A reductio ad absurdum to eliminate a negation is not valid in Intuitionistic logic. But that is not the only place in logic where arguments by reductio are considered invalid. And historically, reductio arguments have been criticized on many grounds since even before Descartes and the Port Royal Logic regarded them with disfavor. ## 1. The principles of _Explosion_ and _Reductio_ There is a close relation between reductio arguments and the principle: _Explosion._ (p & ~p) implies q. I will give a similar name to its (putative) dual: _Implosion._ q implies (p v ~p), There are two forms of reductio arguments (where **f** is the falsum, which may or may not be a sentence of form (p & ~p)): (_Reduc-Int_) if X, p implies **f** , then X implies ~p (_Reduc-Elim_) if X, ~p implies **f** , then X implies p For example, Intuitionistic logic has both _Explosion_ and _Reduc-Int_ , but Intuitionists sharply reject _Reduc-Elim_. In their honor I tend to refer to the first as the _good Reductio_ and to the latter as the _bad Reductio._ In classical logic where (p& ~p) is the _falsum_ and (p v ~p) the _verum_ , there is not much to discuss. The questions I want to address are about these principles in contexts where very little is assumed about negation. ## 2. Assuming only a very minimal background logic. What I would like to do here is to start with just the &-v fragment common to these logics. The syntax will be the sentential syntax with connectives &, v, ~, and a propositional constant **f** , but the logic will have no principles governing ~ or **f**. (Our quest is to find out what principles re ~ and/or **f** will be needed to arrive at, eventually, the bad Reductio.) For semantics I take it that any admissible interpretation will assign to each sentence as semantic value (“the proposition expressed”) and element of a lattice (“the family of propositions expressed”), with &, v, ~ paired with respectively the meet ∧, the join v, and a special unary operation *. But to begin we have no principles pertaining to *, we only have the lattice laws: **Definition 1.** x ≤ y iff x ∧ y = x **Definition 2.** x ≤ y iff x v y = y. **Lattice laws.** x ≤ x v y, y ≤ x v y, x ∧ y ≤ x, x ∧ y ≤ y if x ≤ z and y ≤ z then x v y ≤ z, if z ≤ z and z ≤ y then z ≤ (x ∧ y) The definitions are equivalent, but I will sometimes use one, sometimes the other. The defined relation ≤ on the family of propositions corresponds to _implication_ for the sentences expressing those propositions. On the side of the sentences, the logical principles assumed at the outset are therefore the corresponding principles, with “⊢” for “implies”: p ⊢ p v q, q ⊢ p v q, p & q ⊢ p, etc. This logic I will call _lattice logic._ When examining proofs I will write (L) if the move is valid by the lattice logic, or the lattice laws, alone. **Terminology.** We note the following principles for reference, to fix terminology. _Disjunctive Syllogism_ (DisjSyl) : ~p & (p v q) implies q _Material Modus Ponens_(MatMP): p & (~p v q) implies q In classical logic these are equivalent, but that is due to the principle of _Double Negation_ , which we won’t assume at any point. Once we begin to focus on the semantic side I will give the corresponding lattice-theoretic formulation of all the above logical principles, and will (without expecting any confusing) give them the same names. ## 3. Disjunctive Syllogism entails Explosion This is the famous argument by C. I. Lewis. **[T1]** p & ~p ⊢ q, by DisjSyl 1. p & ~p assume 2. p 1, (L) 3. p v q 2, (L) 4. ~p 1, (L) 5. q from 3, 4 by DisjSyl How could the last move be justified in lattice logic? We can continue for 4. with * 5-1 ~p & (p v q) 4, 3, (L) * 5-2 (~p & p) v (~p & q) 5-1, distribution * 5-3 ~p & q 5-2, emptiness of p&~p * 5-4 q 5-3, (L) It would be interesting enough if we could justify Reductio with just distribution as extra principle. But that is not so. The move to 5.3 appears to rest on principle that (p & ~p) v q implies q or in lattice terms (x ∧ x*) v y ≤ y, and since the converse is obvious, that is equivalent to (x ∧ x*) v y = y. But _that_ is by Definition 2 the same as (x ∧ x*) ≤ y, which is exactly the principle of _Explosion_. So the ‘proof’ would assume _Explosion_ , which begs the question. ## 4. Material Modus Ponens entails Explosion **[T2]** p & ~p ⊢ q, by MatMP Adaption of the previous argument: 1. p & ~p assume 2. ~p 1, (L) 3. ~p v q 2, (L) 4. p 1, (L) 5. q from 3, 4 by MatMP ## 5. Given distribution, Explosion, DisjSyl, and MatMP are equivalent The arguments above establish that _Disjunctive Syllogism_ and _Material Modus Ponens_ , in the context of just the basic lattice laws, each entail _Explosion_. The converses can be proved if we add _Distribution_ to the lattice logic. Now it will be most convenient to focus on the semantic side. We have the following results for distributive lattices: **[T3]** _Explosion_ implies _Disjunctive Syllogism_ 1. x ∧ x* ≤ y Assumption (which includes that the lattice is bounded below, with special element 0 (bottom), and (x ∧ x*) = 0). 2. (x* ∧ x) v y = y Definition 2, (L). 3. (x* v y) ∧ (x v y) = y 2, distribution 4. [x* ∧ (x v y)] v [(y ∧ (x v y)] = y 3, distribution 5. [x* ∧ (x v y)] v y = y (L) 6. [x* ∧ (x v y)] ≤ y 5., Definition 2. **[T4]** _Explosion_ entails _Material Modus Ponens_ : 1. x ∧ x* ≤ y Assumption (which includes that the lattice is bounded below, with special element 0 (bottom), and (x ∧ x*) = 0). 2. (x* ∧ x) v y = y Definition 2, (L). 3. (x v y) ∧ (x* v y) = y 2, distribution, and (L) 4. [x ∧ (x* v y)] v [(y ∧ (x* v y)] = y 3, distribution 5. [x ∧ (x* v y)] v y = y (L) 6. [x ∧ (x* v y)] ≤ y 5., Definition 2. **Corollary.__** In distributive lattices, _Disjunctive Syllogism_ is equivalent to _Material Modus Ponens_ (although _Double Negation_ is not valid). ## 6. When the good Reductio is valid Since the good Reductio is formulated using the propositional constant **f** we need to restrict the interpretations of the syntax to ones that give **f** a semantic value. So from here on we will restrict our attention to lattices that are bounded below, with least element 0 (bottom), that is to say, for all elements x, 0 ≤ x. And the proposition expressed by **f** is 0. **Definition 3.** Lattice elements x, y are _disjoint_ iff x ∧ y ≤ 0 (equivalently, iff x ∧ y = 0). The good Reductio is (_Reduc-Int_) if X, p implies **f** , then X implies ~p For the lattice of propositions, that means that for all elements x, y: (_Reduc-Int_) If x ∧ y ≤ 0 then x ≤ y*. Though we have no special principles yet pertaining to operation *, we know that it is a map of the lattice into itself. Hence it is clear what _Reduc-Int_ says: y* is an element which is at least as weak as any element disjoint from y. It follows from this that the lattice must also be bounded above, with largest (weakest) element 1 (top). For all elements are disjoint from 0, and if the lattice does not have a top then no element is at least as weak as any element disjoint from 0. 1 is disjoint from 0, and is at least as weak as any element whatsoever, hence 1 = 0*. Moreover, only 0 is disjoint from 1, so 1* = 0. A traditional definition of negation is that ~p is the logically weakest statement such that (p & ~p) ⊢**f**. For lattices in general there need not exist a largest (weakest) element in the family of elements disjoint from a given element (other than 0 or 1). So the good Reductio does not suffice to identify the operation *. For that we need to add _Explosion_. **[T5]** If _Explosion_ and _Reduc-Int_ both hold in lattice L then x* is the greatest (weakest) element disjoint from x. This connects very well with a major topic in lattice theory: the _pseudo-complement_. Its definition amounts to the combination of _Explosion_ and the good _Reductio_ : **Definition 4.** Unary operation * on a lattice is a _pseudo-complement_ iff, for all elements x, y, (y ∧ y*) = 0 and if x ∧ y ≤ 0 then x ≤ y*. **Corollary**. If _Explosion_ and _Reduc-Int_ both hold in lattice L then x* is a pseudo-complement. ## 7. When the bad Reductio is valid The pseudo-complement has many of the familiar features of negation in general, but it specifically lacks Excluded Middle. Now, look at (_Reduc-Elim_) if X, ~p implies **f** , then X implies p also called _Indirect Proof_ , which is, for the lattice of propositions, (_Reduc-Elim_) If x ∧ y* ≤ 0 then x ≤ y, equivalently, if x ∧ y* = 0 then x ≤ y. We note that _Implosion_ entails that the lattice has a top element, call it 1, and that for any element x, 1 = x v x*. _Implosion_ entails the bad Reductio, given distributivity. Precisely: **[T6]** If the lattice is distributive, bounded above with top 1, and (x v x*) = 1 for all elements x, then _Reduc-Elim_ holds in that lattice. To prove: If x ∧ y* = 0 then x ≤ y. 1. x ∧ y* = 0 Assume 2. x = x ∧ 1 (L) 3. x = x ∧ (y v y*) 2, Given (_Implosion_) 4. x = (x ∧ y) v (x ∧y*) (L), Distribution 5. x = (x ∧ y) v 0 1, 4, (L) 6. x = (x ∧ y) 5, (L) 7. x ≤ y. 6, Definition 1. **Notice** that this argument does not depend on any special properties of *. _Implosion_ has much to answer for. ### Share this: * Share on X (Opens in new window) X * Share on Facebook (Opens in new window) Facebook * Like Loading... ### _Related_

In Graz-Mariatrost ist heute ein Haus teilweise eingestürzt. Laut Polizei dürfte eine nicht ordnungsgemäß abgestellte Gasflasche zu der Explosion geführt haben. Eine Person wurde ersten Informationen zufolge verletzt.

In Axams sind am Donnerstagabend nach ersten Informationen bei einer Explosion zwei Menschen verletzt worden. Eine Person mit dem Rettungshubschrauber in die Innsbrucker Klinik geflogen.

Unbekannte Täter haben in den frühen Morgenstunden einen Bankomat in Gmunden gesprengt. Die Explosion war so massiv, dass dadurch das Fluchtfahrzeug nicht mehr fahrtauglich war. Die vier Männer flüchteten deshalb zu Fuß.

In der Stadt Salzburg läuft aktuell ein großer Polizeieinsatz. Bei einer Bank in Maxglan kam es zu einer Explosion. Die Maxglaner Hauptstraße, Neutorstraße und noch weitere Straßen sind gesperrt.

GAZA: Hamas is open to an agreement to end the war in Gaza that would see all hostages released and secure a five-year truce, an official said Saturday as the group's negotiators held talks with mediators. A Hamas delegation visited Cairo to discuss with Egyptian mediators ways out of the 18-month war while, on the ground, rescuers said Israeli strikes in Gaza killed at least 35 people. **Read more:**_Explosion at Iranian Shahid Rajaee port leaves over 14 dead_ Nearly eight weeks into an Israeli aid blockade, the United Nations says food and medical supplies are running out in the territory. The Hamas official, speaking to AFP on condition of anonymity, said the Palestinian militant group "is ready for an exchange of prisoners in a single batch and a truce for five years". A statement later from Hamas said its delegation had left Cairo on Saturday evening. The latest bid to seal a ceasefire follows an Israeli proposal Hamas rejected earlier this month as "partial". The new proposal calls instead for a "comprehensive" agreement to halt the war ignited by the group's October 7, 2023, attack on Israel. The rejected Israeli offer, according to a senior Hamas official, included a 45-day ceasefire in exchange for the return of 10 living hostages. Hamas has consistently demanded that a truce deal must lead to the war's end, a full Israeli withdrawal from the Gaza Strip and a surge in humanitarian aid. An Israeli pullout and a "permanent end to the war" would also have occurred -- as outlined by then-US president Joe Biden -- under a second phase of a ceasefire that had begun on January 19 but which collapsed two months later. Hamas had sought talks on the second phase but Israel wanted the first phase extended. Israel demands the return of all hostages seized in the 2023 attack, and Hamas's disarmament, which the group has rejected as a "red line". "This time we will insist on guarantees regarding the end of the war," Mahmud Mardawi, a senior Hamas official, said in a statement. "The occupation can return to war after any partial deal, but it cannot do so with a comprehensive deal and international guarantees." Later on Saturday, senior Hamas official Osama Hamdan reiterated that "any proposal that does not include a comprehensive and permanent cessation of the war will not be considered." "We will not abandon the resistance's weapons as long as the occupation persists", he said in a statement. ## 'The house collapsed' Israel pounded Gaza again on Saturday. Mohammed al-Mughayyir, an official with the territory's civil defence rescue agency, told AFP that the death toll had risen to at least 35. In Gaza City, in the territory's north, civil defence said a strike on the Khour family home killed 10 people and left an estimated 20 more trapped in the debris. Umm Walid al-Khour, who survived the attack, said "everyone was sleeping with their children" when the strike hit and "the house collapsed on top of us." Elsewhere across Gaza, 25 more people were killed, rescuers said. There was no immediate comment from the Israeli military on the latest strikes but it said that "1,800 terror targets" had been hit across Gaza since the military campaign resumed on March 18. The military added that "hundreds of terrorists" were also killed. Qatar, the United States and Egypt brokered the truce which began on January 19 and enabled a surge in aid, alongside exchanges of hostages and Palestinian prisoners held by Israel. With Israel and Hamas disagreeing over the ceasefire's next phase, Israel cut all aid to Gaza before resuming bombardment, followed by a ground offensive. ## Gazans 'slowly dying' Since then, according to the health ministry in the Hamas-run territory, at least 2,111 Palestinians have been killed, taking the overall war death toll in Gaza to 51,495 people, mostly civilians. The Hamas attack that triggered the war resulted in the deaths of 1,218 people on the Israeli side, also mostly civilians, according to an AFP tally based on official Israeli figures. Militants also abducted 251 people, 58 of whom are still held in Gaza, including 34 the Israeli military says are dead. Israel says the military campaign aims to force Hamas to free the remaining captives. On Friday, the UN's World Food Programme (WFP) said the hot meal kitchens it was supplying with food in Gaza "are expected to fully run out of food in the coming days". On Saturday, AFP footage showed queues of people waiting for food in front of a community kitchen. "There is no food in the free kitchen, there is no food in the markets... There is no flour or bread," said north Gaza resident Wael Odeh. **Read more:**_World breathes sigh of relief as Trump spares Fed, IMF_ A senior UN official, Jonathan Whittall, said Gazans were "slowly dying". "This is not only about humanitarian needs but also about dignity," Whittall, head of the UN's Office for the Coordination of Humanitarian affairs in the Palestinian territories, told journalists.

Die Explosion einer Gaskartusche hat am frühen Samstagabend zu einem Großeinsatz der Feuerwehr in Hernals geführt. Durch die Explosion stürzte eine Trennwand zur Nachbarwohnung ein. Verletzt wurde niemand.

PESHAWAR: Mufti Munir Shakir, the founder of the banned Lashkar-e-Islam, was killed after a bomb ripped through a mosque in the Urmar area of Peshawar on Saturday. According to Lady Reading Hospital spokesperson Muhammad Asim, Mufti Munir was brought to the hospital in an injured condition, with severe damage to his leg due to the explosion. _Explosion reported on N-40 Nushki_ The spokesperson said that excessive blood loss led to his death at the hospital. Urmar Police Station House Officer Noor Muhammad told the BBC Urdu that as Mufti Munir entered the mosque for Asr prayers, an already-planted explosive device detonated, initially injuring four people, including him. They were shifted to the hospital for treatment. Peshawar SSP Masood Ahmed told the BBC Urdu that initial investigation suggested Mufti Munir was attacked with a hand grenade. _Deadly blast targets FC convoy in Noshki, five dead, more than 30 injured_ He said that the police were actively investigating the incident. "Mufti Munir’s life was under threat, and he had been receiving threats from the Khorasan branch of the Islamic State," the SSP said. A few days before his death, Mufti Munir had said in a speech that he was receiving threats on WhatsApp. He had submitted screenshots of the said threats as evidence to the local police station. So far, no militant group has claimed responsibility for his killing. ## A rising trend of attacks on religious scholars In recent weeks, religious figures have been targeted in Khyber Pakhtunkhwa. A few weeks ago, _Maulana Hamidul Haq_ was attacked in a bombing at Jamia Haqqania in Akora Khattak. On Friday, Maulana Abdullah Nadeem, a leader of Jamiat Ulema-e-Islam, was also attacked inside a mosque in South Waziristan, leaving him injured. _Funeral prayers for Maulana Hamid-ul-Haq offered_ Mufti Munir’s funeral prayers were held on Sunday in Bagh-e-Naran, Hayatabad. ## Who was Mufti Munir Shakir? Mufti Munir received his early education in the Makhizai area of Kurram Agency, where his father was a teacher. He belonged to the Bangash tribe. Fazal Hameed, a longtime friend of Mufti Munir since their student days, told the BBC, "I have known Mufti Munir since 1989. We lived together in Banuri Town, where he was a year junior to me." According to Fazal Hameed, "Mufti Munir studied in Banuri Town for a year, then moved to Darul Uloom Orangi, where he studied under Mufti Taqi Usmani. Later, he completed his studies at Jamia Farooqia under Maulana Samiullah Khan." Mufti Munir first gained prominence when he founded the banned militant group Lashkar-e-Islam. In 2004 and 2005, clashes erupted in the Bara region of Khyber Agency between Lashkar-e-Islam and another banned group, Ansar-ul-Islam, after which Munir Shakir went into hiding. Following his disappearance, the leadership of Lashkar-e-Islam was handed over to Mangal Bagh. Under Mangal Bagh, violent clashes between the two groups continued, leading to a large-scale military operation in the area. After remaining in hiding for some time, Mufti Munir resumed teaching in Hangu before eventually relocating to the Armar area of Peshawar. In recent years, Mufti Munir gained popularity on social media. His TikTok account had around 93,000 followers. Khyber Pakhtunkhwa Chief Minister Ali Amin Gandapur condemned the incident and has sought a report from police officials regarding the attack.

In Dornbirn ist in der Nacht auf Donnerstag ein Pelletofen explodiert. Ersten Informationen zufolge stellte eine Frau gegen 2.00 Uhr starken Rauch in ihrer Wohnung fest. Als sie den Ofen überprüfte, kam es zu der Explosion.

993 Einsätze hat die Wiener Berufsrettung zum Jahreswechsel absolviert. In der Donaustadt ist ein 14-Jähriger bei der Explosion einer Rakete schwer verletzt worden, in der Innenstadt ein 20-Jähriger durch einen Böller.

Im Bezirk Villach-Land hat ein 13 Jahre alter Bursche beim Hantieren mit Feuerwerkskörpern schwere Verletzungen an einer Hand erlitten. Der Bub zündete Pulver in einer Kunststoffdose. Dabei kam es zu einer Explosion.