HU Logic

Predicate Logic: instantiation and generalization rules All of the rules of implication and the rules of replacement we learned in propositional logic are applicable in predicate logic.   However, we cannot apply them to any line without first getting rid of the quantifiers.   So, we need some additional rules, implicational rules, that allow us to get rid of the quantifiers.   We also need rules that allow us to add quantifiers.   So, the following four rules are new implication rules, and so must be applied to whole lines not parts of lines. Instantiation Rules UI (x)Fx Fy Use this version of the rule if you anticipate using UG later in the proof (the ‘y’ in ‘Fy’ is a variable and not a constant) (x)Fx Fa Use this version of the rule if there is an individual with F (the ‘a’ in ‘Fa’ is a constant and not a variable—it names something specific in the universe of discourse) EI ( ∃ x)Fx Fa (Beware: the existential name must be ...

Let me try to explain the exportation rule a few more times. Here's the rule (I am using the triple bar to symbolize logical equivalence and the arrow to symbolize conditionals): (p & q) -> r  ≡ p -> (q -> r) Explanation 1: Look at the left side of the triple bar. It tells us that if p and q are true, then r is true. Look at the right side of the triple bar. It tells us that if p is true, then if q is true, then r is true. Even in plain English that sounds nearly identical.  Explanation 2: Look at the left side of the triple bar. It tells us, according to the meaning of conditionals, that if p is false then the whole conditional is true. Look at the right side of the triple bar. It tells us that if p is false the whole conditional is true (just remind yourself of the truth-table for p -> q). Explanation 3: Look at the left side of the triple bar. It tells us that if q is false the whole conditional is true. Look at the right side of the...

(I may be completely misreading expressions on faces in class, but I have the impression that DS is tripping some of you up. Even if it is not, this brief post may be helpful.) Here is the rule called Disjunctive Syllogism (DS) p v q ~p :. q Let's show that DS is a valid rule: REMEMBER : An argument is valid if, and only if, the truth of the premises guarantees the truth of the conclusion. Here is a proof of DS in English: If p v q is true, then either p is true or q is true (or both). If ~p is true, then p is false. So, q must be true. We can show that DS is valid by using indirect proof : 1. p v q 2. ~p          / q      3. ~q                             AIP      4. ~p & ~q                    2, 3, CONJ      5. ~(p v q).             ...

We have been studying propositional logic (PL) for some time now. PL is about whole or complete propositions and their relationships to each other in terms of truth-preservation. That is, PL investigates the relationships between sets of true propositions to other propositions. How can we be sure that if we are given a set of true propositions (called the premises) that we will through a series of steps wind up with another true proposition and not a false one (i.e. how can we be sure that the relationship between the premise set and the conclusion is, in fact, a valid one). Put differently, we want to make sure that the logic we are using will never take us from truths to falsity. We have been using our rules to show that if we start with a set of (assumed to be) true propositions we can derive another proposition that must be true if the premises are. But we have only studied a finite number of arguments. There is an infinite number of arguments. So, how can we be sure that there i...

Kristen had a wise request the other day. She asked for a list of logical terms that students should know. I'm going to restrict this list to terms (and phrases) related to propositional logic.  Here ya go: Valid Invalid Truth-value Negation Conjunction Conjunct Disjunction Disjunct Conditional Antecedent Consequent Biconditional Rules of Implication: MP, MT, HS, CD, ADD, SIMP, CONJ, DS Rules of Equivalence: TRANS, IMPL, EQUIV, DM, DN, TAUT, EXP, DIST, COM, ASSOC      What is the difference between an implication rule and an equivalence rule? Conditional Proof Indirect Proof Simple Proposition Complex Proposition Truth Table

Here is a super quick illustration of the importance and value of conditional proof: Suppose I am trying to show that God’s existence makes a difference to our lives, and in particular, God’s existence makes a difference to ethics (to how we should live). Now one way to go about doing that would be to first attempt to show that God does, in fact, exist. But doing that requires a decent amount of work, and it’s not really relevant to what I am interested in at the time. What I am interested in showing is something different, namely, that IF God exists, then how we should live is significantly different from how we should live if God does not exist. Here’s a helpful way of showing what I am interested in: 1.      If God exists, then ethics has the features x, y, and z. 2.      If God does not exist, then ethics does not have features x, y, and z. To attempt to “prove” those conditionals, I do not need to waste tim...