# Questions tagged [propositional-logic]

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### What is the difference between negation-eliminiation ¬E and contradiction-introduction ⊥I?

I don't understand the difference between the rules negation elimination and contradiction introduction. I am using the Open Logic Project's natural deduction proof checker proof checker. The rules on ...
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### Proof of the principle of explosion

The Wikipedia page uses the following proof for the statement: P (Premise) ¬P (Premise) P ∨ Q (Disjunction Introduction) Q (Disjunction Syllogism) There are two issues, to me: this proof ...
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1 vote
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### How do you separate premises in logic?

In logic, you are tasked to identify the premise and conclusion of an argument, I personally find that very straight forward. However, when writing these arguments in standard form people separate the ...
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### Can these statements really be arguments?

https://learn.saylor.org/mod/book/view.php?id=64908&chapterid=57428 I was doing practice 12,14 and I am very confused as to how these are arguments. They seem to be statements 12:"The reason ...
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225 views

### An idea to prevent the principle of explosion

The principle of explosion is the law according to which any statement can be proven from a contradiction. This principle is proved as follows. Assume that P and ¬P are true. Let Q be an arbitrary ...
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### What would the correct answer be for Mendelson Exercise 1.4 (g)

In Elliot Mendelson’s “Introduction to Mathematical Logic”, he states, “Sentences may be combined in various ways to form more complicated sentences. We shall consider only truth-functional ...
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### How can we formalize the claim that these two languages are equally expressive?

Let T be a stripped-down version of propositional logic, whose only connectives are ¬ and ➝, and suppose T can prove all the usual theorems that can be formed from only these two connectives. Let T’ ...
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### Reference Request for Worksheets for Logic

I wanted to ask, could someone point me to a resource that gives problems to analyze and use logic to detect arguments, fallacies, etc. (problems like those in Copi's Intro to Logic) with an answer ...
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### What is a definition, written in symbolic logic, for a person living nearby?

Students often need some axioms and/or definitions to play with in order to learn formal logic. What is a definition of a neighbor written in the style of symbolic logic? By neighbor, we mean a person ...
1 vote
133 views

### If a proposition is necessarily true, does it follow that it's a tautology?

If □P, does it follow that P is a tautology? I know in K modal logic, the law of NEC states ⊢ P; therefore □P. The corresponding conditional of the previous argument is If ⊢ P then □P. Now ⊢P iff P is ...
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### Is there only one viable definition of the logical connectives? [closed]

Is there a way to prove there's only one viable definition of the connectives, which is their truth functional definitions?
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### What is the proper form of universal instantiation?

Definitions C is a specific constant iff ∃! x [x=C] C is a general constant iff ∀x [x=C] C is an arbitrary constant iff ∀x [x=C] ∨ ∃! x [x=C] Consider the commonly accepted form of the rule of ...
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### Can the entirety of first order logic be reduced to the propositional calculus?

I've been wondering, whether or not first order logic can be reduced to the propositional calculus. Rosser's system RS_1, described by Irving M. Copi in 'Symbolic Logic', has 5 axioms or postulates: ...
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57 views

### Which is better, an empirical basis for the meanings of the logical connectives or a relevance logic definition of them, and why?

em·pir·i·cal / emˈpirikəl/ (also em·pir·ic) • adj. based on, concerned with, or verifiable by observation or experience rather than theory or pure logic: they provided considerable empirical evidence ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
33 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.
139 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 ...
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806 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?
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### 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 ...
731 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 ...
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 ...
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1 vote
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### 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 ...
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1 vote
42 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 ...
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?
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### 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 &...
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39 views

### 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 ...
1 vote
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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, ...
159 views

### Tomassi Logic p186, translating arguments into propositional logic

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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 "...
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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 &...
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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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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