is affirmation of the consequent always invalid?

Examples that I've seen usually go something like "If the lamp is off, then the room is dark. The room is dark, therefore the lamp must be off." However, what about the following example &...
Kalcifer's user avatar
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1 answer
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Propositional Logic: Truth table for a certain argument form

I'm confused about evaluating a certain argument form with the truth table approach. I think below argument matches the corresponding truth table I made, but if my truth table is right it means the ...
Damon Fernandez's user avatar
1 vote
0 answers
52 views

Missing two syntactical expressions of rules of inference in sentential logic

I have a table of the rules of inference in propositional logic. Among the entries are an Associative and a Commutative. The Associative rule is expressed with disjunction, but the commutative is ...
J D's user avatar
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3 votes
2 answers
117 views

Implicature justification for translating "P unless Q" as (¬Q → P)

In The Laws of Truth, Smith translates utterances of the form "P unless Q" as (¬Q → P) and takes the further suggestion that (Q → ¬P) to be an implicature of the utterance. The justification ...
user51462's user avatar
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4 votes
3 answers
284 views

Confused On The Definition Of A Proposition

One definition I encountered was something that is either true or false. (for example, I ate vegetables yesterday is a proposition). Another definition I encountered is the meaning of a sentence (for ...
HelpMePlease's user avatar
2 votes
0 answers
69 views

Propositions vs sentence types and tokens and the context insensitivity of PL

I came across the following explanation for the context insensitivity of the language of propositiional logic (PL) on page 34 of The Laws of Truth by Nicholas Smith: Because glossary entries pair ...
user51462's user avatar
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1 vote
1 answer
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Can a valid argument be said to be unsound if the set of premises is unsatisfiable (inconsistent)?

I'm asking in a strict propositional logic sense. Suppose that I have a set of premises that is logically unsatisfiable (or inconsistent), i.e. they can not be all True simultaneously, that argument ...
teaboo's user avatar
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3 votes
0 answers
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Difference between Propositions and Statements

I have read the answers to many questions like mine I reached these two definitions of propositions (I don't know which one is correct) A proposition is a statement that can be true or false. If this ...
Credence's user avatar
3 votes
0 answers
78 views

If the modern understanding of the categorical/hypothetical distinction is better than Kant's, does this undermine the categorical imperative?

In first-order logic (FOL), we have dedicated connectives for conditionals and disjunctions. Again in FOL, we can turn a conditional into a disjunction, though. Frege's logic has a judgment stroke, or ...
Kristian Berry's user avatar
0 votes
1 answer
82 views

Trivialism vs Alethic Nihlism

What are the similiarities and differences between the two theories (as well as arguments for and counterarguments against). From what I know, trivialism states that everything is true (and I believe ...
HelpMePlease's user avatar
0 votes
0 answers
46 views

Proof of A & B from ¬(A → ¬B) using rules of inference

I have been tasked with proving A & B from −(A → −B). However, I'm only allowed to use the following rules: ModusPonens, ModusTollens, ConditionalProof, DoubleNegation, AndIntroduction, ...
Noah Clarkson's user avatar
2 votes
3 answers
129 views

Natural language into logic and proof

I'm working through Logic by Paul Tomassi, and there is one particular problem I'm stumped with. The problem is on pg 186 and involves representing an argument in English as a sequent and then ...
Crest's user avatar
  • 21
0 votes
0 answers
39 views

Are these formulas correct in modal logics?

I'm looking to try to simplify 2 different formulas to be easier to say but i'm not sure on the exact rules of propositional calculus. I am starting with the two formulas (1) and (2) and i'm ...
Richard Bamford's user avatar
3 votes
1 answer
62 views

What is the relationship between possible worlds and a valuations?

A propositional formula is something like this, A&~B, which uses letters to represent propositions. The letters are called propositional variables. Compare the following two sets of terminologies ...
David Gudeman's user avatar
12 votes
7 answers
4k views

How does "if p, then q" compare to "p only if q"?

How do the statements if p then q and p only if q compare
Marsha's user avatar
  • 129
1 vote
1 answer
116 views

What is the difference between a tautological corresponding conditional and (P v ~P)?

The Wikipedia article on the corresponding conditional contains the following sentence: An argument is valid if and only if its corresponding conditional is a logical truth. Some sources use "...
user51462's user avatar
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2 votes
1 answer
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How to show DeMorgan Law in intuitionistic logic using weak excluded middle?

I am trying to show in intuitionistic logic that ~(A & B) > (~A v ~B) using the deduction theorem and weak excluded middle (~A v ~~A). I already proved (~~A & ~~B) > ~~(A&B) and ~(A &...
Clio's user avatar
  • 21
0 votes
1 answer
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Help reconstructing argument

I saw the following argument in Paul Guyer's text "Kant" (Routledge). I am trying to reconstruct it, yet am not sure the of the form of the argument. Can anyone provide help? If whenever ...
rux23's user avatar
  • 117
5 votes
2 answers
172 views

Justification of the material conditional truth function in Introduction to Formal Logic

Pages 150-151 of §18.3 of Introduction to Formal Logic by Peter Smith provide two justifications for the truth table of the material conditional. In the first justification (paragraph (a) - (c) on pg. ...
user51462's user avatar
  • 481
0 votes
2 answers
105 views

First use of exportation/importation in formal logic?

Who is the logician who first used exportation/importation, namely, ((p ∧ q) → r) ⇔ (p → (q → r))? Gödel used it in his 1939 Logic lecture, but it doesn’t seem to have been known from the Aristotelian ...
Speakpigeon's user avatar
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-1 votes
2 answers
310 views

Is there a proof of exportation/importation from more obviously true implications such as Modus ponens?

Is there a proof of exportation/importation, namely, ((p ∧ q) → r) ⇔ (p → (q → r)), from more obviously true implications such as the Modus ponens, Transposition, de Morgan etc. I don’t believe that ...
Speakpigeon's user avatar
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2 votes
3 answers
829 views

Truth-functional vs non-truth functional conditionals

I'm struggling to understand truth functionality. I know that a connective is truth-functional if the truth value of a compound statement formed with that connective is completely determined by the ...
user51462's user avatar
  • 481
-1 votes
1 answer
194 views

Translating English statements to logical expressions

this is my first questions so I apologize for any formatting mistakes. Given the following propositions: c: I will return to college. j: I will get a job. and given the sentence: "There is no ...
JoeyVeeStallion's user avatar
1 vote
1 answer
81 views

Intersection of the Gettier problem and knowing-what or knowing-how

From what I can tell, it seems like the Gettier problem comes down to Smith not knowing that the man who has ten coins in his pocket is going to get the job. What about Smith knowing what the ...
Kristian Berry's user avatar
3 votes
3 answers
234 views

How do you prove that a logic system is sound?

I am aware of the fact that a logic system must be sound, in order to be useful. However, I am not sure, about how, after setting up or coming up with the basic logic axioms that make up my system, I ...
Joselin Jocklingson's user avatar
2 votes
1 answer
70 views

Sentential Interpretation in P. Suppes (1957)

Patrick Suppes gives a working definition of sentential interpretation, based on a sentence maintaining its form. By working definition, I mean an incomplete definition that is needed for someone to ...
Then-Brief-864's user avatar
-4 votes
1 answer
146 views

Hi! I'm 99% sure my formal argument is valid, but can you check? [closed]

I wrote this argument, and while i'm sure it is valid, it has been awhile since I've done basic logic.Thanks!
Anon1313's user avatar
0 votes
1 answer
46 views

How understand abstraction when some cases can’t be abstracted?

Like the liar sentence “this sentence is false” is said not to be a proposition. So not all sentences can be abstracted into props. Can infinite sentences be abstracted into propositions. Can infinite ...
J Kusin's user avatar
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2 votes
3 answers
183 views

Philosophy book written using logic statements

I would like to translate a philosophy text into logic axioms and propositions. Then, I would like to use prolog to check if the text is logically consistent. However, I find it difficult to translate ...
Tereso del Río Almajano's user avatar
-1 votes
3 answers
173 views

Translation of Arguments from Propositional Logic to Predicate Logic

How exactly does this work? What can we assume stays the same, what changes? Take for an example this (valid) argument: A & ~C ~C > ~D ~D > B ∴ B Now let us take rewrite it according to ...
WolandBarthes's user avatar
0 votes
1 answer
65 views

Semantic consequence and Sound Argument

Is that correct to say that semantic consequence is equivalent to the concept of sound argument in classical propositional logic? If it is the case, arguments or theories with contradictory premises ...
Cesar Tacla's user avatar
0 votes
3 answers
994 views

Proof for "⊢ (A → ¬¬A)"

I've spent 4.5 hours on this, with no exaggeration. I clearly have no idea what I'm doing here, and it's become a serious time sink. If any of you could help in proving this, I would be eternally ...
BeepBoop69's user avatar
-1 votes
1 answer
109 views

Help with proving: P, ¬(Q ∧ P) ⊢ ¬Q

Here's the issue, there's no usage of derived rules allowed. So no DeMorgan's Law. All that's allowed is the basic TFL elimination/introduction rules, IP, (e)X(plosion), and ⊥. I'm absolutely lost on ...
BeepBoop69's user avatar
0 votes
1 answer
102 views

What does it mean, intuitively and then also precisely, that a particular English word is not truth functional?

What does it mean, intuitively and then also precisely, when we say that a particular English word is not truth functional? Let me present some examples. Example 1 As far as I can tell from a book I ...
xoux's user avatar
  • 291
-2 votes
1 answer
107 views

Proofs of propositional logic truth tree rules in natural deduction?

It is a great irony of natural deduction that some of the most seemingly obvious inferences are also some of the trickiest to prove! So far, I haven't been able to prove the following, and I'd greatly ...
Spailpín's user avatar
0 votes
1 answer
205 views

How to interpret "P ⟺ Q is true if and only if the first-order logic sentence P ↔ Q is logically necessary"?

I'm learning about the two truth-functional connectives "material conditional" and "material biconditional". I came across this particular snippet in a book: An important fact ...
xoux's user avatar
  • 291
1 vote
1 answer
77 views

Prove that if S tautological consequence of P, S tautological consequence of Q, then S tautological consequence of P | Q

Consider the following argument: S is a tautological consequence of P. S is a tautological consequence of Q. Therefore, S is a tautological consequence of P | Q. I wish to give an informal proof of ...
xoux's user avatar
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