Show that the argument form with premises (p ∧t) →(r ∨s), q →(u ∧t), u →p, and s and conclusion

step 1 So here we're asked to show if the argument given is valid. So we know that the argument given h

Show that the argument form with premises (p ∧t) →(r ∨s), q...

Show that the argument form with premises $(p \wedge t) \rightarrow$ $(r \vee s), q \rightarrow(u \wedge t), u \rightarrow p,$ and $\neg s$ and conclusion $q \rightarrow r$ is valid by first using Exercise 11 and then using rules of inference from Table 1 Show that the argument form with premises $(p \wedge t) \rightarrow$ $(r \vee s), q \rightarrow(u \wedge t), u \rightarrow p,$ and $\neg s$ and conclusion $q \rightarrow r$ is valid by first using Exercise 11 and then using rules of inference from Table $1 .$ .

Show that the argument form with premises $(p \wedge t) \rightarrow$ $(r \vee s), q \rightarrow(u \wedge t), u \rightarrow p,$ and $\neg s$ and conclusion $q \rightarrow r$ is valid by first using Exercise 11 and then using rules of inference from Table 1
Show that the argument form with premises $(p \wedge t) \rightarrow$ $(r \vee s), q \rightarrow(u \wedge t), u \rightarrow p,$ and $\neg s$ and conclusion $q \rightarrow r$ is valid by first using Exercise 11 and then using rules of inference from Table $1 .$ .

Key Concepts

Disjunctive Syllogism

This inference rule allows one to conclude that if one component of a disjunction is false, then the other must be true. In the context of this argument, given a disjunction and a negation of one disjunct, one can directly infer the truth of the remaining disjunct. This method is essential when dealing with premises that include disjunctive statements.

Rules of Inference

Rules of inference are fundamental tools in formal logic that allow one to derive conclusions from premises. These include rules such as modus ponens (if 'p' and 'p ? q' are true, then 'q' is true), conjunction elimination, and others. They provide the step-by-step logical structure necessary for rigorously demonstrating that an argument is valid.

Validity in Propositional Logic

This concept involves ensuring that if all the premises of an argument are true, then the conclusion cannot be false. In proving an argument’s validity, one demonstrates a necessary logical progression from the premises to the conclusion, meaning that any interpretation that makes the premises true will automatically make the conclusion true.

Conditional Proof

A conditional proof is a method used to establish the truth of an implication. It involves assuming the antecedent (the 'if' part) and then logically deriving the consequent (the 'then' part) under that assumption. This method is particularly useful when the conclusion is an implication, as it shows directly that assuming the antecedent leads to the consequent.

Transcript

00:01 So here we're asked to show if the argument given is valid.

00:05 So we know that the argument given has these premises and the conclusion is this.

00:11 So we're asked to use exercise 11 to show that it's valid and then to use table 1 of the rules of inference to show why it's valid.

00:23 So we can say that this argument is valid if we also say that the same premise, and adding and q if we add q as a premise would make conclusion r would make this new argument valid so let's say that they both are valid now how can we show this we'll do a step and reasons now we'll use table one the rules of inference to show why this is bad at so step one we'll use a process so we use if q, then u and t.

01:23 Reason, this is a premise.

01:29 Can't do anything just yet, so let's put another premise.

01:35 If u then p, another premise.

01:42 Now let's look back at the first step.

01:48 We know we could simplify this, so let's use one compound proposition in our simplification.

01:55 So if q, then you.

02:00 So simplification of one.

02:06 Step four, now we could use steps two and three.

02:12 So we could rewrite this as if p, then you, and if you then q.

02:21 So using hypothetical syllogism, if p or if q, then p, or if q, then p.

02:33 Reason, hypothetical syllogism using two and three.

02:48 Now step five, let's use the second compound proposition that we did when we simplified number one...