New answers tagged

1

Contrary to @Mauro ALLEGRANZA's reply, there are cases where mathematicians rely on implication where the premise is false. This occurs in certain theorems where either the statement, or the proof, can be rendered 'cleaner' (in the sense of having fewer assumptions in the statement, or better proof structure) by exploiting this feature of implication. A ...


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I'm not sure the fully understand your question, but I'm trying to answer, reading the title-question: Do mathematicians care about implications where the hypothesis is always false? as regarding the well-known issues about the truth-functional definition of the "if..., then..." connective, and specifically regarding the fact that "if P, ...


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It comes down to how you interpret the meaning of the sentence "x has radius y". (As has been pointed out in the comments, one can meaningfully define the radius of a triangle, but for the sake of argument we take it be defined only for circles.) Consider the following two interpretations: (1) x is a circle that has radius y. (2) x is a circle that ...


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Taking your questions in a different order. #3. Are the laws of thought logical absolutes? #4. If so, in what sense are they absolute (logically)? The idea that logic is concerned with the 'laws of thought' is quite old-fashioned. Since Frege it has become more common to think of logic as being concerned with the relationships of consequence between ...


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1: Everything happens for a reason 2:nothing happens without a reason. In 1, maybe Reason is there but nothing happens but in 2 there has to be a reason for things to happen. I definitely see a difference between the two. So I translated the propositions into French and,again, perceived a difference between the two: 1 (il y a une raison pour tout) has a ...


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Regarding the example, we can apply Russell's analysis of definite descriptions to the statement "the Radius of Triangle has lenght 6" (of the general form "the F is G") : exists x ((Rad(x,T) and for all y (Rad(y,T) → x=y)) and x=6). Thus, mimicking Russell's example regarding "the current Emperor of Kentucky is gray", if we ...


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In Lukasiewicz three valued logic, Neither the LNC nor the LEM is a necessary truth for all propositions. They are contingent statements, true for some propositions but not necessarily true for all. It may be of interest to note that neither is ever actually false. They are, however, equivalent as in classical logic. This is true for the standard negation, ...


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We may introduce two predicates, thinghood, τ(x), and happening for a reason, ρ(x). Then we can translate the statements into the standard first-order language as follows: ‘Everything happens for a reason’ ∀x(τ(x) → ρ(x)) ↔ ∀x(¬τ(x) ∨ ρ(x)) ‘Nothing happens without a reason’ ¬∃x(τ(x) ∧ ¬ρ(x)) ↔ ∀x(¬τ(x) ∨ ρ(x)) We see that they are logically equivalent. ...


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The following proposition: A and B then C Can be expressed (not always) as A + B = C In the second form, A is implicated in C. There's no sequence in time. The system A is part of C, or it has a relationship with C. That's the meaning of implication. In the first form, A is causal of C, meaning that when A happens, C could follow. The word follow expresses a ...


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Axioms are not in actual practice/history, chosen at the beginning of mathematising, but are part of a specific type of project, to work backwards to find logical grounding. It began with geometry, as any "proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., ...


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Welcome- This quotation is from Spinoza's 'Ethics' Part One- Concerning God. It is from Proposition 8- Note or Explanation or Scholia [they are synonymous] #2. Here's the English: Note II.—No doubt it will be difficult for those who think about things loosely, and have not been accustomed to know them by their primary causes, to comprehend the demonstration ...


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Given statements, no M is B S is not M Drawing Venn's diagrams for the above: Possibility 1: Possibility 2: S being B is a possibility. S not being B is also a possibility. Smith given bail is a possibility. But also, Smith not given bail is a possibility. https://en.wikipedia.org/wiki/Affirmative_conclusion_from_a_negative_premise


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Implication ("If-then"), Material Implication, Material Equivalence, Logical Implication, Logical Equivalence! P -> Q means: P 'implies' Q, where 'implies' can be rendered into the following conditional form: "If P, then Q". The symbol (->) denotes "material implication", which sets up a sufficient condition between P and ...


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The Liar's Paradox; Aristotle's Laws of Thought: Laws of Non-Contradiction (LNC), Excluded Middle (LEM); Plus, the Law of Bivalence (LBi) Let: X := "This statement (X) is false". QUESTIONS to Consider: Q1. What is a proposition, i.e., what is the logical definition of a proposition? Is X a proposition? Q2. Is it possible for a proposition to be ...


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Don't need to go far to describe the problem, which is purely conceptual. A fact is an observation performed in the past (or the present, which is just our short-term memory, that is, a recent past). Not in the future. A situation that will occur in the future can never be a fact, including the sun coming out tomorrow. It is highly probable, but it is not ...


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By definition, it is either true now or it is false now that the slide will be finished tomorrow. The fact that something is true doesn't imply that we know it is true. A fact is a state of affairs we know to be the case. It is certainly not the case that anyone would know for a fact that some slide will be finished tomorrow. We don't even know that there ...


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No, person B is stating a future contingent until the truth is actually determined. Some other statements about the future might arguably be facts though, like "the sun will rise at such time tomorrow". If person B says "I want to finish my slide tonight" that can be a fact, if it is actually true. What matters is if it's true or not. ...


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The syllogism has two negative premises. The fallacy is “exclusive premises“. Also, the syllogism draws an affirmative conclusion from a negative premise. See, e.g., Wikipedia > Syllogism.


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no M is B s is not M s is B Observe that #1 is equivalent to "If x is M, then x is not B." So, #1 and #2 do not imply #3.


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The Cartesian Circle is an example of circular reasoning being problematic. One thing leans on another, and there's no structural integrity created. For me the solution to the trilemma is in the total structure, beginning where we are, and forming strange loops, both 'jumping out' into meta-thinking (at risk of infinite regress), circular reasoning but with ...


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Informally, circular reasoning is seen as analogous to trying to lift yourself up by your own bootstraps. I can lift you up, or you can lift me up, but we can't both lift each other up at the same time, because it leaves us no foundation of support. More formally, the way the concept of proof is constructed in systems of structured logic makes it impossible ...


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Material Implication, Logical Equivalencies, Converse, Inverse, Contrapositive Given the material conditional P -> Q, P is termed "antecedent" (condition) and Q is termed "consequent" (consequence), where P -> Q reads "P implies Q" and means that P is a sufficient condition for Q which in its turn means that Q is a ...


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Given the material conditional P -> Q, P is referred to as "antecedent" and Q is referred to as consequent in this form (forward implication from P to Q). P -> Q means "P materially implies Q" which is stated as the following material conditional (if-then) statement: "If P, then Q". The material conditional P -> Q ...


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Material Implication, Logical Equivalencies, Proof by Contraposition: I demonstrate that the contraposition of a material implication (X), called the contrapositive of that implication (X*), is logically equivalent to that material implication (X): X ≡ X*. Given the material conditional P -> Q, P is referred to as "antecedent" and Q is referred ...


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The underlying rule that justifies this reduction is distributivity of disjunction over conjunction: A v (B ^ C) ⟚ (A v B) ^ (A v C) (~p v r) ^ (~q v r) ⊨ (~p ^ ~q) v r is an instance of the backwards direction of this equivalence with A = ~p, B = ~q and C = r. Note that we can also go the other direction. You can verify with a truth table that the two ...


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Presumably, you've a rule that allows you to substitute provably equivalent expressions. My guess is that's the rule you would apply, since you can prove: [(~p v r) & (~q v r)] <-> ((~p & ~q) v r). Left to right direction: ASSUME: [(~p v r) & (~q v r)] SHOW: ((~p & ~q) v r)   SUPPOSE: ~((~p & ~q) v r) (for proof by contradiction)  ...


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Let us start from material implication, then develop material equivalence, and explain what it means for something to be a sufficient condition, a necessary condition, and a necessary and sufficient condition. The logical meaning of material implication is a sufficient condition. Therefore, (P -> Q) -> (P => Q).The material implication 'P -> Q' ...


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Perhaps you're looking for objections to characterizing natural language conditionals with the material conditional ("->"). If so, two examples come to mind. First, from Van McGee: (1) If that creature is a fish, then if it has lungs, it is a lungfish (2) That creature is a fish (3) Hence, if it has lungs, it is a lungfish If we treat "if......


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It is called "the principle of explosion", which states: "Ex falso sequitur quodlibet" (Latin: From falsity anything follows). Given the conditional (if-then statement): P --> Q: reads "P (materially) implies Q", where the forward arrow from antecedent P (the 'if-part') to consequent Q (the 'then-part') is the logical ...


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It would be worthwhile distinguishing between a conditional sentence in the object language and a conditional in the metalanguage. Some deductive arguments have a conditional in the object language, e.g. those of the form modus ponens or modus tollens. Some arguments do not, e.g. those of the form conjunction elimination, disjunction elimination, etc. But ...


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**The Laws of Non-Contradiction, Excluded Middle, and Bivalence** The Law of Non-Contradiction (LNC): ~ [X & ~X]. Nothing can both be and not be. A proposition X and its logical negation ~X cannot both be true together. A proposition X cannot be both true and false. The joint affirmation of contradictories is denied! Something cannot both be and not be....


1

The logical opposite of "I believe X is true" is not "i believe X is false" but "I do not believe X is true". While "X true" and "X false" are contradictory, and can't be both part of your beliefs without contradiction, not believe that X is either true or false is valid, and equivalent to "I don't know ...


0

In your example with coins in a jar, the law of the excluded middle requires that you accept the proposition "the number of coins in the jar is even or the number of coins in the jar is odd". It does not require that you believe or disbelieve either half of that proposition in isolation nor does it speak to the validity of either half of that ...


6

You seem to confuse belief (which is subjective) and the actual truth value of a proposition. The LEM only applies to the latter, not to the former. If you wish to stay inside a mathematical framework, one might view probabilities as being degrees of belief. This is the subjective probability interpretation, or the Bayesian view. In your example, we would ...


1

The definitions are relevant to a Logical system and can be extended to Formal mathematical theories. See the Wiki's page (some lines below your quote): Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon ...


0

You wrote... But how did humans develop formal systems and the notion of syntactic consequence in the first place? Wouldn't they have had to develop such systems based on what semantic consequences they agreed with? ... and you are entirely correct: As has been pointed out in the comments, the study of semantic consequence lead to the notion of syntactic ...


2

It is misleading to speak of an "equivalence thesis". Mathematicians typically use material implication (MI) as a conditional because it is useful to do so. In simple contexts it obeys the implicational rules that we expect a conditional to obey. It is possible to show that if you require a truth-functional dyadic connective for bivalent classical ...


0

In logic, the value "neither true nor false" is used a lot in three-valued logic, interpreted as "unknown", "unknowable/undecidable", or "irrelevant". This can lead to logical conclusions and reasoning including unknowns. Reasoning with unknowns is common in everyday life. As an example, if you know one shop sells ...


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This law cannot be false or true, it is just a definition. It already causes problems when applied to non-atomic physical entities over time, such as the ship of Theseus paradox. In a lot of contexts, this definition is part of arguments and proofs, like many other definitions. If any of the definitions are rejected for the context of the argument, then the ...


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Does philosophy fall apart without the law of identity? Well, without the Law of Identity, a lot of philosophers have a lot of explaining to do. Rene Descartes decided, "I think, therefore I am." If it is not true that each thing is the same with itself and different from another, then he cannot say conclusively that he is the thing that is doing ...


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Suppose P → Q. For contradiction, suppose P ∧ ¬Q, meaning we have both P and ¬Q. By P → Q and P, we have Q. But this contradicts ¬Q, so we actually have ¬(P ∧ ¬Q). Conversely, suppose ¬(P ∧ ¬Q). Suppose P. Then, we cannot have ¬Q, as that would imply P ∧ ¬Q, contradicting ¬(P ∧ ¬Q). So we have ¬¬Q, which is equivalent to Q by the law of excluded middle. This ...


0

Suppose (A <-> ~A) for proof by reductio. Unpacked, this is to assume ((A -> ~A) & (~A -> A)). From this both (A -> ~A) and (~A -> A) follow. Substituting for the conditionals, that implies both (~A v ~A) and (A v A). Suppose A. Then (A & (~A v ~A)) entails ~A. Hence, (A & ~A), i.e. contradiction. Repeat this to show that (A v A)...


3

For proofs involving biconditionals, you often need to prove both directions independently. That is, you'll need to prove (P -> Q) -> ~(P & ~Q) and also ~(P & ~Q) -> (P -> Q). Let's start with the former. First, assume (P -> Q). You must derive ~(P & ~Q). Suppose (P & ~Q) for a reductio argument. In a reductio argument, you ...


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It's not a fallacy, because it's not a logical proof. The point of comedy isn't to prove or disprove something using logical rigour, it's to entertain - and, in as close as it comes to the point of the question - to make the audience think about something they may have never questioned. There's no question that there's a long history of racism, sexism, ...


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At the hypothetical beginning of time before God had his first thought absolutely nothing besides God exists. Not even as much as the idea of "nothingness" exists. The absolute minimum unit of knowledge from information theory is a single binary digit. Nothingness can be thought of as represented by a single zero-bit. On this basis the absolute ...


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The theorems had serious implications. They pretty much killed Logical Positivism, thus proving -- again -- that it is impossible to have a 100% rational system of beliefs (rational means explainable through logic and reason alone). The latter was known at least since Descartes' cogito ego sum, which, strictly speaking, limited our knowledge to the existence ...


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Let me see if I have your overall argument correct: Prior to the existence of the Universe, logic constrained how the universe could be structured or it did not If logic constrained how the universe could be structured, then God could create within those constraints If God could create within those constraints, then God could create ex nihilo Hence, if ...


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"Creatio ex nihilo" occurs all the time. The results are called fiction. These non-real objects do have value, and a major portion of the economy is driven by them. If you want to know whether it's "possible" for real physical objects to come from "nothing", that would depend on the physical laws of the universe. (You basically ...


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In (X iff ~X) the truth value of X is unstable (if true, then false; if false, then true). So it looks similar to the Liar Paradox. (This statement is false.) As far as I know, there is not yet any satisfactory "solution" to the Liar Paradox. the file I have received to start this problem has a contradiction symbol as step one Usually, ...


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The original post mostly had the right idea. Negation Elimination subproofs are required, we just need to nest them. Assume H and then assume A, aiming to derive contraditions to eliminate the assumptions. 1| H > (A > B) Promise 2| ~K & ~B Promise 3|_ ~A > K Promise 4| |_ H Assumed 5|...


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