Can someone help me figure out how to solve the following natural deduction proofs in FOL formatting! Step by step preferably. Im at a loss. Would be super helpful! 1)Ax(B(x)->AyF(y,x)),C(a)->ExB(x) |- C(a)->ExF(a,x)

2)Ex(D(x)/G(x)), Ax(G(x)->F(x)) |- Ex(D(x)/F(x))

3)~Ex(F(x)/\D(x)), Ax(C(x)/D(x)) |- Ax(F(x) ->C(x))

4)Ax(C(x)->(B(x)/~D(x))), D(a) |- Ex~C(x)

5)Ex(F(x)/\Ay(C(y)->R(y,x))) |- Ax(C(x) ->Ey(F(y)/\R(x,y)))

6)Ax(G(x)->Ay(H(y)->R(x,y))), H(b) |- Ax(G(x) ->R(x,b))

7)Ax(~B(x)<->~C(x)) |- Ax(C(x)->B(x))

8. T |- AxB(x)->Ax(B(x)/C(x))

Hi, I'm studying for my Introduction to Symbolic Logic final, and I realized I'm confused by necessary equivalency. The definition I was given is "two sentences are necessarily equivalent if they have the same truth value in every case." I get that, but I'm confused on how this applies to written sentences, particularly facts. One of the practice exercises is determining whether the following pairs of sentences are necessarily equivalent and I'm stuck on "1. Thelonious Monk played piano. 2. John Coltrane played tenor sax." Both of these sentences are true, but I feel like they aren't necessarily equivalent because Thelonious Monk playing the piano does not guarantee that John Coltrane played the tenor sax. It's possible that there's a world where Thelonious Monk plays piano and John Coltrane doesn't play tenor sax. And, wasn't Thelonious Monk actively playing for like a good decade before Coltrane was? A similar example I'm also confused on was "1. George Bush was the 43rd president. 2. Barack Obama was the 44th president." Both of those things are true, but neither of them entail the other. I guess I'm not sure if necessary equivalency requires one sentence to entail the other, and if made up cases (someone else COULD'VE been the 43rd or 44th president) can be used to show that two sentences aren't necessarily equivalent. Any help would be greatly appreciated! Thank you :)

Are Aristotle and medieval logicians committed to logical monism ?

How do I demonstrate validity using a diagram?

For example, I'm an Asian person who was raised in the US. As a result I sound and "act" very American. I also have a lot of Asian American friends. Whenever someone asks my friends or myself "where are you from," I notice that a lot of them purposefully say and push something like "I'm from New Jersey" or "I'm from my mom's womb."

Despite us knowing that what the person is actually asking is "You don't look like the average American that I'm used to seeing. Where is your ethnic heritage from?" some of us choose to purposefully not know this. If someone is asking where in the US we're from, that is often made specific in the context as well.

What is the name of that error when you purposefully feign ignorance?

I believe that Pinocchio's nose would grow after a short time (maybe 5 secs or so).

The only condition for the nose to grow is to tell a lie. I think that only referring to the nose does not prompt it react. The nose would only grow after the lie has been fulfilled, in this case only after "now" has passed, because his nose wouldn't have grown in that moment.

I also think Pinocchio's perception of "now" would affect it in a way that only after his "now" passed that it would grow. If he said "My nose is about to grow" it wouldn't grow because it has no reason to be trigged, only after Pinnochio's perception of "about to" passed it would grow....

What do you think?

Im doing this question which states: Demonstrate that (∃y)(∀x)Ryx does not follow from (∀x)(∃y)Ryx. a) Try a truth-tree first. b) If that doesn’t work, construct an interpretation where (∀x)(∃y)Ryx is true but (∃y)(∀x)Ryx is false.

Im honestly so confused on how to complete both of them. From what I know, constructing the truth tree will be overly complex, long, and almost infinite, and Im not quite sure what it means to construct an interpretation? Do I recreate another truth tree? Do I just write down in english what I think the interpretation might be (idk i missed most of my classes for this cuz I prioritize sleep rather than attending class, and thats my biggest regret) but ive been pondering on this question for hours and its getting me no where. Please help me

can somebody tell me the mistakes made in this proof?

I am having trouble understanding what the law of excluded middle means, and I think it's because I don't understand what negation means. The law of excluded middle says that either a proposition or its negation are true.

Let's suppose that we try our best to break the LEM. Suppose that, in some silly world, being tall means you're over 1.8 meters in height, and being "not tall" means you're less than 1.6 meters in height. Suppose that Jack is 1.7 meters in height. So, he's not tall and he's not not tall.

Consider the proposition "Jack is tall." This proposition is false, since Jack is not over 1.8 meters in height.

If the negation of this proposition is "Jack is not tall," then the negation is false, since Jack is not under 160 centimetres in height. Thus, we have succeeded in breaking the LEM.

If the negation of this proposition is "It is not true that Jack is tall," then the negation is true, since it is indeed not true that Jack is over 180 centimetres in height. Thus, despite my best efforts to break the LEM, it holds.

Which of the two interpretations of that proposition's negation is the correct one? Or are they the same statement?

Is there a formal logic system that can effectively capture and represent hermeneutics and/or other theories of literary criticism and methods in humanities?

LNC : Law of Nonctradiction.

I’m sort of lost on which rules of implication or replacement to use as well as how many steps it will take for me to reach the conclusion above and need some advice. Thank you and I appreciate the assistance.

I’m going through Peter Smith’s Introduction to Formal Logic (again). 

I think this exercise is hard: show that the parse trees of wffs are unique* 

I have a hard time following the answer provided by Smith. Do you have any resource that explains this better? Or, alternatively, could you do it? 

Here is how Smith shows it, from the Answers to Exercises: 

(c*) Show that parse trees for wffs are unique.

(0) The wffs of a particular PL language are determined as follows. Having explicitly specified its

atomic wffs, then:

(W1) Any atomic wff of the language counts as a wff.

(W2) Ifαandβarewffs,sois(α∧β).

(W3) Ifαandβarewffs,sois(α∨β).

(W4) Ifαisawff,sois¬α.

(W5) Nothing else is a wff.

The ‘extremal’ clause (W5) ensures that every wff must have some constructional history, some parse tree starting from atoms, recording a way it can be built up according to the principles (W2) to (W4).

One immediate consequence is that since brackets are always introduced in matching left/right pairs, every wff must have the same number of left-hand and right-hand brackets.

Suppose then that we look at a parse tree for a wff (at this point in the argument, we are not assuming uniqueness, just relying on the fact that there is at least one parse tree). When an occurrence to a binary connective, say ∧, is first introduced at some point on a branch of this parse tree, it is in a (sub)formula of the form (α ∧ β), where α and β are wffs. And hence (since α is balanced), this connective ∧ is preceded by one more left bracket than right bracket (and succeeded by one more right bracket than left bracket).

Now suppose that, as we go up the parse tree, this expression of the form (α∧β) becomes part of a longer formula formed using a binary connective, perhaps ((α ∧ β) ∨ γ) or (γ ∨ (α ∧ β)). In this sort of case, that occurrence of ∧ will now be preceded by two more left brackets than right brackets (and succeeded by two more right brackets than left brackets). And as a binary connective gets buried deeper by the application of more connectives, it will acquire a greater excess of left brackets on its left (and symmetrically, a greater excess of right brackets on its right).

And so it goes. Generalizing, we have . . .

(1) If a binary connective ∧ or ∨ is the main connective of a wff of the form (α∧β) or (α∨β) then the relevant occurrence of the connective ‘∧’ or ‘∨’ is preceded by exactly one more left-hand bracket than right-hand bracket.

Any other occurrence of a binary connective in that wff will be preceded by at least two more left-hand brackets than right-hand brackets.

(2) You know that if a wff starts with a negation, it must have the form ¬α, with α a wff.

And if it starts with a left bracket and ends with a right bracket, you now have a way of assigning it the form (α ∧ β) or (α ∨ β), with α and β wffs – count brackets until you find the only binary connective which is preceded by exactly one more left bracket than right bracket.

(3) So now we have method of disassembling a complex wff stage by stage, building a parse tree downwards as you go. Here’s one way of describing it:

(i) If a wff γ at a ‘node’ on the tree starts with a negation, it must have the form ¬α; continue the branch of the tree downwards from that node by writing α beneath.

(ii) If a wff γ at a node on the tree starts with a left bracket and ends with a right bracket, it must have the form (α ∧ β) or (α ∨ β). Then the relevant occurrence of ∧ or ∨ is the only occurrence of a binary connective which is preceded by one more left bracket than right bracket. Find it! Take the preceding part of γ, minus its initial left bracket: that is to be α. Take the succeeding part of γ, minus its final right bracket: that is to be β. Then, from the node with γ, continue the parse tree by writing α beneath to the left, and β beneath to the right.

Hello!

I am an undergraduate student currently taking Intro. to Formal Logic. My course is using this as our text, and currently we are learning Proofs (§1.4, §1.5, §1.6)--I am having trouble locating supplemental materials to help me better understand proofs and the rules the textbook/my professor want me to use to solve them.

I've watched quite a few youtube videos from William Spaniel's Logic 101 series, but they are not matching up with what is in my text. We are permitted to use the Logic Daemon to check our proofs.

Does anyone have recommendations for videos walking through these types of proofs? Or other learning materials--I am not understanding it based on the textbook alone, and the class overall is not helping. I really appreciate any help!

The error in this case being that the sentence has no error. It doesn't feel quite like a paradox of self reference, since the statement is true in any perspective

The argument:

P1: The testimony of the trustworthy is reliable

P2: John is trustworthy

C: Therefore, the testimony of John is reliable

-----

Moreover, what is "the testimony of the trustworthy" or "the testimony of John" considered? They're the subjects in their respective sentences, but are they considered proper names? Or descriptions?

I need help with deriving ⊢ ((∀x)Fx ∨ ~(∀x)Fx). I have been working on this for hours without success. I'm attaching the attempt I made at solving this along with the rules we're using for my class.

edit: thank you to everybody who responded! I was able to figure it out with all of your help :)

https://preview.redd.it/d5xjq1unz53e1.png?width=2440&format=png&auto=webp&s=7af9ff075822c385291cef11b3b8357b6d55ae9b

https://preview.redd.it/pmikvztnz53e1.png?width=1546&format=png&auto=webp&s=c1e5a625f88e0e7375445ca8afa23179384a68fe

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https://preview.redd.it/tlx1mztnz53e1.png?width=956&format=png&auto=webp&s=28ebd3ba5c8a3dd8369c2257ba3501fc08e7ddee

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I want to make known this strange logic theory of Dr. Koza Uchitelievich Cantero-Rada. He is an expert in proto-indoeuropean studies (PIE) and ancestral indoeuropean drum theory (AIDT). With this knowledge he propose the use of a new connector:

To provide a truth table for the onomatopoeic connector (denoted by ↻), we first need to specify how this connector is defined in the context of formal logic. In the original proposal, ↻ is a connector that reflects an interaction between propositions with a sort of "resonance" or "onomatopoeic effect," meaning that its logical behavior should reflect some specific semantic or phonological property of the propositions involved.

Assumptions:

1. The connector ↻ could be seen as a connector that does not behave traditionally like standard logical connectors (such as ∧, ∨, →, ↔, etc.), but instead adds an "effect of resonance" or contextual influence.
2. One way to conceptualize this connector could be that it modifies the truth of a proposition according to the "affect" of the other proposition it is connected to, something akin to a non-binary interaction in which the effect of one proposition can alter or influence the other proposition.

Proposal for the ↻ Connector:

In simple terms, we could conceptualize the connector ↻ as a way to modify the truth of one proposition according to the "affect" or "interaction" with another proposition, almost like a resonance effect.

1. If both propositions are true (V), ↻ keeps them true, with a greater emphasis on the "mutual resonance."
2. If one proposition is true and the other is false, ↻ could produce a "modulation" or "resonance" effect, leading to a more complex proposition that depends on the nature of the interaction.
3. If both propositions are false (F), ↻ could imply a kind of "nebulization" of truth, where the resulting proposition is also false, but with a "void" that reflects the lack of resonance between the propositions.

The truth table could look something like this (based on the previous proposal):

Explanation of the table:

• P ↻ Q = V when both P and Q are true, indicating that the resonance between the two propositions reinforces the truth.
• P ↻ Q = F when P is true and Q is false, or vice versa, indicating that the connector introduces a modulation effect on the truth, affecting the resulting proposition.
• P ↻ Q = F when both are false, reflecting the lack of resonance between the propositions and producing an empty or null truth.

Also;

If you want the ↻ connector to have a more complex or nuanced interpretation, additional rules can be introduced, such as:

• Modifying the truth table so that the connector acts differently when P or Q are contingent, i.e., in situations where there is no absolute truth (neither completely true nor completely false).
• Incorporating more semantic dimensions, such as context or tone (in linguistic theory), which could influence how the truth values of the connected propositions are interpreted.

As you can see the ↻ connector is far from being a traditional logical connector, but it could be a creative and flexible connector in extended logic, especially if we consider that it introduces a form of resonance or modulation between propositions based on certain linguistic or phonological principles. The truth table above is just a basic proposal that would need to be further expanded and justified according to the semantic principles guiding this new connector.

If you want more information you can consult his research institute AIDTRI. Thanks for your interest.

Hi guys,
i'm a cybersecurity student and on 20th december i have my math logic exam. There are some topics that i haven't understand at all.

Do you have any suggestions to learn this (also with exercises) in a good way? some solved exercises or usefull material?

(resolution is like hell :( )

PREDICATE LOGIC. Syntax and semantics of predicate logic. Deductive systems of predicative calculus: calculus of sequents. Predicate normal form and Skolem's form. Semidecidability of predicative logic. Translation from natural language.

RESOLUTION. Unification algorithm. Methods of propositional and predicative resolution.

BINARY DECISION DIAGRAMS (OBDD). The representation of Boolean functions with OBDD. Reduction of an OBDD. Logic operators and the Apply function.

FORMAL VERIFICATION OF PROGRAMS. Hoare's triples. Rules of computation for partial correctness of programs. Calculus rules for total correctness of programs.

MODAL LOGICS. Syntax and semantics of modal logics. Examples of modal logics. Kripke's model.

LOGIC FOR SECURITY. Syntax and semantics of BAN logic. Analysis of the Needham-Schroeder Protocol.

The title is probably kinda confusing so let me explain. So, natural language (like english) is kinda vague and can have multiple different meanings. For example there are some words that are spelled the same way and only the way of telling them apart is from context. But formal logical languages are certain in the sense that there is only one meaning a logical formula can have (assuming you wrote it correctly). But when we're first teaching logic to people, we use natural language to explain the more formal and rigid logical language.

What i don't understand is how we're able to go from natural language (which can be vague sometimes) to a logical one thats a lot more rigid. Like how can you explain something thats "certain" and "rigid" in terms of "vague" and "uncertain" things? I just don't understand how we're able to do the jump.

Sorry if the question doesn't make sense.

In Natural Deduction systems, how do we prove the rules of inference? If we can't prove them, doesn't that effectively renders them to axioms?

• hello does someone have the answer sheet for the second edition of language, proof and logic. im in my first year of the bachelor AI and we do not get any of the answers except for the ta's

Sorry i know that this is not a good question but maybe if you responsabile me you will respond to a lot of people, i love logic and i love math but Idk where i can start study logic or if there are some website that can help with that. I apologize for my english and good night or morning