Good afternoon,  I've looked at some other answers and I feel those don't quite answer what I'm trying to ask, so I'll just ask it here.  I'm in a logic course and I'm either misunderstanding something or my teacher said something wrong.  The example given was: All cats are animals  Some cats are animals  He argued this was valid from the Aristotelian stand point, as cats really do exists, but invalid from the Boolean stand point (as an existential fallacy) because "All cats are animals" doesn't imply existence, so we can't conclude that "some cats are animals".  The next example was:  All unicorns are animals  Some unicorns are animals.  He argued this was invalid from both the Aristotelian and Boolean stand points because, for the Aristotelian, unicorns don't exist and therefore it commits an existential fallacy. But that it's also invalid from the Boolean stand point because of the same reasons the previous example was invalid.  My confusion comes from the fact that I thought the only difference between Boolean and Aristotelian stand points, for these examples, would be whether the subject (either cats or unicorns) actually existed.  I would have figured the first example would be valid from both Aristotelian and Boolean view points, and the second would be only invalid to the Aristotelian view point.  Can someone please explain?  Thank you!
103720
In this hypothetical:  Firefighters always tell the truth, while politicians always tell lies.  Suppose three people, who are either a mix of firefighters and politicians, all politicians, or all firefighters, surround you and say "every politician in this circle will have a firefighter to their immediate right."   So is the instance where all three are firefighters possible? It would seem if "every politician in this circle will have a firefighter to their immediate right" defaults to true if there's no politicians it would be, otherwise it wouldn't. Which is it?   Edit for clarification in comments: The proposition "every politician in this circle will have a firefighter to their immediate right" cannot be proven, as there are no politicians in the circle. So if the proposition defaults to true, the firefighters were telling the truth and this specific scenario is possible, otherwise it is not, as this would mean the firefighters lied.
113904
I've been seeing arguments of this form for awhile:   (agreed true premise) Some predicate P was applicable to behavior/event/thing A in the past. Variations: P(A) may have been true at some particular past time t, or over all past time T. (fallacious consequence) Therefore P(A) is true now. Variations: P(A) must be true now and all futures.   But I haven't known what to call them (e.g., when "calling out" their utterers). The problem seems to me to be due to ignoring (empirical) context and quantity (but has nothing to do with logical quantification). Some examples:   Mass immigration was "good for America" in its (past) history. Therefore mass immigration must be "good for America" now. (P=good for America, A=mass immigration) (even more logically problematic) Humans have been burning fossil fuels for centuries, and "there was no crisis then." Therefore, to claim that fossil fuel combustion is "a crisis now, one must believe that we've been in crisis for centuries." (P=not crisis-producing, A=fossil fuel combustion)   Does this fallacy (or cognitive bias, but this failure "feels logical" to me--ICBW) have a widely-accepted name? Given the extensive and systematic attention to logic and rhetoric over millenia, I suspect this has been previously taxonomized.
94629
(p → q) ∧ ¬p ⇒ ¬q  I have tried to use the rules of inference in many ways to prove this argument to be valid. However, I cannot seem to find a way to validate the argument, but I am afraid I have missed something.  Using a truth table I find that when p and q is 0, then p → q is 1, ¬p is 1 and ¬q is 1. Would that not make the argument valid?  I apologize in advance if I have misunderstood the whole concept. Any answer would be much appreciated. Thank you
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states that "...For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system".  What does it mean that a statement is true if it's not provable?  What is the difference between true and provable?
58684
Abbreviate: Existential Statement to ES, Existential Instantiation to EI, Universal Instantiation to UI, and Universal Statement to US.   Source:  A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley           [p 468:] [...] Notice that the line involving existential instantiation is listed   before the line involving universal instantiation. There is a reason for this. If the   order were reversed, the existential instantiation step would rest on the illicit assumption that the something that is both an A and a G has thesamename as the name   used in the earlier universal instantiation step. In other words, it would involve the   assumption that the something that is both an A and a G is the very same something   named in the line Ad ⊃ Cd. Of course, no such assumption is legitimate. To keep   this mistake from happening, we introduce   [...]   [p 469:] These two restrictions can easily be combined into a single restriction that requires that   the name introduced by existential instantiation be a new name that has not occurred   in any previous line, including the line adjacent to the last premise that indicates the   conclusion to be derived.   I recognise the possible problem that I coloured grey in the above quote; but how is it a genuine problem? And why does it necessitate EI always to precede UI? I agree that writing EI before UI is visually convenient, but does the grey above exaggerate?  To me, the illicit assumption can be evaded simply by knowledge of EI vs UI. Only US (Line 1) are true for ANY value of x; so only US empower you to choose whatever Constant (ie: letter) you need, such as the specific Constant  needed  to match the Constants in any subsequent PS.  One must start always by applying EI to the ES (Line 2) which generates the first Constant, before one can even know which Constant needs to be matched from applying UI to the US.   So knowing the previous sentences, how can someone be confused by whether the 'something' in EI is identical to what is named in the UI? If they are not identical, then EI and UI would use different Instantial Letters to name the 'something'!
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There are many refutations of John Searle's t against Strong AI. But they seem to be addressing the structure of the thought experiment itself, as opposed to the underlying epistemic principle that it is trying to illustrate.   This principle is "syntax is not semantics" (See  by John Searle): At the end of the day, computer software, even the most advanced AI conceivable, manipulates symbols according to a set of syntactic rules, regardless of their meaning.   Anybody who has studied formal logic knows that rules like De Morgans laws or the laws of idempotency ( e.g. A ^ A = A ) are independent of the meaning of the symbols being processed.   This idea, that syntax is independent of semantics, and therefore a computer can function perfectly  without ever knowing the meaning of what it is computing seems like a much stronger argument against AI, and Mind-Body functionalism in general, than Searle's original Chinese Argument.   What are the main refutations advanced by proponents of functionalism and strong AI specifically of the "syntax is not semantics" argument?
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Excerpt from a mathematical proof that serves to prove by contradiction, using the Well Ordered Principle, that predicate P(n) is true for all nonzero integers n.     If n belongs to set C i.e. n >= c, then P(n) is false.      This implies the contrapositive,      If P(n) is true, n has to not be a part of set C i.e n &lt; c.   I know that the above implication is valid but why so? How should I know that n cannot take on 2 values simultaneously -- n being both >= c and &lt; c, thus P(n) being both true and false simultaneously? (Yes I am thinking Dialetheism, maybe even quantum logic but I know nothing other than that quantum logic defies certain aspects of propositional logic.) What is the basis behind a predicate (or any proposition) being either true or false? Or would the simultaneity be possible just that I wouldn't be dealing with a proposition to begin with?  Also, if you don't mind, please do let me know if you feel my understanding of concepts and terminologies (i.e. discrete, predicate, contrapositive, assume) are erroneous. I am extremely new to discrete mathematics and could really use the extra guidance.  Thank you.
53285
I found this proposition and don't see exactly as to why it is true and even more so, why the converse is false:  Proposition 1. The equivalence between the proposition $z \in D$ and the proposition $(\exists x \in D)x = z$ is provable from the definitory equations of the existential quantifier and of the equality relation. If $D = \{t_{1},t_{2},...t_{n}\}$, the sequent $z = t_{1} \vee z = t_{2} \vee z \vee ... z = t_{n}  \vdash z \in D$ is provable from the definition of the additive disjunction $\vee$.  The converse sequent $z \in D \vdash z = t_{1} \vee z = t_{2} \vee ... z = t_{n}$ is not provable.  The author goes on to say:  "We adopt the intuitionistic interpretation of disjunction. With respect to it, one can characterize a particular class of finite sets"  On the first part of the proposition, well I am not sure what the point is as if we take an element z in D, then we could just call this element x and hence this x = z.  Is there something more to this?  On the second part of Proposition 1 since $z = \text{ some } t \in D$ since t is in the set of axioms and $t = z$, then $t \vdash z$ as z is derivable from t.  For the converse if $z \in D$ then why wouldn't z = some $t \in D$?  Is this because of the Incompleteness theorem?  That perhaps there D as a set of axioms has some consequence which can not be proven by the set of axioms in D?  Or perhaps I am way off here.  Any ideas?  Thanks,  Brian
71276
From reading this  on Hume, having read the first seven sections of the Enquiry Concerning Human Understanding (which covers the relevant section), a further question hit me. Hume dismisses the occurrence as "singular ... scarcely worth our observing", yet he brings the counter example up instead of others. Why does he bring up a counterexample which he himself dismisses as trivial? Is it to bolster his argument by suggesting that the only counterexample so important as to be worth mentioned is itself highly trivial? Or is it merely because its the only concrete, simple counterexample Hume conceived of? Responses with links to Hume scholars would be appreciated.
57027
I am reading Velleman's "How to Prove it", and in one example he proves  $\neg \exists x P(x)  \implies \forall x \neg P(x)$    in the following way:    Suppose $\neg \exists x P(x)$. Let x be arbitrary. Suppose $P(x)$. Since we have a specific $x$ for which P(x) is true, it follows that $\exists x P(x)$, which is a contradiction. Therefore $\neg P(x)$. Since $x$ was arbitrary, we can conclude that $\forall x \neg P(x)$, so $\neg \exists x P(x)  \implies \forall x \neg P(x)$    What makes me confused is the line in bold text, that is "Suppose $P(x)$". The reason is that he writes that he is doing proof by contradiction, but $P(x) \not\iff \neg\forall x \neg P(x)$. He is not treating all possible cases where $\forall x \neg P(x)$ is false by assuming $P(x)$, does he? So how can this be a valid proof by contradiction?    Edit: I deleted my first example since it was clearly not representative for the problem.  Edit 2: So the thing that I am opposing I guess is that $\neg \forall x \neg P(x)$ is not $P(x)$. And from what I have undestood, a proof by contradiction should use the negation of the consequent which I do not think that Velleman does.
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It's possible to imagine something, for example a table, we see one everyday and can bring it in front of our minds eye (although it's a moot point whether we can see it - I certainly don't). But of course this is a real object so we have a referent. But we need not have a referent: to imagine a unicorn means we are hybridising our referents.   But to imagine nothing in its proper sense (ie not the lack of something or just space) seems impossible to me, we cannot avoid our sense of ourselves.
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I have an argument with the conclusion: 'The sky is blue and it is not the case that the sky is blue'.  Without knowing the premises of this argument, am I right in saying this argument is invalid since the this conclusion is necessarily false?
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There are things which are illogical/logically impossible (like saying that 2+2=4 and 2+2=5. Without changing anything in the axioms of mathematics or logic, this would be a contradiction and would be inconsistent and illogical/logically impossible.   There are other types of logic systems apart from classical logic, like paraconsistent logics or even trivialism, that allow these contradictions to occur, prove them as right and work with them.   We can make a paraconsistent or trivialist system and work with it. For example, with trivialism, in theory, we would be able to derive and state everything we would want (since literally everything, even including illogical/logically impossible inconsistencies and contradictions), but we as humans (or as brains), are limited and can't conceive everything we want (at least to what I know). Therefore, no matter how many trivialist models we create and how much time we would work with them, we would never find or conceive many illogical/logically things because they are just that: impossible. There are impossible things to describe and conceive. For example, Russel's set is the set of all sets that do not contain themselves. If Russel's set contains itself, then it cannot contain itself, since it only contains sets that don't contain themselves. But if Russel's set does not contain itself, then it must contain itself, since it contains all sets that don't contain themselves.  There are quite a few logic bombs like this. You cannot ever compute the contents of Russel's set, and there are more formal, mathematical ways to present it. All of them have in common that you can't actually compute what the set is, whether you do it by hand, in your head, or on a computer. It's just a statement that cannot be fully logically processed.  If you take every possible state the human brain can be in, none of them include the computation of Russel's Set's contents. That is, not only can the contents not be computed, they cannot even be represented. No stimulus can cause us to comprehend Russel's Set, since such comprehension is not possible to begin with. It does not have a solution. Even if we try to solve it using trivialism, we would just be able to write a solution that does not make sense and prove it has sense and it is the real solution, but we could not be able to have a solution that would make sense "outside" the realm of trivialism (for example in classical logic), even though, using trivialism, we could prove that such solution would have sense in whatever context and logic system.   But what about hypercomputational machines (for example oracle-like machines)? I've read about some models of hypercomputation which are compatible with paraconsistent or trivialism logics. I've also read there are some models of hypercomputation (particularly those oracle-like models which use a black box) where, essentially, the hypercomputer is an algorithm that cannot exist. It might be because such an algorithm is fundamentally forbidden by logic itself (which is hided in a black box). Would any of these be capable of computing/"conceiving" these impossible things I wrote before? Do you know of anything that would help?
28219
I am reading nested quantifiers. I am confused in between two cases,   1. Existential Quantifier before Universal Quantifier  2. Universal Quantifier before Existential Quantifier   I would be very thankful if someone highlights the difference between them and also give an example.   Thanks in advance.
20540
Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction.   My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?  That, to me, seems the exact opposite of logical. It sounds like we say "so, I'll have a really big problem if this thing isn't true, so out of convenience, I am just going to act like it's true".
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As is well known, the General Set Comprehension Principle (any class is a set) leads to the Russell Paradox (the class $x \notin x$ cannot be a set). As a result, set theories must restrict the Comprehension Principle to avoid self-reference. For example, in the case of ZFC, this is done by enumerating a small list of "safe" comprehension schema such as Separation.  Does General Comprehension lead to other types of paradox than Russell's? Or is this basically the only thing that can go wrong?
23797
let D be a deductive system. The system's axiom set, A, includes all propositions that are not tautology, and the only rule of inference is $\frac{\alpha \vee \beta}{\alpha \wedge \beta}$. I need to answer:  Show with induction on the proof that $ \vdash \alpha \Longrightarrow \nvDash \alpha $, i.e if $ \alpha $ can be proven from the empty set, then exists a model $ M $ such that for every $ \varphi \in A $, $ M(\varphi)=T $ and $ M(\alpha)=F$. Is the system strongly sound? Is $X\nvDash \alpha \Longrightarrow X\vdash \alpha$? Prove that the empty set isn't consistent in D    I've already asked about this part of the question here but I still have a problem with it, because every axiom, $a\in A$,  $\vdash a$ and $A \models a$, in contradiction to what I'm supposed to prove here. Moreover because the axiom set includes all what isn't tautology it also includes all that is contradiction, which means that there is no model $ M $ such that for every $ \varphi \in A $, $ M(\varphi)=T $ which means for every proposition $\psi$, $A\Longrightarrow \psi$. This one was fairly easy base on (1) the system isn't sound so it can't be strongly sound. I'm not even sure how to approach this, I don't even have an intuition whether this is correct or not. Assuming this is wrong, I need to find a set $X$ and proposition $\alpha$ such that $X\nvDash \alpha$ but $\alpha$ cannot be proven from $X$, and if it's correct I need to show that for every set, $X$, and every proposition $\alpha$ such that $X\nvDash \alpha$, $\alpha$ can be proven from $X$ On the surface this looks pretty easy, both $(\varphi \wedge \psi)$ and $\neg(\varphi \wedge \psi)$ are axioms, as they aren't tautology, hence $\vdash (\varphi \wedge \psi)$ and $\vdash \neg (\varphi \wedge \psi)$ and the empty set isn't consistent but this seems too easy, like I'm missing something.   EDIT: I might have come up with a solution for (3), if $X\nvDash \alpha$ it means that $\alpha$ isn't a tautology and if it isn't a tautology it is an axiom in D and therefore can be proven from $X$