symbolic logic ii - lecture 3look/phi520-lecture3.pdfpl-semantic consequence of a set of formulas i...

Propositional LogicGrammar

In our formal language, PL, we will have a grammar that will allowfor certain strings of symbols to count as well-formed formulas, or“formulas” or “wffs”.

Following Sider, we will just have the following PrimitiveVocabulary:

I Connectives: →, ∼I Sentence letters: P,Q,R . . . with or without numerical

subscripts

I Parentheses: (,)

Brandon C. Look: Symbolic Logic II, Lecture 3 1

subscripts

I Parentheses: (,)

subscripts

I Parentheses: (,)

We will have the following Definition of a wff:

(i) Every sentence letter is a wff. (That is, each sentence letter isan atomic sentence.)

(ii) If φ and ψ are wffs, then p(φ→ ψ)q and p∼ φq are also wffs.(iii) Only strings that can be shown to be wffs using (i) and (ii)

are wffs.

Note: in our language, PL, we will no longer have the logicalconnectives ∨, ∧, and ↔. Why not? Because we can define thoseconnectives in terms of ∼ and →.

We will have the following Definition of a wff:

(i) Every sentence letter is a wff. (That is, each sentence letter isan atomic sentence.)

(ii) If φ and ψ are wffs, then p(φ→ ψ)q and p∼ φq are also wffs.(iii) Only strings that can be shown to be wffs using (i) and (ii)

are wffs.

Note: in our language, PL, we will no longer have the logicalconnectives ∨, ∧, and ↔. Why not? Because we can define thoseconnectives in terms of ∼ and →.

Redefining the other logical connectives in terms of ∼ and →:

φ ψ p(φ ∧ ψ)q p∼ (φ→∼ ψ)qT T T TT F F FF T F FF F F F

φ ψ p(φ ∨ ψ)q p(∼ φ→ ψ)qT T T TT F T TF T T TF F F F

φ ψ p(φ↔ ψ)q p∼ ((φ→ ψ) → (ψ → φ))qT T T TT F F FF T F FF F T T

Note: we will only use the logical connectives ∼ and → because itmakes metalogic easier. But we could also have used just ∼ and ∨or ∼ and ∧.

In fact, as we will see, there are are two cases of logicalconnectives that are in themselves sufficient to ‘translate’ ourstandard set of logical connectives.

Propositional LogicThe semantic approach to logic

Let’s look at the way Sider introduces this in 2.2: “A semantics fora language is a way of assigning meanings to words and sentencesof that language. The central concept to be used in characterizingmeaning will be that of truth . . . Philosophers disagree over how tounderstand the notion of meaning in general. But the idea that themeaning of a sentence has something to do with truth-conditions ishard to deny, and at any rate has currency within logic. On thisapproach, one explains the meaning of a sentence by showing howthat sentence depends for its truth or falsity on the way the worldis.” (pp.22-23)

Some of the philosophy behind this. Consider the following:

I “Schnee ist weiß” means “Snow is white”.

I “Snow is white” is true iff snow is white.

What’s going on here? A German-speaker means by his sentencewhat an English-speaker means by his: namely, that snow is white.That is, there is a fact of the matter about the way the world is,and this — mirabile dictu! — has something to do with truth andmeaning.

Some of the philosophy behind this. Consider the following:

I “Schnee ist weiß” means “Snow is white”.

I “Snow is white” is true iff snow is white.

What’s going on here? A German-speaker means by his sentencewhat an English-speaker means by his: namely, that snow is white.That is, there is a fact of the matter about the way the world is,and this — mirabile dictu! — has something to do with truth andmeaning.

Now, consider this:

I The slithy toves did gyre and gimble in the wabe.

This looks meaningful but isn’t. Why not? One answer: ‘toves’and ‘wabe’ (at the very least) do not refer to anything in theworld; nor is ‘gimble’ any kind of action.

Suppose, however, we agree to rename nostril hairs ’toves’;’gimbling’ is just swirling around; and a ’wabe’ is a rush ofmucous. So, when I blow my nose . . . You get the idea.

The point is, we have now a certain interpretation that allows usto assign a truth-value to some event or fact of the matter.

Now, consider this:

This isn’t exactly what interpretation is going to mean for us,however. Rather, for us, an interpretation will consist in thefollowing:

1. A domain — that is, some collection of objects. (The domainalways has at least one object)

2. A name for each object in the domain. (Objects could havemore than one name.)

3. A list of predicates. (Though we’ll talk about predicates morenext chapter.)

4. A specification of the objects of which each predicate is trueand of the objects of which each predicate is false. (E.g. thepredicate ‘is a student’ is true of each of you; ‘blond’ is falseof me.)

5. An interpretation may include an atomic sentence letter, inwhich case the interpretation specifies a truth value of thatsentence letter.

Propositional LogicSemantics of propositional logic

In introductory logic, you were exposed to truth tables as a way ofunderstanding the truth values associated with the various logicalconnectives. And you were told how to use truth tables in yourevaluations of logical equivalence, validity, and so on.

Truth tables are pictorial representations of combinations of truthvalues.

Now, we will give a metalinguistic account of the truth values tobe assigned for different possible combinations of truth values forour atomic sentences.

Definition of Interpretation: A PL-interpretation is a function I,that assigns either 1 or 0 to every sentence letter.

Nothing mysterious here: 1 = True; 0 = False.

And, for p(φ→ ψ)q, we can represent the truth values in thefollowing way:

→ 1 01 1 00 1 1

Definition of Valuation: For any PL-interpretation, I, thePL-valuation for I, VI , is defined as the function that assigns toeach wff either 1 or 0, and which is such that, for any sentenceletter α and any wffs φ and ψ:

VI(α) = I(α)VI(φ→ ψ) = 1 iff either VI(φ) = 0 or VI(ψ) = 1VI(∼ φ) = 1 iff VI(φ) = 0

Some more definitions from Sider:

Definition of Validity: A formula φ is PL-valid iff everyPL-interpretation, I, VI(φ) = 1.

We write “|=PL φ” for “φ is PL-valid”.

Definition of Semantic Consequence: Formula φ is aPL-semantic consequence of a set of formulas Γ iff for everyPL-interpretation, I, if VI(γ) = 1 for each γ in Γ, then VI(φ) = 1.

Look closely: what is being said is that φ is a consequence just incase if all the premises are true, then it is true.

Sequent proofs in propositional logic

In introductory logic, you learned how to prove logical formulas aretrue or consequences of other formulas by means of naturaldeduction or the tree method. Now, we are going to use a differentapproach: sequent proofs.

A sequent looks like this:

Γ ⇒ φ

This simply means that φ is a conclusion or consequence of thepremise Γ, where Γ is to be thought of as a set of premises,{γ1, γ2, γ3, . . . γn}.

Sequent proofs in propositional logic

In introductory logic, you learned how to prove logical formulas aretrue or consequences of other formulas by means of naturaldeduction or the tree method. Now, we are going to use a differentapproach: sequent proofs.

A sequent looks like this:

Γ ⇒ φ

This simply means that φ is a conclusion or consequence of thepremise Γ, where Γ is to be thought of as a set of premises,{γ1, γ2, γ3, . . . γn}.

As with natural deduction, there are rules that allow us to makecertain inferences or that allow us to go from one sequent or set ofsequents to another.

These sequent rules are supposed to represent natural languagesequent rules that preserve logical correctness. (That is, if the‘from’ logical sequents are true, then the ‘to’ logical sequentsought to be true, too.)

Definition of a Sequent Proof: A sequent proof is a series ofsequents, each of which is either of the form φ⇒ φ, or followsfrom earlier sequents in the series by some sequent rule.

Some additional examples:

Show that Q, (P ∧ Q) → R ⇒ P → R

1. (P ∧ Q) → R ⇒ (P ∧ Q) → R RA2. P ⇒ P RA3. Q ⇒ Q RA4. P,Q ⇒ P ∧ Q 2,3 ∧ I5. P,Q, (P ∧ Q) → R ⇒ R 1,4 → E6. Q, (P ∧ Q) → R ⇒ P → R 2,5 → I

Show that ∼ (P ∧ Q),P → Q ⇒∼ P

1. ∼ (P ∧ Q) ⇒∼ (P ∧ Q) RA2. P → Q ⇒ P → Q RA3. P ⇒ P RA4. P,P → Q ⇒ Q 2,3 → E5. P,P → Q ⇒ P ∧ Q 3,4 ∧ I6. P,∼ (P ∧ Q),P → Q ⇒ (P ∧ Q)∧ ∼ (P ∧ Q) 1,5 ∧ I7. ∼ (P ∧ Q),P → Q ⇒∼ P 6 RAA

symbolic logic ii - lecture 3look/phi520-lecture3.pdfpl-semantic consequence of a set of formulas i...

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