10

First, your premises are inconsistent: your second premise implies that turtles do not see other turtles, or themselves, yet, according to the first premise, they see everything. So, taking y=x, we can deduce both ∀xSxx and ∀x¬Sxx. After that you can deduce whatever you want directly using the law of explosion, contradiction implies anything. If you fix ...


6

The following truth table shows that ~(A↔B) → ~(A→B) is not a tautology: If A is False and B is True then the antecedent is True but the consequent is False making the conditional False. Because the truth table does not show a tautology, one should not be able to derive a natural deduction proof of the result. Michael Rieppel. Truth Table Generator. ...


4

...it seems that any inductive reasoning can be done with deductive reasoning by adding in some assumption that a particular pattern continues to hold. You got it exactly right. The assumption is the Uniformity Principle. Some philosophers have accepted the principle as supported by common observation; others have dropped wet blankets all over it. ...


4

You can't derive ~(A→B) from ~(A↔B). Consider: A = I'm in Paris. B = I'm in France. ~(A↔B) is true, because being in Paris is not equivalent to being in France (I could be in France but not in Paris). But ~(A→B) is false, because if I'm in Paris then necessarily I'm in France. So you can't derive ~(A→B) from ~(A↔B).


3

The Principle of Identity is the Principle of Identity, or The Principle of Non Contradiction is the Principle of Non Contradiction. It is not right to say, of the propositions making up this disjunction, that "the law of excluded middle requires one to be negated". I'm not exactly sure what is causing the confusion, but it seems to me that you think the ...


3

It is not the case in general that inductive reasoning is deductive reasoning from a hidden premise. In particular, the idea that a series of observations that instantiate some pattern can be projected or extrapolated over unknown cases by appeal to a general principle of the uniformity of nature does not hold water. The principle of the uniformity of nature ...


2

By hand, I get: 1. |_ ~(~P&Q) & ~(P&Q) A 2. | ~(~P&Q) &E 1 3. | ~(P&Q) &E 1 4. | |_ Q A 5. | | | P A 6. | | | P&Q &I 4,5 7. | | | ~(P&Q) & (P&Q) &I 3,6 8. | | ~P IP 5-7 9. | | ~P & Q &I 4,8 10. | | (~P&Q) & ~(~P&Q) ...


2

Using DC Proof 2.0 (another proof editor and checker)


2

For the first link here is a screenshot of how to enter the premise and conclusion: Note that the FOL (First Order Logic) button is on, not the TFL (Truth Functional Logic) button. The default is TFL. That would trigger a premise not being well formed message. Note that "(y)" is entered as "Ay" without parentheses and with and "A". Note there are no ...


2

Whenever I see inductive arguments being used, it seems as though they can be redone by simply making certain assumptions and rephrasing the argument as a deduction from those assumptions. For example, in this Khan Academy video, Sal says that if you're predicting the population of a town in the future based on the past, that's inductive reasoning. ...


2

Some of the rules that might be present in another system may have to be derived separately in this proof checker as they are needed. Here is the question: I'm looking for Hypothetical Syll, Constructive Dilemma, Communication, association, distribution, transportation, material implication, material equiv, exportation, and tautology (even though it's ...


1

This sounds like a HW question, so you need to edit your question to contain your argument for finding answers 3 and 4 correct, so that instead of just giving you answers, you are forced to defend your reasoning and learn from the question. This also seems to be a duplicate post here. It would help to keep in mind that deductive inference is that which has ...


1

Initially, mathematics was the systematic deduction of the logical consequences of axioms, axioms understood as constituting together, somehow, a model of some particular aspect of the real world. The most historically glorified example of that is probably Euclid's geometry: Euclidean geometry is an example of synthetic geometry, in that it proceeds ...


1

| 1. P > (S v R) |_ 2. ~((~P v ~Q) v (R v ~L)] To conclude S from the first premise you need to derive P and ~R from the second. So, judging by your rule abbreviations , your proof should look somewhat like> | 3. ~(~P v ~Q) ^ ~(R v ~L) DM 2 De Morgan's | 4. ~(~P v ~Q) SIMP 3 Simplification | 5. ~~P ^ ~~Q ...


1

Here is a solution to compare with what you have. Also you might find the proof checker helpful to check the other proofs you are asked to do: For this proof checker DeM is De Morgan rule, ∧E is conjunction elimination, DNE is double negative elimination, →E is conditional elimination and DS is disjunctive syllogism. Kevin Klement's JavaScript/PHP Fitch-...


1

Deduction and induction are not about observation, but certainty in inference. It may be tempting to define deduction as moving from general to specific claims, and induction vice versa, but this is not entirely accurate. Let's take an example to show the difference between deduction and induction. DEDUCTION: If a man is in a kitchen, then he is in the ...


1

The problem of induction discovered by the Scottish philosopher David Hume is quite well known. Induction is allegedly a process that starts with observations, uses them to derive a theory and then shows the theory is true or probable or good or something similarly vague. The problem of induction is that such a process is impossible. Any set of observations ...


1

Only if what genes do is induction... There is a long history from theological arguments to Kant, of considering human logic an absolute limit on our thinking, perhaps inferior to the logic that may actually control the world, but the best we can relate to. On a more modern level, logic does not appear to result from observation. Babies are given pause by ...


1

You are tacitly assuming that if you want X, you ought to take the means to obtain X. Whether or not Hume would allow the premise, if you do allow it, then "I want X" is a claim about an ought (about a value rather than a fact), so therefore your whole syllogism adheres to Hume's principle: reason can't get you a conclusion about ought from premises about is....


1

The premise ∃x∃y∀z(x = z ∨ y = z) says that there exists at most two distinct values in the domain. The conclusion ∀x∀y(¬x = y → ∀z(x = z ∨ y = z)) says that for any two distinct values in the domain, there does not exist a third. Clearly we need to utilise existential elimination and universal introduction. ∃-elim here will be tricky. The 'trick' is ...


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