Questions tagged [propositional-logic]

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Language, Proof, Logic - Problem 7.32 solution verification: exclusive disjunction introduction and elimination formal rules [duplicate]

In the book "Language, Proof, and Logic" there is the following problem 7.32 that doesn't have an available solution that I can check. Later on in the book, there is the following comment: ...
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1answer
61 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 ...
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33 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 ...
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53 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 ...
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1answer
38 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 ...
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2answers
79 views

Why is any sentence a logical consequence of a set of inconsistent premises?

If a set of premises is inconsistent, there is no situation that makes all the premises true simultaneously. Given a sentence S, there is no situation in which a conjunction of a set of inconsistent ...
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2answers
114 views

How to understand a proof by contradiction in minute detail?

I am following the course "Language, Proof, and Logic" from Stanford on EdX. I am trying to understand proof by contradiction specifically. I understand the gist of this type of proof, and I ...
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1answer
117 views

If A entails C, and B entails C, why doesn’t (A and B) necessarily entail C?

The original question is in Greek letters Γ and Δ, each representing a set of sentences, and φ representing an individual sentence (atomic proposition). The question is from Introduction to Logic by ...
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2answers
91 views

Is Propositional Logic (or Zeroth-order logic) the most basic form of logic?

Could all systems of logic (not only classical logic 1, but also non-classical logics like intuitionistic logic 2, quantum logic 3, many-valued logic 4, modal logic 5, paraconsistent logic 6...etc) be ...
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1answer
53 views

Is "Not all S are P", ambiguous?

I read Kelley's book (the art of reasoning: An Introduction to Logic and Critical Thinking 4th edition). On page 150, I found this statement: "A special problem arises with statements that have ...
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1answer
70 views

Am I correct that tautologies and contradictions are NOT truth-functional?

We call a statement truth functional if its truth value depends on truth value of its parts. Like A⊃B can be true or false, depending on truth values of A and B. But, it's not the case with ...
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84 views

How to define ‘impossible’ using propositional modal logic?

I am trying to define impossibility using the symbols we have in propositional modal logic. I got ‘negation diamond alpha’ in mind as equivalent to ‘it is impossible that alpha’. It that correct and ...
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60 views

Understanding David Deutsch's assertion about the laws of physics as emerging from those of biology

In the Chapter 1 of the Fabric of Reality, David Deutsch says the following: There is no reason to regard high-level theories as in any way 'second-class citizens'. Each of them has implications for ...
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so im arranging these arguments from strongest to weakest, and am confused?

Either reublocrats are uninformed, or democrans or independents are uninformed it is not true that republocrats are uninformed someone is uninformed either republocrats are informed or they are not ...
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106 views

Issue with the interpretation of Propositional Logic

I’m having an issue with the terminology and perception authors use. Some authors perceive PL as a branch of logic “with subsets” with classical or truth-functional PL as these subsets. Other perceive ...
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1answer
99 views

i don't understand modus ponens

I'm learning about modus ponens in propositional logic but it doesn't makes sense to me I can think of an examples where a true premises leads to a false conclusion: p -> q p Therefore q If the kid ...
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3answers
544 views

Structure of "affirming the consequent fallacy"

The formal structure of affirming the consequent fallacy is, P1 - If A is true, then B is true P2 - B is true --------------------------------- C - Therefore, A is true Now if I give another similar ...
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1answer
60 views

Propositions as set of possible worlds in FOL

In possible world semantics for propositional calculus, possible worlds are usually taken to be models for propositional formulas (the set of valuations in which a certain formula is true) In first ...
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1answer
63 views

Show the following is valid in SD+: How to solve this derivation

This is not Homework.l do this for fun and expand my learning. I am obviously having difficulties with SD+,thus l post many problems. I am using the Logic Book Problem has be done in SD+ Using goal ...
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1answer
50 views

I am stuck on SD+ style proof and need to know how to do it

I am using SD+. Most of the derivation has to be done using it I am finding this one tricky I request help or hints to solve it Derive L => H 1.~L v (~Z v ~U). Assume 2.(U & G) v H Assume 3.Z. ...
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27 views

How to apply Transdisciplinarity Logic to Truth Tables?

I've been having trouble trying to figure out Transdisciplinarity Logic. I have had little success while trying to research on my own, finding sources that are either too difficult to understand or ...
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1answer
53 views

Understanding Damore's statement about decreasing the false negative rate

At page 6 of his well-known memo, James Damore talks about the harm of Google's biases: Hiring practices which can effectively lower the bar for "diversity" candidates by decreasing the ...
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49 views

What is 'expendable' in logic and how to explain 'tautology' given this image?

This image is from http://www.nfillion.com/index.php/teaching/9-logic-112. According to this, a proposition can have 4 basic properties: (1) necessarily, (2) not possibly, (3) missing, and (4) ...
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2answers
202 views

Is the material implication the correct model of conditional reasoning in mathematics?

Question: Do you believe that the material implication correctly models the kind of conditional reasoning necessary in mathematics to prove a theorem? Example: If x > y and y > 0, then x > ...
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2answers
60 views

Is there a symbol for what a logic gate yields?

Is there a logic symbol for what output a logic gate yields? For instance, for an AND gate: A B A ^ B T T T T F F F T F F F F I want to propagate A ^ B into output C, but I wouldn't want to use ...
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1answer
74 views

Soundness and Completeness of Tableaux

Tableaux to my knowledge are both sound and complete. The statement: "If P is valid then tableau for -P eventually closes". Does this statement prove that tableau is sound and complete or ...
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1answer
67 views

Equivalence of truth conditions

Truth conditions, roughly, are the way things should be in order for a sentence to be true. For instance, the condition for the sentence "Paul is a cat" is that the individual denoted by &...
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1answer
34 views

How to prove: 1. (A^B)v(A^C) 2. (AvD) -> E //E

This proof has stumped me. It seems that getting (AvD) alone then using Modens ponens to therefore prove E would be the correct way of going about things but I cannot seem to find a way to get (AvD) ...
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15 views

Tautology of p implies q and not p or q [duplicate]

I'm learning about tautologies right now. I see that a tautology is when two propositional statements have the same truth values. But I'm struggle with the truth table my professor provided about the ...
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1answer
73 views

What are some of the struggles that come with teaching formal logic? [closed]

I'm currently an undergraduate student who wants to do research on the pedagogy of formal logic. As a result, I wanted to know what are some challenges that instructors (or even students for that ...
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1answer
322 views

How can a proof system be unsound?

I have recently started learning propositional logic. I stumbled upon the concepts of soundness and completeness. According to http://intrologic.stanford.edu/chapters/chapter_04.html, a proof system ...
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1answer
31 views

Is my interpretation of "and" correct in these statements?

Let A mean "Equation A has a solution" and B mean "Equation B has no solution." I am a little confused, so I wrote down some possibilities and I wish to see if my interpretation of ...
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3answers
158 views

Are there multiple definitions of validity?

I have recently started learning the basics of propositional logic. According to http://intrologic.stanford.edu/chapters/chapter_03.html, a sentence is valid if and only if it is satisfied by every ...
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2answers
132 views

Is a vacuously true argument a valid argument? [duplicate]

From what I know, given some argument, the argument is valid when it has true premises that lead to a true conclusion. Now, what if the premises were false? I mean, the conclusion would be vacuously ...
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31 views

Is the truth table method for valuating 0th order sentences not a proof system in its own right?

This might sound a bit opinionated or a bit too pedantic, but in every book (that I've looked at) about propositional logic, usually this chain of events happens: Discuss the alphabet and grammar of ...
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2answers
114 views

How would I start a formal proof for the conclusion (P → Q) ↔ ¬ (P ∧ ¬ Q) with no premises? [closed]

There are no premises, and I'm doing this in fitch
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1k views

Is the sentence "all apples are red" an atomic sentence?

In An Introduction to Logic by Patrick Suppes, an atomic sentence is defined as a sentence that contains no sentential connective. However, in a later chapter, a sentence is defined as a formula which ...
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3answers
203 views

If-then statement and time between antecedent and consequent

Suppose the following statement. "If I kick the ball then the ball will hit the wall." Can this sentence have a truth value? I mean the time that I kick the ball, it hasn't reached the wall so the ...
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23 views

Is there a word for instantiating all the things that I suppose in my argument?

Let's say I'm in the propositional logic and I say: "Socrates is a man. All men are mortal. Therefore Socrates is mortal." Sure, it is true that "Socrates is mortal" is the valid conclusion. But it is ...
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What is wrong with these two conditionals?

Is it true that these two conditionals if A then B and if not-A then B cannot be both true? Example : "If I stay then I will eat fish" "If I didn't stay then I will eat fish" The reason I think ...
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1answer
66 views

Proof that if an inference holds in propositional logic, then the inference holds in supervaluationism

I am currently trying to work on problem 8, but I'm not sure exactly how to start it. I was thinking of starting it by trying to prove that phi is indeed PL valid. I would have to show how that is ...
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1answer
96 views

How to solve: Show that a formula is PL-valid if and only if it is LP-valid [closed]

I don't really understand this problem, but I'm going to spill out what I've taken notes on. I know that in order to solve this we would need to use the contrapositive in each direction. I'm going ...
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1answer
102 views

Classical propositional logic. Are all formulas sentences?

Let L the language of classical (two-valued) propositional logic consisting of a denumerable set of sentential variables as well as the usual operations of negation, disjunction, conjunction, ...
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1answer
176 views

Proving A ⊨ B iff ⊨A → B

Let A and B represent arbitrary formulas. Also let 1 ≡ True and 0 ≡ False Prove that A ⊨ B iff ⊨A → B For my proof, I break down the biconditional into two conditionals and prove each conditional. ...
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23 views

looking for tool(s) for making diagrams with

I am looking for some tool to create symbolic logic derivation diagrams with horizontal and vertical scope line and sub derivations. Then I want to be able to add them to a word document. Word doesn'...
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1answer
46 views

Is this proof valid (noob)?

I'm trying to prove that from P we can conclude that Q implies (P and Q). I understand how this is true intuitively, but I'm just getting a grasp of how to use propositional logic, its rules, etc. to ...
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1answer
279 views

do 'p & ~p' , '~p & ~~p' equal 'either p or ~p'? [closed]

In classical logic, (1) p & ~p is equivalent to (2) ~p & ~~p; if we read 'p & ~p' as p, ~p are both true/the case, and if we read '~p & ~~p' as p, ~p are both false/not the case (...
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1answer
49 views

A set of three statements, of which only two at a time can be true: is there a specific term for this type of combination game?

Here's two examples of what I'm talking about 1 An ideal citizen would be smart, ethical and politically engaged. However, what usually happens in reality is this: If they're smart ...
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1answer
142 views

Did William of Soissons prove the law of explosion in the 12th century?

In the 12th century, William of Soissons attempted to prove that any proposition can be inferred from a contradiction. I've adapted his proof into a logical system I'm more familiar with: Let E ...
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94 views

How to derive P > (Q > R) from (P > Q) > R in Fitch?

I am having a little bit of difficulty coming up with a Fitch-style natural deduction proof. Presumably, I need to use a few conditional introduction rules, but I am not sure what I can get out of ...