8

In propositional logic sentence and formula are synonyms. An atomic sentence is a formula without propositional connectives [see Suppes, page 12]. In predicate logic, sentence and formula have different meaning: a sentence is a formula with no occurrences of free variables [see Def. page 54]. The formal definition of formula for predicate logic is at page ...


7

It only seems like a problem if we think the set Γ is consistent, in which case we feel like it's P which "makes" the premises inconsistent, and so must be used in the proof. But if Γ is already inconsistent, then we don't need to use P at all. Think about it in cases: If Γ is consistent, and Γ, P ⊢ Q ∧ ¬Q, then {Γ, P} is inconsistent, so that Γ ⊢ ¬P If, ...


7

Perhaps the main difference between what might be called a premise and an assumption by different authors is their use in a proof with inference rules. Here is an example of this difference in a natural deduction proof using a Fitch-style of presentation. Note that the first two lines above the horizontal line could be called either premises or original ...


6

The problem is that you've created your logical statements in the form of a causal relationship when no causal relationship exists. Normally we would expect the statement: If I stay, I will eat fish to mean that 'eating fish' is a logical consequence of 'staying', which carries the implication that if you have not eaten fish, you must not have stayed. But ...


6

That's not a fallacy at all, but a deductive argument form, aka modus tollens.


6

This is just normal Modus_tollens or called denying the consequent of classic logic of syllogism The form of a modus tollens argument resembles a syllogism, with two premises and a conclusion: If P, then Q. Not Q. Therefore, not P. The first premise is a conditional ("if-then") claim, such as P implies Q. The second premise is an assertion that ...


5

You said in a comment that you were referring to the material conditional, not other notions of if/then like the antecedent being a cause of the consequent, or the antecedent logically implying the consequent. So let's get rid of the if/then structure and write them explicitly as material conditionals: A) "I stay" -> "I eat fish" B) "I didn't stay" -> "I ...


5

This is, perhaps, one of the earliest surviving derivations of the law of explosion, known by the Latin name ex falso sequitur quodlibet, contradiction implies anything. It seems very counterintuitive, but it is hard to pinpoint where things go wrong. "Conditional exchange" is the definition of the material conditional, which is controversial. Indeed, it ...


5

First, a pro-material conditional case. Mathematicians rarely prove theorems which are genuinely of the form "If P then Q." Instead, results which are phrased colloquially as conditionals generally contain a universal quantifier, e.g. "If x is a prime >2 then x is odd" is shorthand for "For all x (if x is a prime >2 then x is ...


4

The assumption "A ⊨ B" does not mean that A is a tautology. You have assumed f(A)=1, that means that A is true for an interpretation f. The same for the second case : "When ⊨A → B, B must be true since ⊨A is a tautology" is wrong. A → B is a tautology: this means that, whenever A is true also B is, and this is enough to conclude with A⊨B.


4

First of all, a premise is a statement. As such, a premise is therefore explicit. A premise is a statement which is assumed as true for the purpose of an argument, where the conclusion will be considered as following from the given premise (see note on assume). As such a premise may be actually true or actually false. In many cases, it doesn't matter for ...


4

Why isn't it a problem that line 5 does not depend on line 3, when it is the negation of line 3 that we are proving? Because the rule of RAA states that whenever you have Γ, R Ⱶ P ^ ~P, then you may infer that Γ Ⱶ ~R. Now, as Γ Ⱶ P ^ ~P when Γ is {P, ~P}, therefore we can add any additional premise, such as ~Q, and obtain the required sequent so as to be ...


4

There is no problem here. Assuming material conditionals, if it is the case that If A then B, and if not-A then B, then B is simply a tautology, as it is true in every possible case. You are right the the above conditionals entail If not-B then A, and if not-B then not-A, but this shows that not-B cannot be true. Given that B is a tautology, this is ...


4

It is a sentence but not an atomic sentence. It is a sentence because the all-quantifier binds the free variables and so closes the formula. It is not an atomic sentence because it contains an implication.


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 ...


3

In standard propositional logic, where “if…then…” is the material conditional, “if A then B” and “if not-A then B” can certainly both be true together. Indeed, “(if A then B) and (if not-A then B)” is equivalent to simply “B”. Your everyday example with staying vs. eating fish is a bit misleading, because it pushes us towards thinking of the subtler ...


3

No, it's the same definition for validity, and you seem to mistake the notation "Φ ↔︎ Ψ" for a sentence. In Φ ↔︎ Ψ, Φ and Ψ are not variables for truth assignments, but variables for sentences (else the book would have used "p" and "q"). So it is not (only) possible to truth-assign "true" to Φ and "false" to ...


3

Yes. The distinction between formulas and sentences in predicate logic is made by specifying that sentences are those formulas in which there occur no free variables. Since there is no such thing as free and bound variables in propositional logic, all formulas are sentences automatically.


3

There are two questions. True or False? If monkeys can fly, then 1 + 1 = 3. The antecedent of the conditional, "monkeys can fly" is false. So is the consequent, "1 + 1 = 3". In classical truth-functional logic the conditional connecting these two sentences also has a truth-value. Wikipedia describes this "material conditional" as follows: The material ...


3

Assume classical propositional calculus. If C is a tautology, then ⊨ C, and thus B ⊢ C for B whatever, and thus ⊢ B → C. If not, if we have both B ⊢ C and ¬B ⊢ C we apply the Deduction Theorem and we get : ⊢ ¬B → C and ⊢ B → C. Now, using Excluded Middle ⊢ ¬B ∨ B, we can use Disjunction Elimination to conclude with ⊢ C. But if C is not a tautology, ...


3

The differences are subtle and the terms are often used interchangeably, but if one wants to make a distinction, it is roughly as follows: Premises are those statements on which the conclusion of the argument depends, that is, the "if" part, provided the argument is of the form "If A1, A2, ..., hold, then B holds". Of course, we also have tautologies ...


3

Soundness If Δ ⊢ Φ, then Δ ⊨ Φ has an implicit universal quantification to it: For all Δ, Φ: if Δ ⊢ Φ, then Δ ⊨ Φ Unsoundness of a proof system then means Not for all Δ, Φ: if Δ ⊢ Φ, then Δ ⊨ Φ This is equivalent to There exist Δ, Φ such that not: if Δ ⊢ Φ, then Δ ⊨ Φ which is in turn equivalent to There exist Δ, Φ such that Δ ⊢ Φ but not Δ ⊨ Φ ...


3

"If the kid is wet in the winter, then it was raining on him" In your example this premise is false, since the antecedent is true and the consequent is false. But if all your premises were true, the conclusion would have to be true as well, that's all it means for an argument to be logically valid (on the other hand, a "sound" argument is ...


2

I don't know of any external, more comprehensive list, but here are some of what I'd claim the most prominent statements that are valid classically but not intuitionistically: (¬A→⊥)→A (reductio ad absurdum) ¬¬A→A (double negation elimination) A∨¬A (tertium non datur) ¬(¬A∨¬B)→A∧B (DeMorgan 1a←) A∨B↔¬(¬A∧¬B) (DeMorgan 2a→+←) ¬(A∧B)→¬A∨¬B (DeMorgan 2b←) (A→...


2

You are confusing two uses of the word argument. In one sense, an argument is an extended discourse with limited aims such as education or persuasion. In the second sense, argument is a synonym for the technical term inference which is the process by which a single proposition can be constructed from a collection of premises (sometimes unstated). So, in ...


2

Yes, that is basically it. Premises are the undischarged assumptions. Here the second line is latter discharged; so it is just an Assumption. In Fitch style proof presentation, it is helpful to mark contexts of assumptions as they are raised and discharged; this is usually done by some form of indentation. |__(1) P Premise | |__(2) Q ...


2

p&~p is equivalent to ~p&~~p, but p&~p and ~p&~~p are not equivalent to pv~p -- in fact, they are contradictory to each other. This can be easily seen by taking a look at the truth table: p | ~p | ~~p | p&~p | ~p&~~p | pv~p --|----|-----|------|--------|----- 1 | 0 | 1 | 0 | 0 | 1 0 | 1 | 0 | 0 | 0 | 1 p&...


2

For some, knowledge is "justified, true belief". Here is how Wikipedia describes it: Justified true belief is a definition of knowledge that gained approval during the Enlightenment, 'justified' standing in contrast to 'revealed'. There have been attempts to trace it back to Plato and his dialogues. By this definition we would not know something unless ...


2

As you say, a sentence of the propositional logic is valid if and only if it is satisfied by every truth assignment. So the sentence "Φ ↔︎ Ψ" might be valid if the sentences Φ and Ψ are such that there is no valuation under which one of them is true and the other false. For example, if Φ is ¬(A ⋀ B), and Ψ is (¬A ⋁ ¬B) then Φ ↔︎ Ψ is valid, since ...


2

If P is valid, then the tableau for -P eventually closes. only states completeness: If P is valid, the tableau will find out. This does not rule out the possibility that the tableau will also close on the negation of some formulas that are not actually valid. Soundness thus has to be expressed separately; it is the converse direction: If the tableau for -P ...


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